Reducible Dehn Surgeries, Ribbon Concordance and Satellite Knots
by
Holt W. Bodish
A dissertation accepted and approved in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in Mathematics
Dissertation Committee:
Robert Lipshitz, Chair
Boris Botvinnik, Core Member
Jon Brundan, Core Member
Dan Dugger, Core Member
Jiabin Wu, Institutional Representative
University of Oregon
Spring 2024
© 2024 Holt W. Bodish
This work is openly licensed via CC BY 4.0.
2
DISSERTATION ABSTRACT
Holt W. Bodish
Doctor of Philosophy in Mathematics
Title: Reducible Dehn Surgeries, Ribbon Concordance and Satellite Knots
In this thesis we investigate knots and surfaces in 3- and 4-manifolds
from the perspective of Heegaard Floer homology, knot Floer homology and
Khovanov homology. We first investigate the Cabling Conjecture, which
states that the only knots that admit reducible Dehn surgeries are cabled
knots. We study this question and related conjectures in Chapter 2 and de-
velop a lower bound on the slice genus of knots that admit reducible surg-
eries in terms of the surgery parameters and study when a slope on an al-
most L-space knot is a reducing slope. In particular, we show that when
gpKq is odd and ą 3, the only possible reducing slope on an almost L-space
knot is gpKq and in that case the complement of an almost L-space knot
does not contain any punctured projective planes. In Chapter 3 we inves-
tigate the effect of satellite operations on knot Floer homology using tech-
niques from bordered Floer homology [LOT18] and the immersed curve re-
formulation [HRW22; Che19; CH23]. In particular we study the functions
n ÑÞ gpPnpKqq, ϵpPnpKqq and τpPnpKqq for some families of p1, 1q pat-
terns P from the immersed curve perspective. We also consider the function
n ÞÑ dimpHzFKpS3, PnpKq, gpPnpKqqq, and use this together with the fibered
detection property of knot Floer homology [Ni07] to determine, for a given
pattern P , for which n P Z the twisted pattern Pn is fibered in the solid
torus. In Chapter 4 we answer positively a question posed by Lipshitz and
Sarkar about the existence of Steenrod operations on the Khovanov homol-
ogy of prime knots [LS18, Question 3]. The proof relies on a construction of
a particular type of surface, called a ribbon concordance in S3 ˆ I, interpo-
lating between any given knot and a prime knot together with the fact that
the maps induced on Khovanov homology by ribbon concordances are split
injections [Wil12; LZ19].
3
This thesis contains previously published material and unpublished coau-
thored material.
4
ACKNOWLEDGMENTS
I want to first thank my advisor Robert Lipshitz for his care, invalu-
able guidance, support and encouragement over the last six years. I want
to thank my committee: Boris Botvinnik, Daniel Dugger, Jon Brundan and
Jiabin Wu. I am grateful for all my low-dimensional topologist friends at
the University of Oregon, especially the One Flew Over the Sutured Nest
group (Gary, Siavash, Jesse and Champ) for the weekly zoom meetings over
the pandemic and for teaching each other cool math. I am grateful to my
coauthor Robert DeYeso III and collaborator Subhankar Dey for many in-
teresting discussions. I also want to acknowledge the mathematicians from
Montana for their advice and support early in my career: David Ayala, Ryan
Grady, Charles Katerba and Eric Chesebro. I would not know about math
research and especially about knots and tangles if it were not for Eric invit-
ing me to participate in an undergraduate research group after he taught my
Calculus II class.
I am forever grateful to my family and friends, especially my parents for
their encouragement and support and my brother Elijah for introducing me
to the wild world of mathematics and his friendship throughout my life. Fi-
nally I want to express my immense and everlasting gratitude and love to
my spouse Masha Korchagina for their devotion, love, and support through-
out my time in graduate school and beyond. I could not have done it with-
out them.
5
DEDICATION
For Masha
6
TABLE OF CONTENTS
Chapter Page
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Genus, Fiberedness, and Concordance . . . . . . . . . . . . . . 15
Dehn Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Heegaard Floer Homology . . . . . . . . . . . . . . . . . . . . . 20
Khovanov Homology . . . . . . . . . . . . . . . . . . . . . . . 22
Ribbon Concordance . . . . . . . . . . . . . . . . . . . . . . . 22
1.2. Chapter 2: Reducible Surgeries on Slice and Almost L-Space Knots 23
1.3. Chapter 3: Knot Floer Homology, Immersed Curves and Twisted
Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Bordered Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 30
Bordered 3-manifolds and the Pairing Theorem . . . . . . . . . 32
Bordered Floer and Satellites Knots . . . . . . . . . . . . . . . 32
1.4. Chapter 4: Non-trivial Steenrod Squares on the Khovanov Homol-
ogy of Prime Knots . . . . . . . . . . . . . . . . . . . . . . . . 40
2. REDUCIBLE SURGERIES . . . . . . . . . . . . . . . . . . . . . . 42
2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Spinc Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Heegaard Floer Homology . . . . . . . . . . . . . . . . . . . . . 46
The Mapping Cone Formula and the ν` Invariant . . . . . . . 48
2.2. Reducible Surgeries on Slice Knots . . . . . . . . . . . . . . . 50
The d-invariants of Reducible Manifolds . . . . . . . . . . . . 50
Multiple Reducing Slopes on Slice Knots . . . . . . . . . . . . 54
2.3. Almost L-space knots and the Mapping Cone Formula . . . . 55
Facts about Almost L-space knots . . . . . . . . . . . . . . . . 56
7
Relative Gradings and Proper Divisors . . . . . . . . . . . . . 61
3. (1,1) PATTERNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2. Bordered Floer Homology . . . . . . . . . . . . . . . . . . . . 75
C 3 ´zFDpS zνpKqq from CFK pKq . . . . . . . . . . . . . . . . . 77
Immersed Curves for knot complements . . . . . . . . . . . . . 79
Properties of Immersed Multicurves for Knot Complements . . 80
CzFApS1 ˆD2, P q for p1, 1q-patterns P Ă S1 ˆD2 . . . . . . . 83
3.3. The pairing theorem for p1, 1q patterns . . . . . . . . . . . . . 86
Computing τpP pKqq from a pairing diagram . . . . . . . . . . 90
3.4. Trefoil patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Introducing the patterns . . . . . . . . . . . . . . . . . . . . . 96
τ of 0-Framed Satellites With Arbitrary Companions . . . . . 97
3.5. Three Genus and Fiberedness . . . . . . . . . . . . . . . . . . 104
3.6. Next to top Alexander grading . . . . . . . . . . . . . . . . . . 107
3.7. n-Twisted Satellites with Generalized Mazur Patterns . . . . . 111
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . 113
3.8. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Immersed Curves for n-Framed Knot Complements . . . . . . 116
(1,1)-Unknot Patterns . . . . . . . . . . . . . . . . . . . . . . 121
The Curves βpi, jq . . . . . . . . . . . . . . . . . . . . . . . . 125
3.9. Three-Dimensional Invariants . . . . . . . . . . . . . . . . . . 130
Three-Genus and n-twisted Satellites . . . . . . . . . . . . . . 130
Fiberedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.10. Thickness and unknotting number of generalized Mazur satellites
with non-trivial companions . . . . . . . . . . . . . . . . . . . 139
3.11. Heegaard Floer Concordance Invariants and Twisting . . . . . 143
4. KHOVANOV STABLE HOMOTOPY TYPE AND RIBBON CONCOR-
DANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2. Khovanov Homology and Ribbon Concordances . . . . . . . . 161
8
4.3. The Base-point action and Reduced Khovanov Homology . . . 162
4.4. Knots and Prime Tangles . . . . . . . . . . . . . . . . . . . . . 164
4.5. Steenrod Operations and Stable Homotopy Type . . . . . . . 167
4.6. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . 170
4.7. Hyperbolic Knots and Invertible Concordances . . . . . . . . . 171
4.8. Satellite Knots . . . . . . . . . . . . . . . . . . . . . . . . . . 172
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9
LIST OF FIGURES
Figure Page
1.1. Type D structure for 0-framed right handed trefoil complement . 33
1.2. A genus-1 doubly-pointed Heegaard diagram. The blue curve is the
β curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3. The pattern in the solid torus determined by the doubly pointed Hee-
gaard diagram in figure 1.2 by the procedure described in the text 35
1.4. Pairing Diagram Example . . . . . . . . . . . . . . . . . . . . . . 36
3.1. Type D structure for 0-framed right handed trefoil complement . 79
3.2. The immersed curve associated to the 0 framed trefoil complement 81
3.3. The essential component of the immersed curve for a knot K with
τpKq ą 0 and ϵpKq “ 1. . . . . . . . . . . . . . . . . . . . . . . . 84
3.4. The genus 1 doubly pointed Heegaard diagram for the pattern P p3,1q 85
3.5. CzFApHq where H is the doubly pointed bordered Heegaard diagram
shown in Figure 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.6. Lift of the pattern P p3,1q to the cover Σ̃ a single connected lift of β
is shown in bold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.7. pairing diagram showing the trefoil pattern P p3,1q paired with 0 framed
trefoil companion . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.8. Pairing diagram for HFKpS3, P p3,1qz pT2,3qq. Intersection points labelled
x and y satisfy Apyq ´ Apxq “ 5 and Apxq “ 0. . . . . . . . . . . . 93
3.9. The disks shown represent all the differentials that lower filtration
degree by one. Cancelling the disks by an isotopy, we end up with
Figure 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.10. The result of cancelling the disks in Figure 3.9. There are two disks
that connect generators of CzFKpα̃, β̃1q of minimal filtration difference.
Cancelling these disks we arrive at Figure 3.11 . . . . . . . . . . . 94
3.11. The result of isotoping β1 in Figure 3.10, we arrive at a complex with
three generators and one differential connecting two generators of min-
imal filtration difference . . . . . . . . . . . . . . . . . . . . . . . 94
10
3.12. Cancelling all intersection points with filtration difference one (disk
in yellow) and intersection points with filtration difference two (disks
in pink) There are three intersection points remaining. . . . . . . 95
3.13. Doubly pointed Heegaard diagram for p3, 1q cable pattern . . . . . 97
3.14. Midway through the isotopy . . . . . . . . . . . . . . . . . . . . . 97
3.15. Doubly pointed bordered Heegaard diagram for the trefoil pattern
P p3,1q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.16. The trefoil pattern P p3,1q in the solid torus . . . . . . . . . . . . . 97
3.17. The lift of the trefoil pattern P p3,1q shown in Figure 3.15 paired with
the right handed trefoil. We have τpP p3,1qpT2,3qq “ Apyq “ 5. . . . 100
3.18. The lift of the trefoil pattern P3 shown in Figure 3.15 paired with the
left handed trefoil. We find τpP p3,1qpT2,´3qq “ Apyq “ 0. . . . . . . 100
3.19. The general case with τpKq ą 0 and ϵpKq “ ˘1. ϵpKq “ 1 is shown
as a dotted arc, and ϵpKq “ ´1 is shown as a solid arc . . . . . . 101
3.20. The general case with τpKq ă 0 and ϵpKq “ ˘1. ϵpKq “ ´1 is
shown as a dotted arc and ϵpKq “ 1 is shown as a solid arc . . . . 103
3.21. The pairing diagram computing HFKpS3, P pp,1qz pT2,3qq . . . . . . . 106
3.22. P pp,1q paired with a fibered knot with τpKq “ gpKq . . . . . . . . 108
3.23. P pp,1q paired with a fibered thin knot K with |τpKq| ă gpKq . . . 108
3.24. The pattern Qi,j. In the box labelled i, there are i full twists on two
strands as shown in the box on the bottom left. In the box labelled
n insert n full twists on j ` 2 strands . . . . . . . . . . . . . . . . 112
3.25. Type D structure for complement of knot K with τpKq ą 0 and ϵpKq “
1, where we replace the dotted arrow from ξ0 to η0 by the appropri-
ate unstable chain . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.26. Type D structure for complement of knot K with τpKq ą 0 and ϵpKq “
´1, where we replace the dotted arrow from ξ0 to η0 by the appro-
priate unstable chain . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.27. The unstable portion of αpK,nq with τpKq ě 0 and ϵpKq “ 1 and
2τpKq ą n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.28. The unstable portion of αpK,nq with τpKq ě 0 and ϵpKq “ 1 and
n ě 2τpKq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
11
3.29. The unstable portion of αpK,nq with τpKq ě 0 and ϵpKq “ ´1
and n ě 2τpKq . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.30. The unstable portion of αpK,nq with τpKq ě 0 and ϵpKq “ ´1
and n ď 0 ď 2τpKq . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.31. The p1, 1q pattern determined by the pair pr, sq “ p4, 2q . . . . . . 122
3.32. The knot in green in S1ˆD2 determined by the p1, 1q pattern with
β curve the blue curve . . . . . . . . . . . . . . . . . . . . . . . . 122
3.33. The lifted pairing diagram for CzFKpα̃pT2,3, 0q, β̃, w, zq . . . . . . . 123
3.34. The piece of the complex CFK 3 0,3FrU,V s{UV pS ,Q0 pT2,3qq that contains
the intersection point d with Apdq “ τpQ0,30 pT2,3qq and d`h gener-
ates HFpS3x q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.35. The pairing diagram for CzFKpαpT2,3, 1q, βpQ0,3qq . . . . . . . . . . 125
3.36. The pairing diagram for CzFKpαpT 0,32,3,´1q, βpQ qq . . . . . . . . . 125
3.37. The p1, 1q pattern that determines the pattern knot Qi,j. Figure 3.31
shows the case i “ 0 and j “ 2 . . . . . . . . . . . . . . . . . . . 126
3.38. The knot in S1ˆD2 determined by the p1, 1q pattern with β “ βpi, jq127
3.39. The knot from Figure 3.38 after an isotopy . . . . . . . . . . . . . 128
3.40. Isotope the j consecutive strands that are bold in Figure 3.39 to ob-
tain this knot, which is Qi,j0 . . . . . . . . . . . . . . . . . . . . . 128
3.41. The isotopy that produces βp0, j ` 1q from βp0, jq. . . . . . . . . 128
3.42. The curve β̃p0, jq for the knot Q0,j . . . . . . . . . . . . . . . . . 129
3.43. twist up the curve β̃p0, jq to get the curve β̃p1, jq for the knot Q1,j 129
3.44. The collapsed β̃p0, jq curve for the knot Q0,j . . . . . . . . . . . . 129
3.45. The collapsed β̃p1, jq curve for the knot Q1,j . . . . . . . . . . . . 129
3.46. The pairing diagram for Q0,jn when j is odd and n ą 0 . . . . . . . 132
3.47. The general pairing diagram for j even and n odd . . . . . . . . . 134
3.48. The general pairing diagram for j even and n even . . . . . . . . 135
3.49. The lifted pairing diagram for HzFKpQ0,j0 pT2,3qq . . . . . . . . . . . 136
3.50. The pairing diagram for Q0,jn pT2,3q when n ă ´1. The Alexander
grading labels of the β arcs are as in Figure 3.51 . . . . . . . . . . 136
3.51. The general pairing diagram showing intersection points with largest
possible Alexander grading when i “ 0 and n “ ´1. . . . . . . . . 137
12
3.52. The top left of the pairing diagram when n ą 0 and i “ 2. The
intersection points connected by a spiral are in Alexander grading g “
gpQi,jn pKqq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.53. The top left of the pairing diagram when n ą 0 and i “ 3. The
intersection points connected by a spiral are in the same Alexander
grading g “ gpQi,jn pKqq . . . . . . . . . . . . . . . . . . . . . . . . 139
3.54. Illustration of two intersection points in the pairing diagram with a
length j`1 vertical differential between them. The red arc is a por-
tion of αpK,nq that exhibits the genus detection of knot Floer ho-
mology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.55. The pairing CzFKpαpU, nq, βpi, jqq when n ă ´1 . . . . . . . . . . 141
3.56. The pairing CzFKpαpU,´1q, βpi, jqq . . . . . . . . . . . . . . . . . 141
3.57. ϵpKq “ τpKq “ 0 and n ă 0 . . . . . . . . . . . . . . . . . . . . . 145
3.58. ϵpKq “ τpKq “ 0 and n ą 0 . . . . . . . . . . . . . . . . . . . . . 145
3.59. The pairing diagram when τpKq ě 0, ϵpKq “ 1 and n ě 2τpKq
and j odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.60. The pairing diagram when τpKq ą 0 ϵpKq “ 1 and n ď 0 ă 2τpKq
and j odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.61. Case τpKq ą 0, ϵpKq “ 1 and 0 ď n ă 2τpKq with j odd . . . . . 147
3.62. Subcomplex carrying the cycle that generates HxFpS3q together with
horizontal differentials from Figures 3.70 and 3.71 . . . . . . . . . 148
3.63. Subcomplex carrying cycle that generates H 3xFpS q and horizontal dif-
ferentials from Figures 3.60, 3.61, and 3.64 . . . . . . . . . . . . . 148
3.64. τpKq ď 0 ϵpKq “ 1 and n ď 2τpKq . . . . . . . . . . . . . . . . . 150
3.65. τpKq ď 0 ϵpKq “ 1 and n ě 0 . . . . . . . . . . . . . . . . . . . . 150
3.66. Subcomplex carrying the cycle that generates H 3xFpS q corresponding
to the cases in Figures 3.59, 3.65, and 3.68 . . . . . . . . . . . . . 151
3.67. Subcomplex carrying cycle that generates HxFpS3q and horizontal dif-
ferentials from Figures 3.69 and 3.72 . . . . . . . . . . . . . . . . 151
3.68. The pairing diagram when τpKq ă 0 ϵpKq “ 1 and 0 ą n ą 2τpKq 152
3.69. The pairing diagram when τpKq ą 0 ϵpKq “ ´1 and n ă 0 ă
2τpKq and j odd . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
13
3.70. Case τpKq ă 0, ϵpKq “ ´1 and n ą 0 ą 2τpKq with j even . . . . 154
3.71. The pairing diagram when τpKq ą 0 ϵpKq “ ´1 and n ě 2τpKq
and j odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.72. τpKq ă 0 ϵpKq “ ´1 and n ď 2τpKq . . . . . . . . . . . . . . . . 156
3.73. The subcomplex that carries the FrV s-free part of the homology be-
fore twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
3.74. The subcomplex that carries the FrV s-free part of the homology af-
ter twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
3.75. A horizontal differential to the intersection point that survives the
spectral sequence to HxFpS3q when i “ 1 . . . . . . . . . . . . . . 158
3.76. Another horizontal differential to the intersection point that survives
the spectral sequence to HFpS3x q when i “ 1 . . . . . . . . . . . . 158
3.77. A horizontal differential to the intersection point that survives the
spectral sequence to HxFpS3q when i ą 1 . . . . . . . . . . . . . . 158
3.78. Another horizontal differential to the intersection point that survives
the spectral sequence to HxFpS3q when i ą 1 . . . . . . . . . . . . 158
4.1. T1 `p T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.2. pT1 `p T2q ` Cl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.3. The clasp tangle Cl . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.4. The numerator closure of the clasp tangle . . . . . . . . . . . . . 166
4.5. Denominator closure of the clasp tangle . . . . . . . . . . . . . . 167
4.6. Denominator closure of T2 `p T2 . . . . . . . . . . . . . . . . . . . 167
4.7. K1#K2 \ Unknot . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.8. K1#K2 \ Unknot after isotopy . . . . . . . . . . . . . . . . . . . 169
4.9. Another isotopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.10. The result of adding a band; the final stage of the ribbon concordance
between K1#K2 and the prime knot pT1 `p T2q ` Cl . . . . . . . . 169
14
CHAPTER 1
INTRODUCTION
1.1 Background
In this section we will introduce some background and notation that we
will use throughout the arguments in this thesis.
Genus, Fiberedness, and Concordance
A knot K is a smooth embedding K : S1 Ñ S3 considered up to
ambient isotopy. We will also write K for the image of the embedding. In
this thesis we are interested in properties of surfaces embedded in S3 or B4
bounded by the knot. Any knot K bounds an orientable surface Σ in S3,
called a Seifert surface. We define the 3-genus, gpKq, of a knot as
gpKq “ mintgpΣq : Σ Ă S3, and BΣ “ Ku
As such, the genus is difficult to compute. It is however a powerful knot
invariant. For example it detects the unknot: gpKq “ 0 if and only if K is
isotopic to the unknot.
Similarly, any surface in S3 “ BB4 can be pushed into B4 so that BΣ “
ΣX S3 “ K. Then, we can define the 4-genus, g4pKq, of a knot K as
g4pKq “ mintgpΣq : Σ Ă B4, BΣ “ ΣX S3 “ Ku
Given a Seifert surface Σ for a knot K, the embedding of Σ into S3
gives rise to a pairing on H1pΣq which gives bounds on the 3- and 4-dimensional
genus of K. The pairing S, called the Seifert pairing, is defined as follows.
Given a curve a representing a class ras P H1pΣq, let a` denote the result of
pushing the curve a off the surface in the positive normal direction of Σ. Let
Spa, bq “ lkpa, b`q. Moreover, given a basis for H1pΣq the Seifert pairing is
represented by a matrix, called the Seifert matrix.
Two important knot invariants that are defined in terms of the Seifert
matrix are the knot signature, σpKq, and the Alexander polynomial, ∆Kptq.
15
The matrix S ` ST is symmetric, so it can be diagonalized over Q. The knot
signtaure, σpKq, is defined to be the signature of this symmetric matrix,
namely the number of positive eigenvalues minus the number of negative
eigenvalues. The Alexander polynomial is defined as ∆Kptq “ detpt1{2S ´
t1{2ST q and it turns out that ∆Kptq is a Laurent polynomial in t that satis-
fies ∆Kp1q “ 1 and ∆Kptq “ ∆ ´1Kpt q. The breadth of the Laurent polyno-
mial ∆Kptq, defined as the difference of the largest and smallest exponents
that occur, is a lower bound for gpKq:
breadthp∆Kptqq ď 2gpKq.
Further, the signature gives a lower bound for the slice genus:
|σpKq| ď g4pKq
A knot is called slice if g4pKq “ 0 and so the signature vanishes on
the set of slice knots. We call two knots K and K 1 (smoothly) concordant if
there is a smoothly embedded annulus S1ˆ I Ñ S3ˆ I so that S1ˆ t0u “ K
and S1ˆt1u “ K 1. For example it is easy to see that a knot K is concordant
to the unknot U if and only if K is slice. The set of knots considered up to
concordance has a group structure given by connected sum. Let us denote
this group by C. Then the signature actually gives a homomorphism σ : C Ñ
Z.
In Chapter 2 and 3 we study knot invariants coming from Heegaard
Floer and knot Floer homology that in some sense generalize the above in-
variants. Knot Floer homology, introduced by Ozsváth and Szabó [OS04b]
and independently by Rasmussen [Ras03a], is a bigraded Abelian group
À
HFKpS3, Kq – HFK pS3z zm,A m , K,Aq and the graded Euler characteristic is
the Alexander polynomial:
ÿ
∆Kptq “ p´1qm dimpH 3zFKmpS ,K,AqqtA.
m,A
The A-grading is called the Alexander grading and the m-grading is the
homological or Maslov grading. An important property that we use repeat-
16
edly in our arguments in Chapter 3 is the genus detection property of knot
Floer homology [OS04a, Theorem 1.2]:
gpKq “ maxtA : HzFKpS3, K,Aq ‰ 0u.
This shows that knot Floer homology is a strictly stronger invariant than
the Alexander polynomial, for which there are non-trivial knots such that
∆Kptq “ 1. In particular, since the knot genus detects the unknot, knot
Floer homology detects the unknot.
Knot Floer homology is the homology of the associated graded of a Z-
filtered Z-graded chain complex defined in terms of a Heegaard diagram for
the pair pS3, Kq. See Section 1.2. From the definition, there is a spectral
sequence with E2 page HzFKpS3, Kq and E8 page H 3xFpS q – F [OS03b].
Then, there is a numerical invariant τpKq, which is defined to be the min-
imal Alexander grading of any cycle homologous to a generator of H 3xFpS q.
The invariant τpKq is a lower bound for the slice genus, |τpKq| ď g4pKq
[OS03b, Corollary 1.3]. Similarly to the knot signature, τ also gives a homo-
morphism τ : C Ñ Z. Although in general τpKq is different from σpKq, it
agrees with σpKq up to a factor for a few classes of knots, for example al-
ternating knots [Pet13]. The invariant τpKq has also proved fruitful in the
discovery of many subgroups of infinite rank in the smooth knot concordance
group [Lev16; Hom14; Hed07].
In addition to the complexity of the surface Σ in S3 or B4 measured
in terms of the genus, we are also interested in the complement of Σ in S3.
In particular, we are interested in when this complement is a product. We
call a knot K fibered with fiber Σ if there is a locally trivial fiber bundle
S3 ´ νpKq Ñ S1 with fiber Σ. It is easy to see that K is fibered if and
only if S3 ´ νpΣq – Σ ˆ I. It is worth pointing out that while it is possi-
ble for a knot to have two or more minimal genus Seifert surfaces that are
non-isotopic, fibered knots have a unique minimal genus Seifert surface up to
isotopy. As with the genus, there is an obstruction coming from the Alexan-
der polynomial of fibered knots, namely if K is fibered then the Alexander
polynomial is monic.
17
In Chapter 3 we will repeatedly use the fact that knot Floer homology
detects when a knot is fibered in S3 [Ni07; Juh08b]: K is fibered in S3 if
and only if dimpHzFKpS3, K, gpKqqq “ 1. Since the trefoil knot and the fig-
ure eight knot are the only genus 1 fibered knots in S3, it follows that knot
Floer homology detects these knots as well as the unknot.
Computing knot Floer homology directly from the definition is extremely
difficult. Although there are combinatorial models, for example the grid ho-
mology of [OSS15] and the nice diagrams of [OSS12] that do not require any
of the analysis that originally goes into the construction, the number of gen-
erators in this construction also makes computations impractical. In the fol-
lowing, we make use of bordered Floer homology and the bordered pairing
theorem to compute knot Floer homology of satellite knots in Chapter 3,
where we apply the genus and fiberedness detection results to some families
of satellite knots and compute τ for these same families, thus giving smooth
slice genus bounds, as well as independence results and constructions of
satellite operators that do not acts surjectively on the smooth concordance
group. See Section 1.3 and Chapter 3 for more details.
Dehn Surgery
Ů
A link in S3 is an embedding L : S1 Ñ S3 up to ambient isotopy.
A fundamental theorem in knot theory and low-dimensional topology says
that any closed, connected orientable 3-manifold can be obtained from S3 by
cutting out a neighborhood of a link L and re-gluing. This cutting out and
re-gluing operation is called Dehn surgery. We will study this operation in
Chapter 2, so we give a brief overview of it here.
Given a knot K in S3, we can thicken K to a tubular neighborhood,
νpKq, which is diffeomorphic to a solid torus. Cutting S3 along the bound-
ary BνpKq, we get two 3-manifolds-with-boundary: the complement of the
knot in S3, S3 ´ νpKq, and the tubular neighborhood of K, νpKq – D2 ˆ S1.
We can then use any homeomorphism h : BD2ˆS1 Ñ BpS3´ νpKqq to reglue
18
the solid torus and obtain a potentially new 3-manifold
pS3 ´ νpKqq Yh pD2 ˆ S1q.
It turns out that the diffeomorphism class of the 3-manifold obtained in
this way only depends on the image of the meridian hpBD2 ˆ tptuq. In fact,
H pS31 ´ νpKqq is generated by a meridian µ of K, and there is a unique ho-
mology class of curve on the boundary BpS3 ´ νpKqq that is null-homologous
in S3 ´ νpKq (the boundary of the Seifert surface from the previous section).
We call this curve a preferred longitude λ. Then µ and λ form a basis for
H1pBpS3´νpKqqq and any simple closed curve (e.g. hpBD2ˆtptuq) is isotopic
to a curve of the form pµ ` qλ, with gcdpp, qq “ 1. This sets up a bijection
between the set of simple closed curves on BpS3 ´ νpKqq and the set of re-
duced fractions p{q together with 8 “ 1{0 (8 corresponds to gluing back
along the meridian, so 8 surgery on any knot K in S3 gives back S3). Since
the diffeomorphism type of the manifold only depends on the isotopy class of
the attaching curve, we will write S3p{qpKq to denote the 3-manifold
pS3 ´ νpKqq Yh pD2 ˆ S1q
where hpBD2 ˆ tptuq “ pµ ` qλ, and we will call it the result of p -Dehn
q
surgery on K, and call p a slope. If L is a link with multiple components,
q
Dehn surgery on L is Dehn surgery on each component of L. We say the
surgery is integral if all the slopes are integral (i.e. q “ 1).
Theorem 1.1.1. [Lic62; Wal60] Every closed orientable 3-manifold can be
obtained from S3 by performing an integral Dehn surgery on a link L Ă S3.
As we will discuss in Chapter 2 the question of which 3-manifolds can
be obtained from Dehn surgery (rational or integral) on a knot in S3 is still
far from resolved. However, recently techniques have been developed that
allowed for the construction or recognition of 3-manifolds that are not Dehn
surgery on a knot.
19
Heegaard Floer Homology
In Chapter 2 we study the question of when a Dehn surgery on a knot
can produce a non-trivial connected sum of two 3-manifolds, called an re-
ducible 3-manifold, from the perspective of Heegaard Floer homology. As
such, we want to understand the Heegaard Floer homology of 3-manifolds
that are Dehn surgery along a knot: HF˝pS3p{qpKqq. We give a brief discus-
sion of the definition of Heegaard Floer homology here, and postpone dis-
cussing the computation of Heegaard Floer homology of Dehn surgeries us-
ing the mapping cone formula of [OS11] until Chapter 2.
Let Y be a closed, connected, oriented 3-manifold and let H “ pΣ, α, β, zq
be a pointed Heegaard diagram for Y . Here Σ Ă Y is a closed, connected,
orientable surface of genus g, Y ´ Σ consists of two handlebodies H0 and
H1, α “ tα1, . . . , αgu is a collection of non-intersecting simple closed curves
that form a set of compressing disks for H0 (we’ll refer to this as the α-
handlebody) and β “ tβ1, . . . , βgu is a collection of non-intersecting simple
closed curves that form a set of compressing disks for H1 (we’ll refer to this
as the β-handlebody). We also require that α and β intersect transversely
and that z is contained in Σ ´ pα Y βq. From this data we construct two La-
grangian submanifolds of the symmetric product SymgpΣq “ Σˆg{Sg, where
Sg is the symmetric group acting on the coordinates of Σˆg, for an appropri-
ate choice of symplectic form on SymgpΣq. The Lagrangian submanifolds are
the tori determine by the α and β curves
Tα “ α1 ˆ ¨ ¨ ¨ ˆ αg
and similarly for Tβ. The Heegaard Floer chain complex CF8pHq is freely
generated over F2 by pairs rx, is where x P TαXTβ and i P Z. The differential
is given by
ÿ ÿ
B8rx, is “ #Mxpϕqry, i´ nzpϕqs
tyPTαXTβu tϕPπ2px,yq|µpϕq“1u
Here π2px, yq denotes the set of homotopy classes of Whitney disks from
20
x to y, µpϕq denotes the Maslov index of ϕ, Mxpϕq is the moduli space space
of J-holomorphic disks in the class of ϕ, and nzpϕq “ #pϕXptzuˆSymg´1pΣqqq.
Moreover, if we let SpincpY q denote the set of Spinc structures on Y (an
affine copy of H2pY ;Zq), there is a map sz : T cα X Tβ Ñ Spin pY q so
that x and y are connected by a Whitney disk if and only if szpxq “ szpyq.
Therefore CF8pHq naturally splits over Spinc structures on Y : CF8pHq “
À 8
sPSpincpY qCF pH, sq. There is also an action by F2rU,U´1s on CF
8pH, sq
given by U rx, is “ rx, i ´ 1s and the U action decreases the grading by 2.
There are other flavors of the Heegaard Floer chain complex for a pointed
Heegaard diagram, C ´ `xFpH, sq,CF pH, sq and CF pH, sq, which are generated
by elements rx, is with szpxq “ s and i “ 0, i ă 0, or i ě 0 respectively.
We will write CF˝pH, sq for any of these flavors of the Heegaard Floer chain
complex, and HF˝pH, sq for the homology of this chain complex. In this no-
tation we have the following theorem
Theorem 1.1.2. [OS04d] The isomorphism class of HF˝pH, sq as a F2rU s
module is an invariant of the 3-manifold Y .
Given this theorem, we will write HF˝pY, sq instead of HF˝pH, sq. In
Chapter 2 we will recall how to compute HF˝pS3p{qpKqq from data associated
to the knot Floer chain complex using the mapping cone formula of [OS11].
In our work in Chapter 2 we are mostly interested in HxFpS3ppKq, sq and
HF`pS3ppKq, sq. Importantly, if Y is a rational homology 3-sphere, and s P
SpincpY q, we have the following structure theorem for HF`pY, sq: HF`pY, sq –
T ` ‘ HFredpY, sq [OS04c, Theorem 10.1] where HFredpY, sq is a finite dimen-
sional U -torsion module (called the reduced Heegaard Floer homology) and
T ` – FrU,U´1s{UF rU s (called the tower summand). An important numer-
ical invariant of rational homology 3-spheres derived from this is the grading
of the element 1 P T `, which we denote dpY, sq. It turns out that dpY, sq P Q
and if Y and Y 1 are Spinc-homology cobordant, that is if they cobound a
Spinc 4-manifold W with H˚pW ;Zq – H˚pYi;Zq, then dpY, sq “ dpY 1, s1q
where s and s1 are restrictions of the same Spinc structure t on W . In our
21
work in Chapter 2 we will investigate the behaviour of the d-invariant for
reducible manifolds.
Khovanov Homology
Khovanov homology is a bi-graded Abelian group, denoted Khi,jpKq,
that categorifies the Jones polynomial in the sense that the graded Euler
characteristic recovers the Jones polynomial V pKq up to a factor of pq ` q´1q
[Kho99]:
ÿ
pq ` q´1qV pKq “ p´1qi dimpKhi,jpKqqqj
i,j
Khovanov homology is an invariant of the isotopy class of a knot or link
in S3 that is combinatorially defined from a diagram of the knot or link.
Khovanov homology is most strikingly useful in its functoriality under knot
cobordisms. Namely given an oriented cobordism Σ : K0 Ñ K1 there is an
induce map KhpΣq : KhpK0q Ñ KhpK1q.
Ribbon Concordance
Let us consider the concordance S1 ˆ I sitting inside of S3 ˆ I so that
the projection map S3ˆI Ñ I restricts to a Morse function of S1ˆI. Then a
concordance is called ribbon if there are only index 0 and 1 critical points for
this Morse function, that is the concordance annulus can be built with only
births and band attachments, no deaths. A fundamental open problem in
knot theory is the Slice-Ribbon conjecture which asserts that a knot is slice if
and only if the knot is ribbon.
Recently progress has been made relating ribbon concordanes between
knots to properties on the induced maps on Khovanov and knot Floer ho-
mology. Work of Zemke in [Zem19] and Levine and Zemke in [LZ19] showed
that ribbon concordance of knots induce injective maps on knot Floer ho-
mology and Khovanov homology, respectively. Classical work of Kirby and
Lickorish [KL79] shows that any knot K is ribbon concordant to a prime
22
knot, which implies that the Khovanov homology of any knot K is a sum-
mand of the Khovanov homology of a prime knot; see Chapter 4. This ob-
servation allows for the propagation of Steenrod squares from the Khovanov
homology of composite knots to the Khovanov homology of prime knots.
1.2 Chapter 2: Reducible Surgeries on Slice and Almost L-Space Knots
This chapter contains unpublished co-authored material with Robert
DeYeso III. Recall that Heegaard Floer homology is the homology of a graded
chain complex CF˝pY q where ˝ P t̂ ,`,´,8u, where CF˝pY q is either a
Z-graded F2-vector space, a FrU s-module or a FrU,U´1s-module for ˝ P
t̂ ,`,´,8u. The chain homotopy type of any flavor of the chain complex
is a smooth 3-manifold invariant. The definition of knot Floer homology is
similar to the definition for Heegaard Floer homology. One starts with a
doubly-pointed Heegaard diagram for the pair pS3, Kq which is the data of
a tuple pΣ, α, β, z, wq so that w and z are in the complement of the α and β
curves and pΣ, α, β, zq is a pointed Heegaard diagram for S3. The knot K is
the union of two arcs: an arc a in Σ ´ α connecting w to z and pushed into
the α-handlebody and and arc b in Σ ´ β connecting z to w and pushed into
the β-handlebody. The chain complex CFK´pS3, Kq is generated over FrU s
by the intersections between the α and β tori, and the differential is defined
as
ÿ ÿ
B´x “ #MpϕqUnwpϕqx y
y ϕPπ2px,yq
indpϕq“1
The chain complex CFK´pS3, Kq has a Z-grading, called the homolog-
ical or Maslov grading, denoted m, and a Z-filtration called the Alexander
filtration, denoted A. The differential decreases the Maslov grading by one
and respects the Alexander filtration. For generators x and y the relative
Maslov and Alexander gradings are defined as
Mpxq ´Mpyq “ indpϕq ´ 2nwpϕq and Apxq ´ Apyq “ nzpϕq ´ nwpϕq
23
Furthermore, MpUxq “ Mpxq ´ 2 and ApUxq “ Apxq ´ 1. Moreover,
setting U “ 0 we obtain the filtered complex CzFKpS3, Kq. The homology of
the associated graded is denoted by HFKpS3z , Kq and the filtration induces
a spectral sequence with E8-page H 3xFpS q. We lift the Maslov grading to an
absolute grading by setting the Maslov grading of this generator to be zero.
There is a related complex, that governs the mapping cone formula
and specializes to the hat and minus complexes above. Define the complex
CFK8pS3, Kq :“ CFK´pS3, Kq b FrU,U´1s. This is a Z ‘ Z-filtered Z-
graded chain complex. We view this complex in the pi, jq-plane where a gen-
erator Unx is plotted at the coordinates p´n,Apxq´nq. Then, given a subset
S Ă Z ‘ Z such that pi, jq P S implies pi1, j1q P S for all pi1, j1q ď pi, jq, we
can form a subcomples CtSu, the set of all generators of CFK8pS3, Kq with
pi, jq coordinates in S. Note that in this language CFK´pS3, Kq “ Cti ď 0u
and CzFKpS3, Kq “ Cti ď 0u{Cti ă 0u. In these terms we can define τpKq as
follows [OS03b]
τpKq “ mints | ι : Cti “ 0, j ď su Ñ Cti “ 0u induces a non-trival map on homologyu
An important computational tool in the theory of Heegaard Floer ho-
mology is the mapping cone formula which relates information about the
knot Floer complex to the Heegaard Floer homology of the 3-manifold ob-
tained by Dehn surgery on K [OS11]. Using the mapping cone formula one
can both compute the Heegaard Floer homology of many 3-manifolds that
are surgery on knots and one can also hope to obstruct certain 3-manifolds
from being realized as Dehn surgery on a knot (or a knot of a particular
type) in S3.
An easy obstruction to a 3-manifold being Dehn surgery on a knot is
that the first homology of the 3-manifold must be Z or Z{nZ. Therefore,
many 3-manifolds, e.g. T 3 and pS1 ˆ S2q#pS1 ˆ S2q, cannot be obtained by
Dehn surgery on a knot in S3. Similarly, it is easy to see from the Wirtinger
presentation for π1pS3 ´ νpKqq that a 3-manifold obtained by Dehn surgery
on a knot must also have weight-one fundamental group, that is the funda-
mental group is normally generated by a single element. If we restrict atten-
24
tion to integer homology 3 spheres (3-manifolds with the same integral ho-
mology as S3) there are examples of irreducible integer homology 3-spheres
that do not arise as Dehn surgery on a knot. The first example was discov-
ered by David Auckly [Auc97], using gauge theory and Donaldson’s Diago-
nalization theorem. Auckly’s construction is sufficiently complicated that he
was not able to verify if the fundamental group was weight one.
More recently, the mapping cone formula and Heegaard Floer homology
has proved useful in studying this question. For example, Hom-Karakurt-
Lidman in [HKL16] showed that many irreducible integer homology spheres
are not surgery along a knot in S3 (see also [HL16]). Their proof involves
in a non-trivial way both the mapping cone formula of [OS11] and the com-
putation of Heegaard Floer homology of some small Seifert fibered spaces.
In particular they find a relation between the d-invariant of an integer ho-
mology 3-sphere Y and the module structure on HFredpY q when Y is Dehn
surgery on a knot K in S3, and show that this relation does not hold for a
family of Seifert fibered spaces, namely the Seifert fibered spaces Σpp, 2p ´
1, 2p`1q. Furthermore, because their examples are relatively small 3-manifolds,
they were able to verify that they have weight one fundamental group.
In a similar direction, one might ask when Dehn surgery on a knot pro-
duces a non-trivial connected sum of two or more 3-manifolds, called a re-
ducible 3-manifold. Perhaps surprisingly, there are numerous examples of
such knots. In [Mos71], Moser showed that S3pqpTp,qq – Lpp, qq#Lpq, pq where
Tp,q denotes the pp, qq torus knot, and by a similar argument one can show
that S3pqpK 3p,qq – Lpp, qq#Sq{ppKq, where Kp,q denotes the pp, qq-cable of the
knot K. In [GS86], the authors propose the following conjecture, called the
Cabling Conjecture, which is still open:
Conjecture 1.2.1. If K is a knot that admits a reducible surgery with slope
p, then K “ Jr,s for some knot J and p “ rs.
Note that this question is asking when does Dehn surgery on a knot
contain an essential separating 2-sphere. This conjecture has been verified
in many cases, and many properties of the resulting reducible manifold are
25
known. See Chapter 2 for a more comprehensive survey. We only mention
a few important results here. If Dehn surgery on a knot is a reducible 3-
manifold, then there must be a non-trivial lens space summand (and so the
surgery slope has to be ą 1) and the slope is integral [GL87]. In particu-
lar there are no reducible integer homology spheres that can be obtained
from S3 by Dehn surgery on a knot. Additionally, the Cabling Conjecture
is known for torus and satellite knots [Mos71; Sch90] so it remains to study
integral surgeries on hyperbolic knots. Moreover for hyperbolic knots the re-
ducing slopes satisfy the restrictive bound 1 ă |p| ď 2gpKq ´ 1 [MS03].
Two other related conjectures that we explore are the Two-Summands Con-
jecture and the Multiple Reducing Slopes Conjecture. The two-summands
conjecture asserts that at most two summands can result from Dehn surgery
along a knot in S3 and the multiple reducing slope conjecture asserts that
any knot has at most one reducing slope. If we consider slopes as curves
on the boundary of the knot complement, we can compute their geometric
intersection in T 2. A theorem of Gordon and Luecke [GL96, Theorem 1.2]
shows that if there are multiple reducing slopes for a given knot, then they
have geometric intersection 1, and since they are integral slopes this implies
they are consecutive integers. Furthermore, no more than three summands
can occur in a Dehn surgery along a knot in S3 [Val99; How02], and if three
summands occur one is an integer homology sphere and the other two are
non-trivial lens spaces.
Recently, Heegaard Floer homology has proved useful in studying the
Cabling, Multiple Reducing Slopes and Two Summand Conjecture, as well
as the general question of when a 3-manifold is surgery along a knot. Since
there is a non-trivial lens space summand in any reducible surgery on a knot
in S3, which has the simplest possible Floer homology in each Spinc struc-
ture, and Heegaard Floer homology satisfies a Künneth formula for con-
nected sums [OS04c, Theorem 1.5], we have the following
Theorem 1.2.2 ([HLZ15] Lemma 2.6). Suppose K satisfies S3ppKq – Lpa, sq#R
where p “ ar and |H1pR;Zq| “ r. Then for any t P SpincpS3ppKqq and any
α P H2pS3 ` 3 ` 3ppKqq we have HF pSppKq, rtsq – HF pSppKq, rt`αrsq as relatively
26
graded FrU s modules.
In [HLZ15], the authors use the above observation to show that if an L-
space knot K has a reducible surgery, then the reducing slope is 2gpKq ´ 1
[HLZ15, Theorem 1.3]. So in particular, L-space knots do not have multi-
ple reducing slopes and do not have properly embedded punctured projec-
tive planes in their complements, since the surgery slope is odd [HLZ15,
Corollary 1.5]. Moreover, they show using the mapping cone formula that
for knots of genus 1 and 2, there are not multiple reducing slopes. Greene,
in [Gre15], verified the Cabling Conjecture when the resulting manifold is a
connected sum of two lens spaces, and in [Mei17] Meier shows that a slice
knot satisfies the two-summands conjecture by comparing the d-invariants
of S3ppKq when K is slice (which are equal to the d invariant of Lpp, 1q by
[NW15, Proposition 1.6]) to Lpr, aq#Lps, bq#Y , where p “ rs and Y is an
integer homology sphere and showing that these are never equal, using the
computation of d-invariants of lens space provided in [OS03a].
In Chapter 2, we use the mapping cone formula and d-invariants to
study reducible Dehn surgeries on slice and almost L-space knots in S3 and
determine obstructions to general knots admitting reducible surgeries. Re-
call that if a knot K is slice, then so is the pp, 1q-cable of K, Kp,1. Then as
above, S3ppKp,1q – Lpp, 1q#S31{ppKq. In particular, in a reducible surgery on
a cable of a slice knot there are two summands, one a lens space that car-
ries all the homology and the other an integer homology 3-sphere. Using the
formula for the d-invariant of surgery along a knot established in [NW15,
Proposition 1.6], it also follows that dpS31{ppKqq “ 0. We show that this
holds in a general reducible surgery on a slice knot
Theorem 1.2.3. For a slice knot K and a reducing slope p, S3ppKq – Lpp, 1q#Y ,
where Y is an integer homology 3-sphere with dpY q “ 0.
The above theorem, when combined with [GS86, Proposition 1.4], gives
the following Corollary, which further restricts the possible reducing slopes
on fibered hyperbolic slice knots. Recall that if K is a hyperbolic knot with
reducing slope r, then 1 ă |r| ď 2gpKq ´ 1. We show
27
Corollary 1.2.4. If K is a fibered, hyperbolic slice knot of genus g and r is
a reducing slope, then |r| ď g.
For a general knot K with a reducing slope r “ pq, we prove the follow-
ing slice genus bound:
Theorem 1.2.5. If S3pqpKq – p q ě
pp´ 1qpq ´ 1q
Yp#Yq, then g4 K “
2
g4pTp,qq.
This generalizes a result of [GS86] (See also [Eis22]) that says that the
3-genus of a knot K with reducing slope pq is bounded below by gpTp,qq “
pp´ 1qpq ´ 1q
2
We also study the question of when a knot can admit multiple reducing
slopes and show that if a hyperbolic slice knot K admits multiple reducing
slopes r and r ` 1 then r ` 1 ď gpKq. In the following Theorem we make
reference to the invariant νpKq. The map v̂ : Ap Ñ Bps s s is the map from the
mapping cone formula of [OS11] (see also Chapter 2).
Definition 1.2.6. [OS11, Definition 9.1] For a knot K Ă S3, let νpKq :“
mints | pv̂sq˚ ‰ 0u.
Theorem 1.2.7. Suppose K is a hyperbolic knot in S3 with νpKq ă gpKq
that admits consecutive reducing slopes r and r ` 1. Further, suppose that
both r and r` 1 surgery split off an integer homology sphere summand. Then
r ` 1 ď gpKq.
In joint work with Robert DeYeso III, we also study the question of
when 2gpKq ´ 1 surgery on an L-space knot can be reducible and prove the
following. This theorem, combined with the work of [DeY21], shows that
thin knots satisfy the Cabling Conjecture.
Theorem 1.2.8. Thin hyperbolic L-space knots do not admit reducible surg-
eries.
28
In a similar direction, building on work of [HLZ15] we show that only
certain slopes on almost L-space knots can produce reducible 3-manifolds,
see Chapter 2, Section 2.3.
An L-space is a rational homology 3-sphere Y so that dimpHxFpY qq “
|H1pY ;Zq|. A knot K is called an L-space knot if S3npKq is an L-space for
all n ě 2gpKq ´ 1. In [BS22] the authors define a generalization of an L-
space, called an almost L-space, which is a rational homology 3-sphere Y so
that dimpHxFpY qq “ |H1pY ;Zq| ` 2 (the next to smallest possible dimen-
sion). Exactly as above we call a knot K an almost L-space knot if S3npKq
is an almost L-space for all n ě 2gpKq ´ 1. It turns out that the full knot
Floer complex of almost L-space knots can be determined, similarly to the
case of L-space knot, see [Bin23]. We use these results to prove the following
theorems.
Theorem 1.2.9. For an almost L-space knot K, the only possible reducing
slopes are gpKq (or ˘gpKq if gpKq “ 3), ˘2, or p2gpKq ´ 2q (and the latter
two only in the case gpKq is even)
Corollary 1.2.10. If K is an almost L-space knot with odd genus gpKq and
gpKq ą 3, then gpKq is the only reducing slope.
Corollary 1.2.11. If K is an almost L-space knot with odd genus gpKq,
then the complement of K does not contain any properly embedded punctured
projective planes.
Proof. Suppose that the complement of K contained a properly embedded
punctured projective plane P . Then BP gives a slope p so that filling along
that slope gives an embedded RP 2, Pp. Consider a tubular neighborhood of
P in S3p ppKq. If S3ppKq ´ Pp were a 3-ball, then S3ppKq would be homeomor-
phic to RP 3. By the RP 3 theorem, [Kro+07, Theorem 1.1], this is impossi-
ble since K is non-trivial. Therefore S3ppKq – RP 3#Y for some 3-manifold
Y with p “ 2|H1pY ;Zq|. However, p is even and is also a reducing slope for
an almost L-space knot with odd genus. This contradicts Theorem 1.2.9.
29
Corollary 1.2.12. Almost L-space knots of genus gpKq ě 2 do not admit
multiple reducing slopes.
1.3 Chapter 3: Knot Floer Homology, Immersed Curves and Twisted
Satellites
In this chapter we study the behavior of certain invariants, coming from
classical 3-manifold topology and Heegaard Floer homology, under the oper-
ation of taking the n-twisted satellite. Given a knot K in S3 and a knot P
in S1 ˆD2, we can form a new knot in S3, called the n-twisted satellite knot
with pattern P and companion K, denoted PnpKq, as follows:
pS3, PnpKqq “ pS3 ´ νpKqq Yϕ pS1 ˆD2, P q,
where the map ϕ takes the meridian of the solid torus, BD2 ˆ tptu, to the
meridian of K and the longitude of the solid torus, S1 ˆ tptu, to nµ ` λ,
where µ and λ are a meridian and longitude for the knot.
In general, computing the knot Floer homology of satellite knots from
the definition is extremely difficult. Our approach to studying the question
of how knot Floer homology behaves under satellite operations uses the bor-
dered Floer homology of [LOT18] and the reformulation of the bordered in-
variants and the bordered pairing theorem for manifolds with torus bound-
ary in terms of immersed curves [HRW17; HRW22; Che19; CH23].
Bordered Algebra
In this section we describe the algebraic preliminaries to understand
the bordered pairing theorem. In what follows we will focus on bordered 3-
manifolds with torus boundary. Let A “ ApT 2q. The algebra A is defined as
follows. Over F it has a basis consisting of two mutually orthogonal idempo-
tents ι0 and ι1 and six other nontrivial elements ρ1, ρ2, ρ3, ρ12, ρ23, ρ123. The
non-zero products in the algebra are given as follows:
ρ1ρ2 “ ρ12 ρ2ρ3 “ ρ23 ρ1ρ23 “ ρ12ρ3 “ ρ123
30
ρ1 “ ι0ρ1ι1 ρ2 “ ι1ρ2ι0 ρ3 “ ι0ρ3ι1
ρ12 “ ι0ρ12ι0 ρ23 “ ι1ρ23ι1 ρ123 “ ι0ρ123ι1
If we let I “ xι0y ‘ xι1y, then a type D structure over A is a unital left I
module N together with an I linear map δ : N Ñ AbI N such that
pµb Iq ˝ pIb δq ˝ δ “ 0
A type A structure is a right unital I module M with a collection of
maps mi`1 : M bAi ÑM , for i ě 0, such that M “Mι0 ‘Mι1 and
n n´2
ÿ ÿ
0 “ mn´ipmipxba1b¨ ¨ ¨bai´1qb¨ ¨ ¨ban´1q` mn´1pxb¨ ¨ ¨baiai`1b¨ ¨ ¨banq
i“1 i“1
(1.1)
and so that for x PM and ai P A
m2px, 1q “ x
mipx, ¨ ¨ ¨ , 1, ¨ ¨ ¨ q “ 0
Given a type A structure M and a type D structure N , we can form a
chain complex, called their box tensor product and denoted M b N . The
underlying vector space is the tensor product M bI N , and the differential is
defined by
8
ÿ
Bbpxb yq “ pmi`1 b Iqpxb δipyqq, (1.2)
i“0
In the case that the type D structure is bounded, as defined in [LOT18,
Section 2], then the above sum is finite and the box tensor complex is well
defined.
31
Bordered 3-manifolds and the Pairing Theorem
Given a closed three manifold Y and a surface F embedded in Y so
that rF s “ 0 P H2pY q, we can cut Y along F to produce two 3-manifolds-
with-boundary Y1 and Y2 so that BY1 – ´BY2 – F . Such a 3-manifold-with-
boundary together with a diffeomorphism ϕ : F Ñ BY is called a bordered
3-manifold. To such a surface, [LOT18] associate a differential graded alge-
bra, ApF q, and to Y1 and Y2 as above they associate a right Type A struc-
ture CzFApY1q and a left type D structure CzFDpY2q. Furthermore, given a
doubly-pointed bordered Heegaard diagram H “ pΣ, αa, β, z, wq for a knot
K in Y1 which becomes null-homologous in Y , we can associate a right type
A structure over ApF q with coefficients in FrU s denoted CFA´pHq, or a fil-
tered type A structure CzFApHq. Then the work in [LOT18, Theorem 1.3,
11.21] shows that there are homotopy equivalences
CxFpY q » CzFApY1qb CzFDpY2q
CzFKpY,Kq » CzFApHqb CzFDpY2q
gCFK´pY,Kq » CFA´pHqb CzFDpY2q
Warning: CFA´pHq depends on the choice of Heegaard diagram for the pair
pY1, Kq, but the result of pairing with C 3zFDpS zνpKqq does not. We will
abuse notation by writing CFA´pY1, Kq when it’s clear we have fixed a Hee-
gaard diagram for pY1, Kq.
Bordered Floer and Satellites Knots
In the satellite knot construction, we have two 3-manifolds with torus
boundary, S3 ´ νpKq and pS1 ˆ D2, P q and the latter is a 3-manifold with
boundary together with a knot P that becomes null homologous in the glued
up manifold. Note that in the n-twisted satellite construction we can either
directly add n twists to the pattern knot P , or we can change the fram-
ing of the knot complement. The bordered pairing theorem implies that
gCFK´pS3, P pKqq – CFA´n pS1 ˆ D2, P q b CFDpS3z ´ νpKq, nq, where
32
ρ2 ρ3
ξ0 λ ξ1
ρ1
µ1 ρ1
ρ23
µ2 κ
ρ123
ρ3
ξ2
Figure 1.1. Type D structure for 0-framed right handed trefoil complement
CzFDpS3 ´ νpKq, nq denotes the type D structure associated to the n-framed
knot complement. So one way to compute the knot Floer homology of n-
twisted satellite knots is to understand CzFDpS3 ´ νpKq, nq and CFA´pS1 ˆ
D2, P q.
In the case of a knot in S3, [LOT18, Theorem 11.26] shows that the
type D structure for the n-framed knot complement is algorithmically ob-
tainable from the knot Floer complex CFK´pS3, Kq. Indeed, they show that
given a horizontally and vertically simplified basis for the knot Floer com-
plex CFK´pKq and a choice of framing n, one can easily write down the
bordered type D structure associated to the n-framed knot complement.
See Figure 1.1 for an example of CzFDpS3 ´ νpT2,3q, 0q. In fact, we can feed
in partial information about CFK´pKq, and still extract some information
about the type D structure. In particular, we will see in Chapter 3, Lemma
3.2.2 that this algorithm allows us to understand a piece of the type D struc-
ture CFDpS3z ´ νpKq, nq only knowing the triple pτpKq, ϵpKq, nq, and this
piece of the type D structure ends up carrying a lot of the information in the
pairing with C 1 2zFApS ˆD ,P q.
More recently [HRW22] showed that the bordered invariants for mani-
folds with torus boundary are equivalent to the data of an immersed multi-
curve in the once-punctured torus T 2 ´ tzu. Therefore for the case of type
D structures associated to n framed knot complements, CzFDpS3 ´ νpKq, nq
33
λ “ αa1
y0
y1
y2
y3
y4
µ “ αa2 z
y5
y6
y7
w
x0 x1 x2
Figure 1.2. A genus-1 doubly-pointed Heegaard diagram. The blue curve is
the β curve
gives rise to an immersed multi-curve, which we will denote αpK,nq, in the
punctured torus T 2 ´ tzu. In Chapter 3, Lemmas 3.2.2, 3.8.2 and 3.8.3, we
recall how this correspondence works. The work of [LOT18] discussed in the
previous paragraph gives a partial structure theorem for one special com-
ponent of the immersed curve in terms of the triple pτpKq, ϵpKq, nq, and we
use that to give a partial structure theorem for αpK,nq in terms of the triple
pτpKq, ϵpKq, nq.
The other ingredient in the bordered pairing theorem are the Type A
structures associated to pattern knots in S1 ˆ D2. In general, these are
difficult to compute since the definition of the type A structure involves
counting holomorphic disks with prescribed boundary conditions in some
symplectic manifold associated to a bordered Heegaard diagram for the pair
pS1 ˆD2, P q.
A p1, 1q-pattern knot is a knot in S1 ˆ D2 that has a genus-1 doubly-
pointed bordered Heegaard diagrams. A genus-1 doubly pointed bordered
34
Figure 1.3. The pattern in the solid torus determined by the doubly pointed
Heegaard diagram in figure 1.2 by the procedure described in the text
Heegaard diagram is a tuple pT 2, αa, βpP q, z, wq, where αa “ tµ, λu is a
preferred framing of the boundary and βpP q is a curve in T 2 that is iso-
topic to the meridian after forgetting the z basepoint, so when we attach a
two-handle to βpP q we obtain the 3-manifold-with-boundary S1 ˆ D2. The
knot P is specified by joining z to w and w to z in the complement of the αa
andv βpP q respectively. See figures 1.2 and 1.3 for an example.
For p1, 1q-patterns P and arbitrary companions K [Che19] showed that
the bordered pairing theorem for computing CzFKpS3, PnpKqq can be re-
formulated as the intersection Floer homology of two curves in the twice-
punctured torus: namely the immersed multi-curve αpK,nq and the β curve
from the p1, 1q diagram associated to P , βpP q. See Figure 1.4 for an exam-
ple of this pairing in T 2´tz, wu where αpK,nq “ αpT2,3, 0q and βpP q is the β
curve from the genus-1 doubly pointed bordered Heegaard diagram in figure
1.2. We review this work in Section 3.2 and use it to prove the main results
of Chapter 3, namely a computation of the Heegaard Floer concordance in-
35
x0ξ0 αpT2,3, 0q
x0ξ1
y0µ1 y0κ
β
Figure 1.4. Executive summary of the pairing theorem for satellites with
p1, 1q patterns. Intersection points in the second and fourth quadrant
correspond to generators of the knot Floer chain complex of the satellite and
Whitney disks as shown connecting two generators and not containing either
basepoint are differentials
variants τpPnpKqq and ϵpPnpKqq for satellite knots with patterns from two
novel families of p1, 1q-patterns. The first family, denoted P pp,1q is a family of
patters so that P pp,1qpUq „ T2,3 (called trefoil patterns) with winding number
p ` 1. See figure 1.3 for a picutre of P p3,1q. As discussed in Chapter 3 these
trefoil patterns are related to the pp, 1q cabling patterns by a finger move
applied to the curve βpp, 1q. The second family is a family of pattern knots
denoted Qi,jn , shown in figure 3.24, so that Q
i,j
0 pUq „ U and the winding
number of Qi,jn is j. The family of patterns Qi,j generalizes both the White-
head doubling pattern and the Mazur pattern, both of which have been used
to probe the structure of the smooth, topological and PL structure of knot
concordance and homology cobordism [Hed07], [Lev16].
On top of understanding the behaviour of the concordance invariants
τpQi,jn pKqq and ϵpQi,jn pKqq as functions of n, we also study the three genus
36
and fiberedness of these families of satellite knots via bordered Floer homol-
ogy. In [HMS08] the authors give a criterion for when a satellite knot fibers
in S3: the n-twisted satellite knot PnpKq is fibered if and only if both K
is fibered in S3 and P is fibered in S1 ˆ D2n . This result can be read as a
way to detect if a pattern knot Pn is fibered in the solid torus, all that is re-
quired is to determine whether or not the satellite knot PnpT2,3q is fibered.
Since knot Floer homology detects fiberedness of knots in S3, to show that
PnpT2,3q is fibered it is enough to compute the rank of the knot Floer homol-
ogy of the satellite knot PnpT2,3q in the top Alexander grading. As we will
see in Chapter 3 this computation is relatively simple for p1, 1q-patterns as
the type D structure associated to the complement of the right-handed tre-
foil is relatively simple. Using this, we determine for which triples pi, j, nq
the pattern knot Qi,jn is fibered in the solid torus:
Theorem 1.3.1. The pattern knots Qi,jn Ă S1 ˆD2 are fibered if and only if
i “ 0 and either j ě 2 and n ‰ 0 or j “ 1 and n ‰ 0,´1.
It is interesting to compare the above result with the following theorem
which classifies fibered unknot patterns.
Theorem 1.3.2. [HMS08, Theorem 5.1] If P is a pattern so that P pUq »
U , then P is fibered if and only if it is braided.
In particular Qi,j0 is never fibered. However, it is interesting that this is
in some sense the only bad slope in that for basically all the other values of
n and j, the pattern Q0,jn is fibered and for no values of n is the pattern Qi,jn
fibered when i ą 0. This raises the interesting question: which (non-braided)
unknot patterns admit infinitely many twist parameters so that the resulting
pattern (no longer of unknot type in S3) is fibered.
Another 3 dimensional invariant of knots is their genus, the minimal
genus of a surface that the knot bounds. A classical result of Schubert [Sch53]
shows that the genus of a satellite with non-trivial companion knot K can
be expressed in terms of the genus of the companion, the winding number of
the pattern, and the genus of the pattern as follows:
37
gpP pKqq “ |wpP q|gpKq ` gpP q.
In particular, since the above equality works for any non-trivial knot K,
we might as well take K “ T2,3. Then the genus of the pattern is gpP q “
gpP pT2,3qq ´ |wpP q|. Since knot Floer homology detects the genus of knots
in S3, we see that in some sense the bordered invariant CzFApS1 ˆ D2, P q
detects the genus of the pattern knots Pn, by pairing with the type D struc-
ture associated to the n-framed trefoil complement CzFDpS3 ´ νpT2,3q, nq,
finding the genus of the resulting satellite and then using Schubert’s formula
[Sch53]. We perform this computation for the family of patterns Qi,jn and
find:
Theorem 1.3.3. For K non trivial, we have
$
&1 n ě 0
gpQi,jn pKqq “
jpj ` 1q
jgpKq ` |n| `
2 %1´ j n ă 0
We can use the same techniques to determine gpQi,jn pUqq
Theorem 1.3.4. For K “ U ,
$
jpj ` 1q
’
’ n` 1 n ą 0
’
’
& 2
gpQi,jn pUqq “ 0 n “ 0
’
’
’jpj ` 1q
’
% |n| ` 1´ j n ă 0
2
This shows that the genus of these twisted patterns depends quadrat-
ically on j (the winding number) and linearly on the number of meridional
twists n added to the pattern. This generalizes work of [PW21], where the
study the genus of the n-twisted Mazur pattern Q0,1. The question of how
the genus of knots that are related by adding full twists grows is also ex-
plored in [BM19].
The question of how satellite knots behave with respect to concordance
is especially interesting, as there are many open questions that are easily
38
stated. For example, if W denotes the Whitehead double pattern, it is un-
known if W pT´2,3q is smoothly slice (it has Alexander polynomial 1, so is
topologically slice by work of Freedman). However, in the last 20 years,
many invariants have been developed that obstruct certain families of satel-
lite knots from being slice. Two of the more successful invariants are τpKq
and ϵpKq coming from knot Floer homology. Recall that, by construction,
the knot Floer homology H 3zFKpS ,Kq admits a spectral sequence to HxFpS3q
and the invariant τpKq is the Alexander grading of the generator that sur-
vives this spectral sequence. In fact there are two spectral sequences to HxFpS3q,
by symmetry of the knot Floer homology, and the invariant ϵ measures how
the above generator interacts with the other spectral sequence. In our work,
we also compute τ and ϵ of the two families of satellites with arbitrary com-
panion knots and patterns P pp,1q and Qi,jn . In particular, we show that for
the patterns Qi,jn as above, we have
Theorem 1.3.5. If K is a knot in S3 with ϵpKq “ ´1, then
τpQi,jn pKqq “ jpτpKq `
jpj ´ 1q
1q ` n.
2
If K is a knot in S3 with ϵpKq “ 1, then
$
’
&jτpKq ` jpj ´ 1qn` 1 n ă 2τpKq
τpQi,jn pKqq “ 2 .
’
%jτpKq ` jpj ´ 1qn n ě 2τpKq
2
If K is a knot in S3 with ϵpKq “ 0, then
$
jpj ´ 1q
’
& n n ě 0
τpQi,jn pKqq “ 2jpj ´ 1q
’
% n` j n ă 0
2
Corollary 1.3.6. For M “ Q0,1 the Mazur pattern, the value of τpQ0,1n pKqq
does not depend on n, only on n relative to 2τpKq.
Theorem 1.3.7. For any knot K, ϵpQi,jn pKqq ‰ ´1.
39
A consequence of the above theorem is that these patterns do not act
surjectively as operators on the smooth or rational homology concordance
group, since for any knot K, Qi,jn pKq is never concordant to a knot L with
ϵpLq “ ´1. This shows that there are infinitely many patterns of arbitrarily
large winding number with this property, and shows that this property is
preserved by adding twists in the clasp region, and adding full meridional
twists to these patterns. Consequently, following a construction of Levine in
[Lev16], the patterns Qi,1n can be used to construct infinitely many knots not
concordant to any knot in S3.
1.4 Chapter 4: Non-trivial Steenrod Squares on the Khovanov Homology of
Prime Knots
This chapter contains previously published material. In 2014, Lipshitz
and Sarkar introduced a stable homotopy refinement of Khovanov homology
[LS14a]. For each knot K and fixed j it takes the form of a suspension spec-
trum X jpKq. The cohomology H˚pX jKqq of this spectrum is isomorphic to
the Khovanov homology Kh˚,jpKq. In subsequent work (e.g. [LS14c]) they
used this refinement to define stable cohomology operations on Khovanov
homology. This lead to a refinement of Rasmussen’s s-invariant [Ras03b]
for each nontrivial cohomology operation, and in particular for the Steen-
rod squares [LS14c]. In Chapter 4 we positively answer a question posed in
Lipshitz-Sarkar [LS18, Question 3]: Are there prime knots with arbitrarily
high Steenrod squares on their Khovanov homology? Explicitly, we prove the
following theorem:
Theorem 1.4.1. Given any n, there exists a prime knot Pn so that the op-
eration
n i,j i`n,jSq : KĂh pP Ănq Ñ Kh pPnq
is nontrivial for some pi, jq. Here KĂh denotes reduced Khovanov homology.
We construct these knots explicitly by ribbon concordances. The main re-
sults we use are a classical result of Kirby and Lickorish that any knot K is
40
ribbon concordanct to a prime knot [KL79], a result of Levine-Zemke [LZ19]
(see also [Wil12]) that if Σ : K Ñ K 1 is a ribbon concordance between knots
K and K 1 then the map induce on Khovanov homology KhpΣq : KhpKq Ñ
KhpK 1q is injective and the fact that there are composite knots with ar-
bitrarily large Steenrod operations on their Khovanov homology [LLS15,
Proof of Corollary 1.4, Page 67]. We are also able to use a result of Liv-
ingston which shows that any knot is ribbon concordant to a prime satel-
lite knot [Liv81] to show that there are prime satellite knots with arbitrarily
large Steenrod squares on their Khovanov homology. Further, using work of
[Kaw89], we also note that since any knot K has an invertible concordance
to a hyperbolic knot, we can propagate Steenrod squares from composite
knots to prime hyperbolic knots as well, without the use of the injectivity
result of Levine-Zemke [LZ19].
41
CHAPTER 2
REDUCIBLE SURGERIES
2.1 Overview
In this chapter we study obstructions to knots admitting reducible surg-
eries. This chapter draws on unpublished co-authored material joint with
Robert DeYeso III, though the included material is all the author’s own.
Introduction
In the following section, we will be making use of the relationship be-
tween knot Floer homology and the Heegaard Floer homology of 3-manifolds
obtained from Dehn surgery along a knot in S3. This relationship was first
worked out by Oszvath and Szabo in [OS11] and we recall their notation and
results here. In the first section we are mostly interested in the d-invariants
of Dehn surgery, which are the gradings of some distinguished generators in
the Heegaard Floer homology. In later sections, we make use of the full map-
ping cone to study reducible surgeries on almost L-space knots and multiple
reducing slopes on general knots.
Let K be a knot in S3, and denote the result of p -Dehn surgery on K
q
as S3p{qpKq. This is the operation of removing a tubular neighborhood of
K, νpKq – S1 ˆ D2, in S3 and replacing it with a solid torus S1 ˆ D2 so
that tptu ˆ BD2 maps to a slope p{q curve on BpνpKqq in terms of a basis
rµs and rλs H1pBpνpKqq;Zq. Here rµs is a curve that bounds a disk in νpKq
and rλs is a curve on BpνpKqq that intersects µ in a single point and provides
the Seifert framing of K (so it is nullhomologous in S3zK). We are inter-
ested in Dehn surgeries that produce essential 2-spheres. A 2-sphere in a 3-
manifold is essential if it is not the boundary of an embedded 3-ball B3, and
we say that M is reducible if it contains an essential 2-sphere. The solution
of the Property-R conjecture in [Gab87] allows us to assume that p ‰ 0 and
q
that the surgery decomposes as a connected sum. All known examples of re-
ducible surgeries on knots in S3 are given by pq-surgery on the pp, qq-cable of
42
a knot K. Letting Cp,qpKq denote the pp, qq-cable of K, we have
S3pqpCp,qpKqq – Lpp, qq#S3q{ppKq. (2.1)
The Cabling Conjecture of Gonzáles-Acuña and Short asserts that these are
the only examples.
Conjecture 2.1.1 (Cabling Conjecture - [GS86]). If K is a knot in S3
which has a reducible surgery, then K is a cabled knot and the reducing slope
is given by the cabling annulus.
The Cabling Conjecture is known to be true for many families of knots:
satellite knots [Sch90], alternating knots [MT92], torus knots [Mos71], genus
1 knots [BZ96], and for knots with symmetries and low bridge number (for
a survey of known results and techniques see [Boy02].) In particular, it re-
mains to study reducible surgeries on hyperbolic knots with genus larger
than one. We will make this assumption throughout the rest of this chap-
ter.
Much is known about reducible surgeries on general knots. Observe that
in the case of cabled knots, the reducing slope is integral and one of the con-
nected summands is a non-trivial lens space. Gordon and Luecke [GL87]
show that this is the case for any reducing slope. In particular, any reducing
slope r satisfies 1 ă |r| since a reducible surgery contains a non-trivial lens
space summand. Due to the theorem of Gordon and Luecke [GL96] that the
geometric intersection number of any two reducing slopes is 1, we see that a
knot admits at most two reducing slopes, which would necessarily be consec-
utive integers. Further work in [MS03] shows that for non-cabled knots, the
reducing slope r satisfies the restrictive bound
1 ă |r| ď 2gpKq ´ 1. (2.2)
It is also known that in a reducible surgery, no more than one of the sum-
mands is an integer homology sphere, and at most two of the summands
are lens spaces [How02; Val99]. It is conjectured that three summands never
arise from Dehn surgery on a knot in S3.
43
Another bound on the reducing slopes, this time for fibered knots with
reducible surgeries with integer homology sphere summands, is proved in
[GS86, Proposition 1.4]. We state it here for convenience, rephrased from the
original source and using the Poincaré Conjecture.
Theorem 2.1.2 ([GS86] ). Suppose K is a fibered knot in S3 of genus g. If
S3r pKq – Lpr, aq#Y for Y a homology sphere, then r ď g.
More recently, progress has been made on the Cabling Conjecture us-
ing tools from Heegaard Floer homology. Hom, Lidman, and Zufelt show in
[HLZ15] that L-space knots admit at most one reducing slope r “ 2gpKq ´ 1.
In [Gre15], Greene shows that the Cabling Conjecture is true for knots that
have surgeries to connected sums of lens spaces. Meier shows in [Mei17] that
reducible surgeries on slice knots (or more generally knots with VipKq “ 0
for all i ě 0) only have two summands. In [DeY21], it is shown that hyper-
bolic thin knots do not admit reducible surgeries, except possibly when such
a knot is also an L-space knot.
In this chapter, we show that slice knots only admit reducible surgeries
of a particular type, and more generally we can bound the slice genus of a
knot in terms of the reducing slope parameters. The form of the reducible
surgery on a slice knot allows us to restrict the possible slopes on fibered,
hyperbolic slice knots, as well as restrictions on multiple reducing slopes
on slice knots. Our techniques mostly involve studying differences of the d-
invariants of a reducible surgery, which are largely affected by the order of
second cohomology of the connected summands. To that end, let Yp denote
a manifold with |H2pYp;Zq| “ p.
Theorem 2.1.3. Suppose K Ă S3 is a hyperbolic slice knot and p, q are rela-
tively prime integers. If pq is a reducing slope for K and S3pqpKq – Lpp, aq#Yq,
then q “ 1, a “ 1, and dpY q “ 0.
For fibered, hyperbolic slice knots, Theorem 2.1.3 together with The-
orem 2.1.2 implies the following, which when compared with Equation 2.2
44
shows that we cut down the possible reducing slopes on fibered, hyperbolic
slice knots by half.
Corollary 2.1.4. If K is a fibered, hyperbolic slice knot of genus g and r is
a reducing slope, then r ď g.
The proof of Theorem 2.1.3 also gives the following slice genus bounds.
Corollary 2.1.5. Suppose K admits a reducible surgery of the form S3pqpKq –
Yp#Yq with p ą q ą 1 and relatively prime. Then g pKq ě pp´1qpq´1qs ą 0.2
Remark 1: Note that by [GS86, Theorem 2.2] and [Eis22, Theorem 8], the
Alexander polynomial of the pp, qq torus knot divides the Alexander poly-
nomial of any knot K which admits a reducible surgery of the form Yp#Yq.
This implies that g3pKq ě g3pTp.qq “ pp ´ 1qpq ´ 1q{2. Corollary 2.1.5 above
shows that the slice genus satisfies the same bound.
Remark 2: It is not known if more than two summands can occur in a re-
ducible surgery on a slice knot. By [Val99] we know that there are at most
three summands, and if three summands occur in a reducible Dehn surgery,
then two of them are lens spaces and one is an irreducible integer homology
three-sphere. Work in [Mei17] shows that slice knots admit only two sum-
mands in any reducible surgery. Corollary 2.1.5 implies that if a slice genus
1 knot K has a reducible surgery with two summands carrying non-trivial
homology, then the reducing slope is 6. This implies that the two summands
conjecture is true for all reducing slopes on slice genus 1 knots except for the
possibility that S36pKq – Lp2, 1q#Lp3, 2q#Y for Y an irreducible homology
sphere with dpY q “ 0. Similarly the only possible reducing slopes for slice
genus 2 knots that could produce more than two summands are r “ 6 and
r “ 10. As far as we know, Heegaard Floer theoretic invariants cannot ob-
struct three summands from appearing in these surgeries.
Next, we investigate how Theorem 2.1.3 may be applied to the problem
of multiple reducing slopes for slice knots. This theorem and its proof are
inspired by [HLZ15, Theorem 1.6]. For the definition of ν, see Definition
2.1.12.
45
Theorem 2.1.6. Suppose K is a hyperbolic knot in S3 with νpKq ă gpKq
that admits consecutive reducing slopes r and r ` 1. Further, suppose that
both r and r` 1 surgery split off an integer homology sphere summand. Then
r ` 1 ď gpkq.
Corollary 2.1.7. Suppose that r and r ` 1 are simultaneous reducing slopes
for a hyperbolic slice knot K in S3. Then r ` 1 ď gpKq.
Spinc Structures
Let SpincpY q denote the set of Spinc structures on Y , and recall that
SpincpY q is an affine copy of H2pY ;Zq. Given a choice of Spinc structure s0
on Y , every other Spinc structure satisfies s “ s0 ` a for some a P H2pY ;Zq.
Furthermore we have an identification SpincpY1#Y2q “ SpincpY1q ˆ SpincpY2q
and the projection maps onto each factor, πY1 and πY2 , intertwine the conju-
gation actions. Therefore, for s P SpincpY #Y q a self-conjugate Spinc1 2 struc-
ture, both πY1psq and πY2psq are self-conjugate Spinc structures on Y1 and Y2
respectively. Next, observe that if p “ |H1pY1;Zq|, then πY1ps ` pq “ πY1psq
for any s P SpincpY1#Y2q. This gives a relation among the d-invariants of
reducible three-manifolds that arise as Dehn surgery along a knot in S3.
For surgeries on knots in S3, we fix an identification of SpincpS3p{qpKqq
with Z{pZ, given by σ : Z{pZ Ñ SpincpS3p{qpKqq which sends ris Ñ σprisq
and satisfies σpri ` 1sq ´ σprisq “ rK 1s P H1pS3p{qpKqq – SpincpS3p{qpKqq,
where rK 1s is the homology class of the dual knot. For more details on this
assignment, see [OS03a, Section 4.1]. We will often abuse notation and write
i or ris for the image of ris under the map σ, and will take ris to be notation
for i pmod pq unless otherwise stated.
Heegaard Floer Homology
Heegaard Floer homology is an invariant of closed, oriented three man-
ifolds that was introduced by Oszváth and Szabó in [OS04d]. We will as-
sume familiarity with all flavors of Heegaard Floer homology, as well as the
46
Z ‘ Z-filtered knot Floer complex CFK8pKq for knots K in S3 defined in
[OS04d] and [Ras03b]. For the readers convenience, we give a brief review of
the structure of HF`pY, sq, the properties of the d invariants, and the map-
ping cone formula, since they will be used in our main arguments in the next
section.
Given a rational homology three-sphere, consider the invariants HxFpY q,
HF`pY q, and HF8pY q. These are a finite dimensional F-vector space, an
FrU s-module, and an FrU,U´1s-module respectively. Further, we have HF˝pY q –
À ˝
sPSpincpY qHF pY, sq for ˝ P tp,`,8u.
For any rational homology three-sphere Y with Spinc structure s, we
have HF8pY, sq – FrU,U´1s. Also, HF`pY, sq decomposes non-canonically
into two pieces. The first is the image of HF8pY, sq in HF`pY, sq. This
summand is isomorphic to FrU,U´1s{UFrU s, which is called the tower and
is denoted T `. The grading of 1 P T ` is an invariant of the pair pY, sq and
is denoted dpY, sq. The rational number dpY, sq is called the correction term
or d invariant. The second summand in HF`pY, sq is the quotient by the
image of HF8pY, sq and is denoted HFredpY, sq. It is a finite dimensional F
vector space annihilated by a high enough power of U .
The d-invariant has many useful properties [OS03a, Section 4]:
• Suppose s is the image of s under conjugation. Then dpY, sq “ dpY, sq.
• For pairs pY1, s1q and pY2, s2q,
dpY1#Y2, s1#s2q “ dpY1, s1q ` dpY2, s2q (2.3)
• d is a homology cobordism invariant: If W : Y Ñ Y 1 is a Z homology
cobordism and there is a Spinc structure on W that restricts to s on Y
and s1 on Y 1, then dpY, sq “ dpY 1, s1q.
By work in [OS11], the d-invariants of pp{qq-surgery along a knot are
related to the d-invariants of Lpp, qq. The latter are determined in [OS03a,
Proposition 4.8], where they show
p p q p2ris ` 1´ p´ qq
2 ´ pq
d L p, q , risq “ ´ dpLpq, rq, rjsq, (2.4)
4pq
47
with r and j the reductions of p and i modulo q, respectively. We also use
the notation ris to denote the residue class of i mod p. This formula to-
gether dpS3q “ 0 allows one to determine the d-invariants of a lens space
recursively.
The work of [LL08, Proposition 5.3] also shows that the d-invariants of
Lpp, aq satisfy the relation
p p q r sq ´ p p q r ` sq “ p´ 1´ 2risd L p, a , i d L p, a , a i . (2.5)
p
The Mapping Cone Formula and the ν` Invariant
In this section, we establish some terminology and notation for the map-
ping cone formula and the ν` invariant. For more details, see [HW16] and
[Gai17]. Material from this section will be used to establish the claimed slice
genus bounds and the bound on multiple reducing slopes.
As above, we write HF ˝ to mean either the plus or hat version of Floer
homology. Let C “ CFK8pS3, Kq denote the knot Floer complex associated
to K. This is a Z ‘ Z-filtered Z-graded chain complex over FrU,U´1s, where
the U action lowers the filtration degree by one and the Z grading by 2. As-
sociated to C are the following quotient and sub-quotient complexes useful
for computing the plus and hat versions of Floer homology of manifolds aris-
ing as Dehn surgery along a knot K. To this end, define:
A`k :“ Ctmaxti, j ´ ku ě 0u and Apk :“ Ctmaxti, j ´ ku “ 0u
as well as
B` :“ Cti ě 0u and Bp :“ Cti “ 0u
where i and j refer to the two filtration degrees. From the definition of CFK8pS3, Kq
the complex B˝ is isomorphic to CF ˝pS3q.
There is an obvious map v` ` `k : Ak Ñ B defined by projection. Sim-
ilarly, there is a map h` : A` Ñ B`k k which projects to Ctj ě ku, shifts
to Ctj ě 0u via multiplication by Uk, and then applies a chain homotopy
equivalence between Cti ě 0u and Ctj ě 0u (both of which compute
48
CF`pS3q, so by general theory are chain homotopic). There are similar
maps for the hat versions.
Just as HF`pY, sq decomposes as a tower and a reduced part, the ho-
mology of the quotient complexes A`k pKq decompose non-canonically as
T ` ‘ Aredk . The maps v`k and h
`
´k are isomorphisms for large values of k
and so represent multiplication by some non-negative power of U , say UVk
and UHk respectively when restricted to the tower summand in each A`k .
The non-negative integers Vk and Hk are concordance invariants of K which
satisfy, by [NW15, Lemma 2.4] and [HLZ15, Lemma 2.5], the relations Hk “
V´k, Hk “ Vk ` k, and
Vk ´ 1 ď Vk`1 ď Vk. (2.6)
Furthermore, for each i, we have [Ras03b, Corollary 7.4]
R V
ď g4pKq ´ iVi . (2.7)
2
The Vi also determine the correction terms or d invariants of surgery along
the knot K:
Theorem 2.1.8. [NW15, Proposition 1.6] For p, q ě 0 and 0 ď i ď p ´ 1.
we have:
dpS3p{qpKq, iq “ dpLpp, qq, iq ´ 2maxtVti{qu, Vt p`q´1´i uu (2.8)
q
Now we explain how the maps vk and hk together with the quotient
complexes A`k determine the Heegaard Floer homology of p{q surgery along
K. Since we are only interested in integer surgery in this paper, we write
the theorem down in this case. The reader interested in the change to the
case of fractional surgeries and a more detailed explanation of the notation
should consult [OS11; Gai17]. To this end, let
A˝
à ˝ à
i,ppKq :“ pn,Ai`pnq, B˝ :“ pn,B˝q.
nPZ nPZ
Then define a chain map D˝ : A˝ ˝i,p i,p Ñ B by D˝i,pptpk, akqukPZq “ tpk, bkqukPZ
where bk “ v˝ ˝i`pkpakq ` hi`ppk´1qpak´1q. Letting X ˝i,p denote the mapping cone
of D˝i,p, we have
49
Theorem 2.1.9. [OS11, Theorem 1.1] There is a relatively graded isomor-
phism of FrU s-modules
H˚pX ˝ ˝i,pq – HF pS3ppKq, iq.
Next, we introduce the ν` invariant. as defined in [HW16, Definition
2.1].
Definition 2.1.10. The invariant ν` is defined as follows: ν` :“ mintk P
Z | v : A`k k Ñ CyF pS3q, v
`
k p1q “ 1u, where 1 P H˚pA
`
k q is a generator with
lowest grading of the tower summand.
Recall that ν`pKq ď gspKq [HW16, Proposition 2.4]. With the mapping
cone formula we can give an alternative definition of ν`. This definition is
equivalent to the one just given since the integers Vk determine the map v`k
on the non-torsion summand of A`k [NW15].
Lemma 2.1.11. ν`pKq “ mintk P Z | Vk “ 0u.
We will also make use of the hat version ν as defined in [OS11, Definition
9.1]
Definition 2.1.12. For a knot K Ă S3, define νpKq :“ mints | pv̂sq˚ ‰ 0u.
Then genus detection of knot Floer homology implies that
gpKq “ maxtνpKq, ts | dimH˚pAps´1q ą 1uu. (2.9)
2.2 Reducible Surgeries on Slice Knots
In this section we prove Theorems 2.1.3 and 2.1.6.
The d-invariants of Reducible Manifolds
Theorem 2.1.3 follows from the more general Theorem 2.2.1 below, which
deals with d-invariants of knots which admit a reducible surgery.
50
Theorem 2.2.1. Suppose K is a knot in S3 such that pq is a reducing slope
with pp, qq “ 1, p ą q and S3pqpKq – Yp#Yq. Then for each 0 ď ℓ ď
pp´1qpq´1q ,
2
the Vi’s satisfy
q´1
ÿ
` ˘
´ “ pp´ 1qpq ´ 1qVℓ`i Vαpℓ`i`pq ´ ℓ, (2.10)
“ 2i 0
Where αpjq “ mintj, pq ´ ju.
Proof. Suppose K is a knot in S3 with S3pqpKq – Yp#Yq. We will write πp
for the projection map πYp : SpincpS3pqpKqq Ñ SpincpYpq and similarly for πq.
As |H2pYp;Zq| “ p we have πpprp`isq “ πpprisq for ris, rp`is P SpincpS3pqpKqq.
Then by additivity of the d-invariants, for any ℓ P Z we have:
dpS3pqpKq, rp` i` ℓsq´dpS3pqpKq, ri` ℓsq “ dpYq, πqrp` i` ℓsq´dpYq, πqri` ℓsq.
(2.11)
Our assumptions on p and q imply that αpℓ` iq “ ℓ` i, so by Theorem 2.1.8
and Equation 2.4 we see that the left hand side difference in equation 2.11
equals
2pℓ` iq ` pp1´ qq ` 2Vℓ`i ´ 2Vαpℓ`i`pq.
q
Summing from i “ 0 to i “ q ´ 1 we see:
q´1 q´1ˆ ˙
ÿ ÿ
dpS3pqpKq, rℓ` i` psq ´ dpS3pqpKq, r `
2pℓ` iq ` pp1´ qq
ℓ isq “
“ qi 0 i“0
q´1
ÿ
` ˘
` 2 Vℓ`i ´ Vαpℓ`i`pq .
i“0
(2.12)
On the other hand, by Equation (2.11) the left-hand side of Equation
(2.12) is equal to
q´1
ÿ
dpYq, πqrℓ` i` psq ´ dpYq, πqrℓ` isq.
i“0
Additionally, this sum is zero because the projection πq induces bijections
between SpincpYqq and both sets tℓ, . . . , ℓ` q ´ 1u, tp` ℓ, . . . , p` ℓ` q ´ 1u.
51
Rearranging the sum in Equation 2.12, we see that
q´1
ÿ
` ˘
´ “ pp´ 1qpq ´ 1qVℓ`i Vαpℓ`i`pq ´ ℓ.
“ 2i 0
Proof of Theorem 2.1.3. We first show that Theorem 2.2.1 implies Y is an
integer homology sphere. Since V0 “ 0 by Equation 2.7 when K is slice, we
have that Vi “ 0 for all i ě 0 due to their non-increasing behavior. Equation
2.10 with ℓ “ 0 implies pp ´ 1qpq ´ 1q “ 0, so either p “ 1 or q “ 1. If p “ 1,
the positive solution of the two summands conjecture in the case where Vi “
0 for all i ě 0 in [Mei17] implies that q was not a reducing slope since Y is
irreducible. Therefore, under the assumption that we have a reducing slope,
it follows that q “ 1 and the reducible surgery is S3ppKq – Lpp, aq#Y with Y
an irreducible homology sphere.
To finish off the proof of Theorem 2.1.3, it remains to show that a “ 1
and dpY q “ 0. Using Equation 2.5, we have
p p p´ 1´ 2risd L p, aq, risq ´ dpLpp, aq, ra` isq “ .
p
Notice that if i is a self-conjugate spinc structure of Lpp, aq, then ri ´ as and
ri ` as are conjugate for any a. Then dpLpp, aq, ri ´ asq “ dpLpp, aq, ri ` asq
using [OS04c, Theorem 2.4]. Equation 2.5 used for the pairs ris, ri ` as and
ris, ri´ as then yields
p´ 1´ 2ris “ ´p´ 1´ 2ri´ as .
p p
This implies that 2i ” a ´ 1 pmod pq, and so the self-conjugate Spinc struc-
ture(s) of Lpp, aq correspond to the integers amongst ra´1s and rp`a´1s.
2 2
Both are realized as self-conjugate Spinc structures when p is even, and pre-
cisely one of them is realized when p is odd (depending on the parity of a).
Recall that S3ppKq “ Lpp, aq#Y with Y a Z-homology sphere, and let
rss P SpincpS3ppKqq satisfy πLprssq “ πLpr0sq ` a. Equation 2.5 yields
dpS3ppKq, r0sq ´ dpS3ppKq, rssq “ dpLpp, aq, πLpr0sqq ` dpY q ´ pdpLpp, aq, πLprssqq ` dpY qq
52
“ dpLpp, aq, πLpr0sqq ´ dpLpp, aq, πLpr0sq ` aq
“ p´ 1´ 2πLpr0sq .
p
Additionally recall that dpS3ppKq, risq “ dpLpp, 1q, risq´2Vαpiq due to Equation
2.8. This implies
dpS3ppKq, r0sq ´ dpS3ppKq, rssq “ dpLpp, 1q, r0sq ´ dpLpp, 1q, rssq ´ 2pV0 ´ Vαps`pqq
“ dpLpp, 1q, r0sq ´ dpLpp, 1q, rssq
“ spp´ sq ,
p
since Vi “ 0 for all i ě 0.
Now either π a´1Lpr0sq “ or πLpr0sq “ p`a´1 . Provided the former, the2 2
two equations above yield spp ´ sq “ p ´ a. However spp ´ sq ě p ´ 1, and
so we must have a “ 1 when πLpr0sq “ a´1 . If we suppose the latter so that2
π r0s “ p`a´1L , then these two equations yield the contradiction spp ´ sq “2
´a. Thus a “ 1, which forces dpY q “ 0 using Equation 2.8 with s “ 0 and
V0 “ 0.
Proof of Corollary 2.1.5. Suppose K is a knot in S3 which admits a reduc-
ing slope of the form r “ pq with S3pqpKq – Yp#Yq. Choosing k “
pp´1qpq´1q ´
2
1 for ℓ in Equation 2.10 shows
q´1 q´1
ÿ ÿ
Vk`i “ 1` Vαpk`i`pq,
i“0 i“0
after rearranging terms. Thus, Vk ą 0 since the Vi’s are non-negative and
non-increasing, and so ν`pKq ě k ` 1 “ pp´1qpq´1q by Lemma 2.1.11. Since
2
ν`pKq is a lower bound for the slice genus of a knot [HW16], the result fol-
lows.
53
Multiple Reducing Slopes on Slice Knots
In this subsection we use the mapping cone formula for HxF and the fol-
lowing lemma to prove Theorem 2.1.6. The lemma below follows immedi-
ately from the Ku:nneth theorem for HxF and the fact that lens spaces are L-
spaces (for the proof and for the analogous statement for HF`, see [HLZ15,
Lemma 2.6] and Lemma 2.3.13.
Lemma 2.2.2. Suppose Y is a three manifold and Y – Lpp, aq#Yq with
|H2pYq,Zq| “ q. Then for any α P H2pY q and s P SpincpY q we have
dimHxFpY, s` qαq “ dimHxFpY, sq.
Proof of Theorem 2.1.6. Suppose S3r pKq – Lr#Y and S3r`1pKq – Lr`1#Z
where r ě gpKq, Y and Z are both integer homology spheres, and Lr is
either a lens space or a connected sum of two lens spaces, and similarly for
Lr`1. We assume the two reducing slopes are consecutive positive integers
r and r ` 1 by mirroring the knot if necessary. Since both surgeries split
off integer homology three spheres, and the complementary summand is an
L-space, we see by Lemma 2.2.2 that
dimpHxFpS3r pKq, iqq “ dimpH 3xFpSr pKq, jqq
for any two Spinc structures i and j on S3r pKq. Similarly,
dimpHFpS3 3x xr`1pKq, iqq “ dimpHFpSr`1pKq, jqq.
Note that if r is odd, we choose representatives ris of Spinc satisfying
´tr{2u ď i ď tr{2u. If r is even, choose representatives with ´tr{2u ă i ď
tr{2u. By the assumption r ě gpKq we have r “ g ` i for 0 ď i ď g ´ 2,
since a reducing slope on a hyperbolic knot is bounded above by 2gpKq ´ 1
by Equation (2.2) and r ` 1 is a reducing slope, we have r ` 1 ď 2gpKq ´ 1.
In this case, the mapping cone formula implies that HxFpS3r pKq, kq – H˚pApkq
for k “ ´i,´i ` 1, . . . , 0, . . . , i ´ 1, i. For all other k between 0 and tr{2u we
have
54
HxFpS3r pKq, kq – H˚pConepApk´i´g ‘ Apk Ñ Fqq.
Note that HxFpS3r pKq,´kq – H˚pConepAp´k ‘ Apg`i´kq Ñ Fq, so it suffices
to consider only those k with 0 ď k ď tr{2u, since by [HLZ15, Lemma 2.3]
H˚pAp q – H˚pAps ´sq and under this isomorphism the maps vs and h´s agree
in the mapping cone.
Now, consider the mapping cone for r`1 surgery. Since r`1 “ g` i`1,
HxFpS3 pr`1pKq, kq – H˚pAkq for k “ ´i´ 1,´i,´i` 1, ¨ ¨ ¨ , i´ 1, i, i` 1 and for
all other k between 0 and tpr ` 1q{2u we have
H 3xFpS p pr`1pKq, kq – H˚pConepAk´i´1´g ‘ Ak Ñ Fqq.
Let nk “ dimpH˚pApkqq. Then by Lemma 2.2.2, since r is a reducing
slope with a Z homology sphere summand, we have
n0 “ n1 “ n2 “ ¨ ¨ ¨ “ ni “ ni`1 ` n1´g ` 1´ 2rankph1´g ‘ vi`1q.
However, r ` 1 surgery reducible implies that n0 “ ni`1, and so
ni`1 “ ni`1 ` n1´g ` 1´ 2rankph1´g ‘ vi`1q. (2.13)
Since rankph1´g ‘ vi`1q “ 0 or 1, we either have n1´g “ ´1 or n1´g “ 1 by
Equation (2.13). The former case is impossible, so rankph1´g ‘ vi`1q “ 1 and
H˚pApg´1q is one dimensional. This contradicts equation (2.9).
Proof of Corollary 2.1.7. This follows immediately from Theorem 2.1.3 and
Theorem 2.1.6.
2.3 Almost L-space knots and the Mapping Cone Formula
In this section we dive a bit deeper into the mapping cone formula and
use it to both count the ranks of knot Floer homology of Dehn surgeries on
almost L-space knots in each Spinc-structure and determine relative grad-
ings of generators of HFredpS3ppKq, rssq. This will allow us to rule out many
slopes p with 1 ă |p| ď 2gpKq ´ 1 from being reducing slopes.
55
Facts about Almost L-space knots
Most of this section comes from [BS22] and [Bin23]. We review the
necessary facts about almost L-space knots that will be used in the proofs.
Since we are interested in producing obstructions to a knot admitting a re-
ducible surgery, we restrict ourselves to considering almost L-space knots
of genus g ě 2 (since genus 1 knots satisfy the cabling conjecture [HLZ15;
BZ96].) Recall that an L-space is a rational homology 3-sphere Y that satis-
fies dimpHxFpY qq “ |H1pY ;Zq|.
Definition 2.3.1. [BS22, Definition 1.9] A closed 3-manifold Y is called
an almost L-space if Y is a rational homology sphere and dimpHxFpY qq “
|H1pY ;Zq| ` 2. A non-trivial knot K Ă S3 is an almost L-space knot if
dimpHxFpS3npKqqq “ n` 2 for all n ě 2gpKq ´ 1.
Theorem 2.3.2. [BS22, Proposition 3.9] Suppose K is an almost L-space
knot of genus g ě 2, then K is fibered and Vg´1 ‰ 0
A corollary of this theorem is that Vi ‰ 0 for all i ď g ´ 1. In particular,
v̂i vanishes on the copy of F coming from the tower summand in A`i . We
will also need the following lemmas:
Lemma 2.3.3. [BS22, Lemma 3.8] For K an almost L-space knot, we have
A 3p pi – F for i ‰ 0 and A0 – F . Moreover A` `i – T for i ‰ 0 and A`0 –
T ` ‘ F rU´1s{U´n2 for some n ě 1.
Proof. The proof of the above lemma follows from [BS22, Lemma 3.8], and
the large surgery formula.
Recall that there is a chain map D˝ ˝ ˝i,p : Ai,ppKq Ñ Bi,p defined by
D˝i,pptk, akuq “ tpk, bkqu
where bk “ v˝i`pkpakq ` h ˝i`ppk´1q˝pak´1q. If we let Xi,p denote the mapping
cone of D˝i,p, then we have the following result
56
Theorem 2.3.4. There is a relatively graded isomorphism of FrU s modules
H pX ˝ q – HF˝pS3˚ i,p ppKq, iq
We can in fact be more explicit. In the following lemma DTi,p denotes
the restriction of the map D`i,p to the tower summand of A`i,p
Lemma 2.3.5. For p ą 0, the map D` ` `i,p : Ai,p Ñ Bi,p is surjective and
À
HF`pS3ppKq, risq – kerpD`i,pq “ kerpDTi,pq ‘ Aredi – i”p kerpv
` `
i ` hi q
Proof. It follows from [NW15] and [Gai17, Corollary 14] that for positive
surgeries the map DTi,p is surjective for all i. Since we are dealing with an
almost L-space knot, the result follows for i ‰ 0 immediately. In the case
i “ 0, it follows from [Gai17, Proposition 15] that kerpD`0,pq “ kerpDT0,pq ‘
Ared0 .
Lemma 2.3.6. In the case p ă 0, the map vp0 vanishes on the summand of
A coming from Aredp0 0
Corollary 2.3.7. For p ă 0 the map cokerpD` `i,pq – T and HF 3redpSppKq, risq –
kerpD`i,pq – kerpDTi,pq ‘ Aredi
Proof. By [Gai17, Proposition 19] and Lemma 2.3.3, the result follows if we
can show that V0pmpKqq “ 0. This can be done by analyzing the 3 cases
in [Bin23]. In each case, the filtered homotopy type of CFK8 is given ex-
plicitly. The proof that V0pmpKqq “ 0 for K an almost L-space knot with
gpKq ě 2 is exactly the same as the proof that V0pmpKqq “ 0 for K an
L-space knot.
To obstruct reducible surgeries on almost L-space knots, first we will de-
termine the rank of HxFpS3ppKq, rssq for each rss. Following [HLZ15], we pass
to a smaller, but quasi-isomorphic model, of the mapping cone. Recall that
for s ě gpKq, the map vs induces an isomorphism on homology, similarly,
for s ď ´gpKq, the map hs also induces an isomorphism on the level of ho-
mology. Now, if we let A˝i,p :“ H˚pA˝i,pq, and similarly B˝ :“ H pB˝i,p ˚ i,pq and
57
vi,p “ pvi,pq˚ and similarly for hi,p, we see that the truncated mapping cone
complex, defined as
˜ ¸ ˜ ¸
à
X ˝
à ˝
i,p :“ Ai,p ‘ Bi,p
1´gďiďmaxtg´1,g´1`pu 1´g`pďsďg´1
With the induced differential, is quasi-isomorphic to the mapping cone
X ˝. Since this model has only finitely many objects, it is simpler to work
with.
Using the truncated mapping cone model, it is a simple task to count
the rank of Heegaard Floer homology of Dehn surgery on an almost L-space
knot in S3.
Lemma 2.3.8. Let K be a knot with a positive almost L-space surgery, and
let p be an integral slope on K satisfying 1 ă |p| ď 2gpKq ´ 1. Let k ” 2g ´ 1
mod p with 0 ď k ă |p|. If p ą 0, then for s P Z with g ´ k ď s ă g ´ k ` p
we have
$
2g´1
’
’2t u` 3 g ´ k ď s ă g and s ” 0 mod p
’ p
’
’
’
2t2g´1& u` 1 g ´ k ď s ă g
rkpHxFpS3pKq, sqq “ pp
2t2g´1’’ u` 1 g ď s ă g ´ k ` p and s ” 0 mod p
’ p
’
’
’
2t2g´1% u´ 1 g ď s ă g ´ k ` p
p
If p ă 0, then
$
Y ]
’2 2g´1’ ` 3 g ´ k ď s ă g s ı 0 mod p
’ |p|
’
’
’
’ Y ]
’
’
2 2g´1’ | | ` 5 g ´ k ď s ă g s ” 0 mod p’& p
rkpHxFpS3 Y ]ppKq, rssqq “
2 2g´1’’ | ` 3 g ď s ă g ´ k ` |p| s ” 0 mod p’ p|
’
’
’
Y ]
’
’
’2 2g´1’ | | ` 1 g ď s ă g ´ k ` |p| s ı 0 mod p’ p
%
58
Proof. This proof is exactly the same as the proof of [HLZ15, Lemma 3.2].
Indeed when p ą 0 Lemma 2.3.3 shows that the maps vpi vanish on the tower
summands of A`i and Lemma 2.3.14 shows that the rank of p surgery on K
in Spinc structure rss is the either the same as the rank of p surgery on an
L-space knot if rss ‰ r0s or is 2 more than the rank of p surgery on an L-
space knot if rss “ r0s. The case for negative surgeries is also the same by
Lemma 2.3.15 since V0pmpKqq “ 0 for K an almost L-space knot, we just
need to add 2 to the rank of HxFpS3ppKq, r0sq.
Now suppose that p is a reducing slope for the hyperbolic almost L-
space knot K. Then we have S3ppKq – Lpa, bq#R where p “ ar and |H1pR;Zq| “
r. Note that in particular this implies that pa, rq “ 1. Since Lpa, bq is an L-
space Lemma 2.2.2 implies that for any s P SpincpS3ppKqq we have
rankpHxFpS3ppKq, rssq “ rankpHxFpS3ppKq, rs` rsq
This shows that in order for p surgery to be reducible there must be
some periodicity, so the rank of HxF is any particular Spinc structure cannot
be the unique Spinc structure with that rank (they at least come in pairs).
We first deal with the case k “ 0, and show that in this case p is not a re-
ducing slope.
Lemma 2.3.9. With the notation as in Lemma 2.3.8, for k “ 0 there is no
possible periodicity among the ranks of HxFpS3ppKq, rssq
Proof. When k “ 0 we see from Lemma 2.3.8 that there is a unique rss,
anmely rss “ r0s, so that g ď s ă g ` p and rankpHFpS3x ppKq, rssqq “
Z ^
2g ´ 1
| | ` 1. The contradicts Lemma 2.2.2p
In a similar way, we handle the cases where 1 ă k ă p´ 1.
Lemma 2.3.10. For 1 ă k ă p´ 1 there is no possible periodicity among the
ranks of HxFpS3ppKq, rssq.
Proof. By Lemma 2.3.8 for 1 ă k ă p ´ 1 we see that there are at least 2
Spinc structures rss satisfying g ´ k ď s ă g and at least 2 Spinc structures
59
rss satisfying g ď s ă g ´ k ` p. In any case either there is an rss with
g´k ď s ă g so that s ” 0 mod p or there is an rss so that g ď s ă g´k`p
and s ” 0 mod p. In the former case Lemma 2.3.8 implies that there is a
Z ^
2g ´ 1
unique Spinc structure with rank 2 | | ` 3, contradicting Lemma 2.2.2.p
In the latter case, there are k consecutive Spinc structures rss, namely those
Z ^
2g ´ 1
s with g ´ k ď s ă g, with rank 2 | | ` 1 and only one Spin
c structure
p
rss with g ď s ă g´k`p with the same rank. Again this contradicts Lemma
2.2.2
Recall that k ” 2gpKq ´ 1 mod |p|. Lemma 2.3.10 implies that if K
admits a reducible surgery of slope p, then k “ 1 or k “ p´ 1 ” ´1 mod |p|.
Therefore we have the following Corollary
Corollary 2.3.11. Suppose K is an almost L-space knot with a reducing
slope p. Then either p|gpKq or p|2gpKq ´ 2 and p is even.
Proof. By Lemmas 2.3.9 and 2.3.10 we see that the only possibilities are k “
1 ” 2gpKq ´ 1 mod p and k “ p ´ 1 ” 2gpKq ´ 1 mod p. In the former
case, we have that p|2gpKq ´ 2 and in the latter case we have that p|2gpKq.
If k “ p´ 1 ” 2gpKq ´ 1 mod p, then by Lemma 2.3.8 we see that the Spinc
structure labelled g must satisfy g ” 0 mod p, so actually p|gpKq and in
that case all the Spinc structures have the same rank. If p|2gpKq´2, then we
see that the Spinc structure labelled g ´ 1 cannot be equivalent to the Spinc
structure labelled 0 mod p. So in particular p does not divide gpKq ´ 1.
p
Hence p is even. Moreover, we have gpKq ´ 1 “ m where m is odd (if m
2
were even then p would divide gpKq ´ 1).
Lemma 2.3.12. Suppose that p is a reducing slope so that p|2gpKq ´ 2.
Then gpKq pis even and S3ppKq – Lp2, 1q#R with |H1pR;Zq| “ 2
Proof. In the proof of Corollary 2.3.11 we found that if p is a reducing slope
then either p|gpKq or p|2gpKq ´ 2. If we are in the latter case then rg ´ 1s
and rg ´ 1 ` rs have to be paired up where r divides p and g ´ 1 ` r is the
unique Spinc structure so that rg ´ 1 ` rs “ 0. By Corollary 2.3.11 we see
60
that g ´ ` p p1 r “ m ` r ” 0 mod p, so r “ . Hence S3ppKq – Lp2, 1q#R2 2
p p p
with |H1pR;Zq| “ , and we necessarily have that p2, q “ 1, so that is
2 2 2
odd. Again by Corollary 2.3.11 we find that gpKq “ pm ` 1 for some m ě 1
2
p
and odd. Since is also odd, we see that gpKq is even.
2
In summary, we showed that for p to be a reducing slope on an Almost
L-space knot, we have either p|gpKq or p|2gpKq ´ 2 and p and gpKq are both
even.
Relative Gradings and Proper Divisors
In this section we use the gradings of elements of HFredpS3ppKq, rssq to
further restrict the possible reducing slopes on almost L-space knots. The
following is an upgraded version of Lemma 2.2.2 (see [HLZ15])
Lemma 2.3.13. For S3ppKq – Lpa, sq#R, there is an isomorphism of rela-
tively graded FrU s-modules HF`pS3ppKq, rssq – HF`pS3ppKq, rs` rsq.
We will use this lemma in conjunction with a computation of the rela-
tive gradings from the mapping cone to rule out divisors of gpKq and 2gpKq´
2 from being reducing slopes.
To extract the gradings of elements in HFredpS3ppKq, rssq we want to
dive deeper into the mapping cone and identify certain cycles in the map-
ping cone that generate HFredpS3ppKq, rssq.
The following is from Gainullin [Gai17], building on work of [NW15]
[HLZ15] and [OS04b]
Lemma 2.3.14. [Gai17, Proposition 15] Suppose p ą 0, then there is an
isomorphism HF`pS3pKq, risq – kerpD`p i,pq and
à à à
kerpD` `i,pq – T τpHi´npq τpVi`npq Ai,red
ně1 ně1
for i ď p´ i
and otherwise
61
` `à à àkerpDi,pq – T τpHi´npq τpVi`npq Ai,red
ně2 ně0
Lemma 2.3.15. [Gai17, Lemma 18] Suppose p ă 0 and K is a knot so that
VipmpKqq “ 0. Then cokerpD` `i,pq – T , and
HFredpS3ppKq, risq – kerpD` Ti,pq – kerpDi,pq ‘ Ai,red
In particular, we can identify the largest graded pieces of each tower in
HFredpS3ppKq, rssq as follows. By Lemma 2.3.3, for each s ‰ 0 A` – T `s .
Define x “ U´maxtVs,Hsu, y “ U´mintVs,Hsus s and z “ U´mintVs,Hsu`1s . So
that zs is the top graded element in kerpv`s ` h`q – F rU´1s{U´mintVs,Hsus .
Knowing HF 3redpSppKq, rssq as a relatively graded FrU s-module is equivalent
to knowing the gradings of the zt for t ” s mod p. In the case s “ 0 by
Lemma 2.3.3 we have A`0 – T ` ‘ FrU´1s{U´N , so we have x0, y0 and z0 as
above in the T ` summand, and we have and element a in FrU´1s{U´N so
that a ‰ Ub for some b. Then by Lemmas 2.3.15 and 2.3.14 understanding
HF pS3red ppKq, r0sq as a relative graded FrU s-module is equivalent to knowing
the grading of both zt for t ” 0 mod p and a. Expressed a different way,
define the auxilliary object
HFpS3| ppKq, rssq :“ cokerpU : HF`pS3ppKq, rssq Ñ HF`pS3ppKq, rssqq
That is, HF pS3} ppKq, rssq picks off the top of each truncated tower in
HF 3redpSppKq, rssq, so when rss ‰ r0s this is equivalent to picking of the z1ts
with t ” s mod p and when rss “ r0s this is picking out both the z1ts for
t ” 0 mod p and the element a P FrU´1s{U´N so that a ‰ Ub for any b by
Lemmas 2.3.14 and 2.3.15. In particular, we see that for rss ‰ r0s and p ą 0,
we have
rankpH|FnpS3ppKq, rssqq “ #t
p
t : t ” s mod p, ă |t| ď gpKq ´ 1, grpztq “ nu
2
62
and for p ă 0 and rss ‰ r0s, we have
rankpHF pS3| n ppKq, rssq “ #tt : t ” s mod p, grpztq “ nu
Moreover, if we let rssgrtop denote the largest grading of an element of
H|FpS3ppKq, rssq and
rss
grbot denote the smallest, then a consequence of Lemma
2.3.13 is that if p is a reducing slope with S3ppKq – Lpa, bq#R, then for for
every n P Z
rankpH|F 3rss ` pSppKq, rssq “ rankpHF
3
| rs`rs pS
gr n gr `n ppKq, rs` rsqtop top
and
rankpH 3 3|F r s |s
grbot`
pSppKq, rssq “ rankpHF rs`rs pS pKq, rs` rsqn gr pbot `n
In summary, we want to understand the relative gradings of the elements
zt and zt`r when these elements exist and when rts and rt ` rs are distinct
non-conjugate Spinc structures. With this in mind the following is [HLZ15,
Lemma 3.8 and Lemma 3.9].
Lemma 2.3.16. For xt and yt as above, we have grpxtq ´ grpytq “ 2|t|.
Lemma 2.3.17. For zt as above we have:
If p ą 0
$
’
’2t t ą 0
’
&
grpzt`pq ´ grpztq “ 2pt` pq t` p ă 0
’
’
’
%0 t ď 0 ď t` p
If p ă 0
$
’
’2t t´ |p| ě 0
’
&
grpzt´|p|q ´ grpztq “ 2pt´ |p|q t ď 0
’
’
’
%2p2t´ |p|q t´ |p| ď 0 ď t
We will us the above Lemma in conjunction with Lemma 2.3.13 to rule
out proper divisors of ˘gpKq and ˘p2gpKq ´ 2q from being reducing slopes.
63
Lemma 2.3.18. Suppose K is an almost L-space knot and p is a slope with
p|gpKq and p ‰ gpKq. If p is a reducing slope then p “ 2r or p “ ´2r and
gpKq is even
Proof. Let K be an almost L-space knot. Suppose gpKq “ pm for some m
with |m| ě 2. Further, suppose S3ppKq – Lpa, bq#R with p “ ˘ar and
pa, rq “ 1. We first deal with the case p ą 0. We will compare the relative
gradings of elements in H 3|FpSppKq, r0sq and H|FpS3ppKq, r´rsq. In Spinc struc-
ture r0s the elements a, z0 zp, and z´p are all non-zero in the mapping cone
and a, zp and z´p survive in H|FpS3ppKq, r0sq by Lemma 2.3.14, where a is the
element in HF pS3red ppKq, r0sq coming from the A`0,red summand of Lemma
2.3.3. Note that Lemma 2.3.17 shows that grpz0q “ grpzpq “ grpz´pq.
Therefore, rankpH|F 3grpz0qpSppKq, r0sqq ě 2. Now consider Spinc structure
r´rs. In the mapping cone for Spinc structure r´rs the elements z´r, zp´r
z2p´r and z´r´p are all non-zero, and zp´r, zp`r and z2p´r are non-zero in
H|FpS3ppKq, r´rsq by Lemma 2.3.14. By Lemma 2.3.17 we see that grpz2p´rq´
grpzp´rq “ 2pp´ rq, grpzp´rq “ grpz´rq and grpz´p´rq ´ grpz´rq “ 2r. There-
fore, the only way for there to be a relatively graded isomorphism is if these
two elements are in the same relative grading, so we need 2pp´ rq “ 2r. This
implies that p “ 2r.
Next, suppose p ă 0. Then as before we have the elements z0, zp and
z c´p in the mapping cone for Spin structure r0s, as well as the element a
coming from Ared0 , but in this case all the zi with i ” p are non-zero in
H 3|FpSppKq, r0sq. Computing relative gradings we see that grpz0q ´ grpzpq “
grpz0q ´ grpz´pq “ 2|p|, so rankH|F 3grpz0q´2|p|pSppKq, rrsq ě 2. Comparing with
Spinc structure rrs, we see that the elements zr, zp´r, z2p´r and z´r´p are all
non-zero and contribute to H|FpS3ppKq, rrsq. Computing relative gradings, we
find that grpzrq ´ grpzr´pq “ 2pr ´ pq, grpzrq ´ grpzr`pq “ ´2p2r ` pq and
grpzr`pq ´ grpzr`2pq “ ´2pr ` 2pq. In rrs, the zi with i ” r mod p are the
only elements that contribute to H|FpS3ppKq, rrsq and so
rrs
grtop “ grpzrq and if
there is a relative graded isomorphism HF`pS3ppKq, r0sq – HF`pS3ppKq, rrsq
necessarily ´2p2r ` pq “ 2p2r ` pq, which implies p “ ´2r. Otherwise there
are not two or more of the zi in the same relative grading in Spinc structure
64
rrs. Therefore, the only possible reducing slopes for an almost L-space knot
that are proper divisors of gpKq are p “ ˘2r with r ě 1 odd.
Corollary 2.3.19. If gpKq is odd, then K does not admit any reducing
slopes that are proper divisors.That is, if p is a reducing slope for an almost
L-space knot with odd genus then p “ ˘gpKq.
Proof. We saw in Lemma 2.3.8 that the only possible reducing slopes for
an almost L-space knot are divisors of gpKq and even divisors of 2gpKq ´ 2
when gpKq is even. So if gpKq is odd, only the former are possible. But by
Lemma 2.3.18 we see that also the only possible reducing slopes are even
divisors of gpKq, so in particular gpKq is even.
Now, we analyze the cases when p “ 2r, p|gpKq with p ‰ gpKq and
r ą 1.
Lemma 2.3.20. Suppose p ą 0 , p|gpKq, p ‰ gpKq and p “ 2r. If r ą 1,
then p is not a reducing slope.
Proof. Suppose p “ 2r and consider Spinc structures r1s and r1 ` rs. By
Lemma 2.3.13 there is a relatively graded isomorphism HF`pS3ppKq, r1sq –
HF`pS3ppKq, r1`rsq. To this end, in Spinc structure r1s the elements z1, z1`2r, z1´2r
and z1´4r are all non-zero in the mapping cone and z1`2r, z1´2r and z1´4r
contribute to H|FpS3ppKq, r1sq. By lemma 2.3.17 we find grpz1`2rq ´ grpz1q “
2, grpz1´2rq “ grpz1q and grpz1´2rq ´ grpz1´4rq “ ´2p2r ´ 1q. We com-
pare this with Spinc structure r1 ` rs (here we use the fact that r ą 1 so
that r1 ` rs ‰ r0s). In that Spinc structure we have the elements z1`r,
z1`3r, z1´r and z1´3r are all non-zero in the mapping cone and by Lemma
2.3.14 all but z1`r survive in H|FpS3ppKq, r1 ` rsq. Computing relative grad-
ings, we find that grpz1`3rq ´ grpz1`rq “ 2p1 ` rq, grpz1`rq “ grpz1´rq and
grpz1´3rq´grpz1´rq “ 2pr´1q. Therefore, in order for there to be a relatively
graded isomorphism we must have 2 “ 2pr ´ 1q and 2p1 ` rq “ 2p2r ´ 1q.
Hence r “ 2. This contradicts Lemma 2.3.18 where we proved that in this
case r is odd.
65
Lemma 2.3.21. Suppose p ă 0, p|gpKq with p ‰ gpKq and p “ ´2r. If
r ą 1 then p is not a reducing slope.
Proof. In the mapping cone the elements z1 and z1´2r are non-zero and sur-
vive in H|FpS3ppKq, r1sq. We compute grpz1´2rq ´ grpz1q “ 2p2 ´ 2rq. In
Spinc structure r1 ` rs, the elements z1´r and z1`r are non-zero and survive
in H 3|FpSppKq, r1 ` rsq. Computing their relative grading difference we have
grpz1´rq ´ grpz1`rq “ 4. Hence either 2p2 ´ 2rq “ 4 or 2p2 ´ 2rq “ ´4.
So either r “ 0, which is impossible, or r “ 2. The latter contradicts
S3ppKq – Lp2, 1q#R.
Lemma 2.3.22. For K an almost L-space knot the slope. If gpKq ą 3 then
p “ ´gpKq is not a reducing slope.
Proof. Let p “ ´gpKq “ ´ar and consider the mapping cone for Spinc
structure r1s
A` A`1´g 1
v` v`
h`
1´g 1
´ h
`
1 g 1
T ` T ` T `
There are two generators in H 3|FpSppKq, r1sq, z1´g and z1. We compute
their relative grading difference:
grpz1´gq ´ grpz1q “ 2p2´ gq
By periodicity, there should be a relatively graded isomorphism between
H|FpS3ppKq, r1sq and H|FpS3ppKq, r1` rsq. When r ‰ 1 are two non zero gener-
ators in H|FpS3ppKq, r1`rsq, z1´g`r and z1`r. Their relative grading difference
is
grpz1´g`rq ´ grpz1`rq “ 2p2p1` rq ´ gq
Hence, in order for there to be a relatively graded isomorphism, we need
either 4p1`rq´2g “ 2p2´gq or 2g´4p1`rq “ 4´2g. In the former case we
66
find that 4 ` 4r “ 4, hence r “ 0 which is a contradiction. In the latter case
we find that g ´ 2p1 ` rq “ 2 ´ g, this implies g “ 2 ` r. Since p “ ´gpKq
and p “ ´ar, we have that ´ar “ 2 ` r So ´rpa ` 1q “ 2. Therefore either
r “ 2 and a “ ´2, which contradicts pa, rq “ 1, or r “ 1 and a “ ´3. This
implies that p “ ´3.
Lemma 2.3.23. If p ą 0, K is an almost L-space knot and p|2gpKq ´ 2 with
p ‰ 2, 2gpKq ´ 2. Then p is not a reducing slope.
Proof. By Lemma 2.3.8, we see that p “ ar with a “ 2 and r “ p odd.
2
Therefore, we will compare relative gradings of elements in r1s and r1 ´ ps.
2
Now in Spinc structure r1s by Lemma 2.3.17 we have grpz1q ` 2 “ grpz1`pq
and grpz1q “ grpz1´pq. In r1´ ps we have grpz1` p q “ grpz1´ p and grpz2 1´ p q`2 2 2
2pp ´ 1q “ grpz1´ 3p q. Therefore, in order for there to be a relatively graded2 2
isomorphism, we would need 1 “ p ´ 1, so p “ 4. This is impossible since
2
pp , 2q “ 1.
2
Lemma 2.3.24. Suppose p ă 0, K is an almost L-space knot and p|2gpKq ´
2 with p ‰ 2´ 2g,´2. Then p is not a reducing slope.
Proof. In this case the elements z1, z1`p and z1´p are all non-zero in the
mapping cone and survive in HFpS3| ppKq, r1sq. By Lemma 2.3.17 we have
that grpz1´pq ´ grpz1q “ ´2pp ´ 1q and grpz1`pq ´ grpz1q “ 2p2 ` pq.
Since p ‰ ´2, the Spinc structures r0s and r1 ´ ps are distinct. We have the
2
elements z1´ p s, z1´ 3p and z1` p . Computing their relative grading differences2 2 2
we find grpz1´ p q ´ grpz1` p q “ ´4 and grpz1´ 3p q ´ grpz
3p
1´ p q “ ´2p ´ 1q.
2 2 2 2 2
Hence in order for there to be a relatively graded isomorphism from r1s to
r1 ´ ps, we would need either 2p2 ` pq “ 4 or 2p2 ` pq “ ´4. In the former
2
case we have p “ 0, which obviously cannot happen, and in the latter we
have p “ ´4, which cannot happen because by Lemma 2.3.8 p is odd.
2
Lemma 2.3.25. Suppose p “ 2 ´ 2g and p ‰ ´2 (so gpKq ‰ 2). Then p is
not a reducing slope.
67
Proof. Since p ‰ ´2, we have that g ě 4 (recall that in the case when p
divides 2gpKq ´ 2, the genus is even) and r1s and r2´ gs are not conjugate.
We have that in each Spinc structure rss ‰ r0s, rg ´ 1s there is just
one non-zero zs. However, we can still compare the grading on this zs to the
bottom of the tower in two Spinc structures in the same orbit, namely r1s
and r2´ gs. In r1s, we have grpx1q “ d1` 1, hence grpz1q “ d1´ 3. In r2´ gs,
we have grpz2´gq “ d2´g ´ 1. So there is no relatively graded isomorphism of
FrU s-modules.
In summary, we have shown that for p to be a reducing slope, p must
satisfy either p “ ˘2, or p “ gpKq, unless gpKq “ 3 in which case p “
˘gpKq, or p “ p2gpKq ´ 2q with p and g even. Hence if g is odd and greater
than 3, then the only possible reducing slope is gpKq.
Corollary 2.3.26. Almost L-space knots of genus gpKq ě 2 do not admit
multiple reducing slopes.
68
CHAPTER 3
(1,1) PATTERNS
In this chapter we study the knot Floer homology of satellite knots with
p1, 1q-patterns from the perspective of bordered Floer homology and the im-
mersed curve reformulation of the pairing theorem for satellite knots with
p1, 1q-patterns with a goal of computing various 3- and 4-dimensional invari-
ants of satellite knots with patterns from two novel families of patterns. In
Sections 3.1-3.6 we study a family of patterns which we denote by P pp,1q.
These patterns satisfy wpP pp,1qq “ p ` 1 and P pp,1qpUq „ T2,3. We call such
patterns trefoil patterns. We compute the three-genus, and bound the four-
genus of these satellites. We show that all patterns in this family are fibered
in the solid torus. This implies that satellites with fibered companions and
patterns from this family are also fibered. We also show that satellites with
thin fibered companions or companions K with τpKq “ ˘gpKq formed from
these patterns have left or right veering monodromy. We then use this to
show that satellites with fibered companion knots K so that |τpKq| ă gpKq
formed from these patterns do not have thin knot Floer homology, using a
recent result of [BNS22].
In Sections 3.7-3.11 we study a family of patterns denoted by Qi,j such
that wpQi,jq “ j and Qi,jpUq „ U (called unknot patterns) and the knot
Floer homology of the n-twisted satellites that are formed from these pat-
terns with arbitrary companions. Recall that an n-twisted satellite knot,
denoted PnpKq, is formed from a companion knot K and a pattern knot
P where the longitude of the solid torus containing P is glued to the curve
nµ ` λ on S3 ´ νpKq. We study how the invariants gpQi,jpKqq, τpQi,jn n pKqq
and ϵpQi,jn pKqq behave under this twisting operation and find closed for-
mulas for then in terms of i, j, n. We also investigate the function n ÞÑ
dimpH 3 i,jzFKpS ,Qn pKq, gpQi,jn pKqqqq and use this and fibered detection of
knot Floer homology to understand when the n-twisted pattern Qi,jn is fibered
in the solid torus.
69
3.1 Introduction
Knot Floer homology, introduced by Rasmussen [Ras03a] and Ozsváth
and Szabó [OS04b], is an invariant of null homologous knots in the three
sphere. Its simplest instantiation takes the form of a bigraded Abelian group,
À
HzFKpS3, Kq – 3zm,A HFKmpS ,K,Aq. Here m is called the Maslov grading
and A is called the Alexander grading. Knot Floer homology contains in-
formation about the knot K and its complement S3zνpKq. For example, it
detects the three-genus [OS04a] and fiberedness of the knot [Juh08c; Ni07],
contains information about the monodromy of fibered knots [Ni20], bounds
the number of disjoint, non-isotopic Seifert surfaces in the knot complement
[Juh08a], and bounds the four-genus of the knot [OS03b]. In this note, we
use these detection properties to investigate three- and four-dimensional in-
variants of satellite knots formed from a family of p1, 1q-patterns.
Recall that, given a knot K Ă S3 and a pattern P Ă S1 ˆ D2, we can
construct a new knot, called the (0-twisted) satellite knot with companion
knot K and pattern knot P , denoted P pKq, by removing a tubular neigh-
borhod of K and gluing in the pair pS1 ˆ D2, P q so that S1 ˆ tptu is identi-
fied with the Seifert longitude of K. A pattern knot P is a p1, 1q-pattern if it
admits a genus-1 doubly-pointed bordered Heegaard diagram, a concept that
we recall in section 3.2.
Our main reason for restricting to p1, 1q-patterns is computational. For
an arbitrary pattern P , the bordered pairing theorem of [LOT18] expresses
HzFKpS3, P pKqq in terms two invariants: C 1 2 3zFApS ˆD ,P q and CzFDpS zνpKqq.
For p1, 1q-patterns, the work of Chen in [Che19] recasts this pairing theo-
rem in terms of Lagrangian intersection Floer homology of two curves in the
punctured torus. This facilitates computation in two ways: it allows one to
vary the pattern within a family and it allows one to compute the decom-
positon into Alexander gradings much more efficiently than with the lan-
guage of the original bordered pairing theorem.
Many of the computations of knot Floer homology of satellite knots
that exist in the literature involve p1, 1q-patterns. For example the cabling
70
patterns studied in [Hom14] (see also [HW19]), Mazur pattens studied in
[Lev16] and [PW21], and Whitehead double patterns studied in [Hed07] are
all p1, 1q-patterns. Given this, it is interesting to compute knot Floer homol-
ogy of satellites where the pattern comes from a family of p1, 1q-patterns. In
[Che19] this project is taken up and he examines the case where P is an ar-
bitrary p1, 1q-pattern P so that P pUq » U , called an unknot pattern, and the
companion knot is the right or left handed trefoil.
In the following, we use the immersed curve pairing theorem as stated
in [Che19] to compute the knot Floer homology of satellites with arbitrary
companion knots K and patterns P from a specific family of p1, 1q-patterns
with the property that P pUq » T2,3. We will refer to such patterns as tre-
foil patterns. In section 3.4 we introduce, for each p ą 1, a trefoil pattern
denoted P pp,1q which is closely related to the pp, 1q unknot cabling pattern.
Our goal is to investigate various three- and four-dimensional properties of
the satellite knots obtained from these trefoil patterns. First, for each p ą 1
and for any knot K, we compute the invariant τpP pp,1qpKqq, an integer val-
ued concordance invariant derived from the knot Floer homology package
first defined by [OS03b], in terms of τpKq and ϵpKq.
Theorem 3.1.1. For the patterns P pp,1q and for an arbitrary companion
knot K Ă S3, we have
• If ϵpKq “ 1, then τpP pp,1qpKqq “ pp` 1qτpKq ` 1
• If ϵpKq “ ´1, then τpP pp,1qpKqq “ pp` 1qpτpKq ` 1q
• If ϵpKq “ 0, so τpKq “ 0, then τpP pp,1qpKqq “ τpT2,3q “ 1.
As shown in [OS03b, Corollary 1.3], the integer τpKq satisfies |τpKq| ď
g4pKq, where g4pKq is the smooth four-genus of a knot (the minimal genus
of a surface properly embedded in B4 with boundary K Ă S3). This gives
the following corollary concerning the slice genus of these satellite knots.
Corollary 3.1.2. For any companion knot K with τpKq ‰ ´1 and ϵpKq ‰
´1, the satellite knots P pp,1qpKq are not slice.
71
Given a pattern in the solid torus, we can associate to it an integer
wpP q, called the winding number of the pattern, by computing the algebraic
intersection between the pattern P and a meridional disk t0u ˆ D2. Given
a pattern P with winding number r, we define a relative Seifert surface for
P to be a surface Σ̃ in S1 ˆ D2 so that the interior of Σ̃ is disjoint from P ,
and the boundary of Σ̃ consists of P together with r coherently oriented lon-
gitudes. A pattern is fibered if the complement S1 ˆD2zνpP q is fibered over
S1 with fiber surface a relative Seifert surface for P . Furthermore, the genus
of a pattern, gpP q, is defined to be the minimal genus of a relative Seifert
surface for P .
For a satellite knot P pKq with a non-trivial companion K a result of
Schubert [Sch53] shows that the three-genus of the satellite knot gpP pKqq
can be expressed in terms of wpP q, gpKq and gpP q:
gpP pKqq “ |wpP q|gpKq ` gpP q. (3.1)
This has the consequence that for any non-trivial knot K, the value of
gpP pKqq is determined by the value of gpKq and gpP q. However, gpP q de-
pends only on the pattern. Hence, we can compute gpP q if we can compute
the three genus of some satellite with non-trivial companion K and pattern
P , for example P pT2,3q. Using the fact that knot Floer homology detects the
genus of knots, we prove
Lemma 3.1.3. For any p ą 1, the trefoil patterns P pp,1q have gpP pp,1qq “ 1.
Now, given the value of gpP q, we can determine gpP pKqq in terms of
gpKq for any non-trivial companion knot K by using equation (3.1). This
gives the following corollary. Note that the case K “ U follows since gpUq “
0 and P pp,1qpUq » T2,3 has genus 1.
Corollary 3.1.4. For any knot K and for any p ą 1, gpP pp,1qpKqq “ pp `
1qgpKq ` 1.
In a similar vein, Hirasawa, Murasugi, and Silver proved in [HMS08]
that a satellite knot with non-trivial companion is fibered if and only if both
72
the pattern and the companion knot are fibered. This has the consequence
that to determine if a pattern P is fibered in the solid torus, it is enough
to determine if the knot P pT2,3q is fibered. Since knot Floer homology de-
tects when a knot is fibered, to show that the pattern P is fibered, it is then
enough to compute HzFKpS3, P pT2,3q, gpP pT2,3qqq and show that it has rank
1.
Theorem 3.1.5. For p ą 1 the pattern knot P pp,1q is fibered in the solid
torus.
One motivation to understand fibered patterns is the result of Ni [Ni06,
Theorem 1.2] that the knot Floer homology of satellites with fibered pat-
terns in the top Alexander grading has the same dimension as the knot Floer
homology of the companion in the top Alexander grading. That is
rkHzFKpS3, K, gpKqq “ rkHzFKpS3, P pKq, gpP pKqqq. (3.2)
This theorem, when combined with the work of Juhasz in [Juh08c; Juh08a]
which relates the knot Floer homology in the top Alexander grading to the
sutured Floer homology of the complement of a Seifert surface for the knot
K has the following consequences.
Proposition 3.1.6. If K is a knot with rkpHFKpS3z , K, gpKqqq ă 4 and P
is a fibered pattern, then for all i ě 1 the knots K and P ipKq have unique
minimal genus Seifert surfaces.
Proposition 3.1.7. If K is a knot with rkpHFKpS3z , K, gpKqqq “ 3 and
P is a fibered pattern, then K and P ipKq admit depth ď 1 taut foliations
transverse to the boundary.
Recall that fibered knots have unique minimal genus Seifert surfaces.
These propositions can be viewed as generalizations of this fact. In particu-
lar, by Theorem 3.1.5, these propositions apply to the patterns P pp,1q.
Finally we study the next to top Alexander graded piece of the knot
Floer homology of these satellite knots. In the case that K is a fibered knot,
73
HzFKpS3, K, gpKq ´ 1q contains information about the monodromy of the
fibration, in the following sense.
Theorem 3.1.8 ([Ni20]). If K is a fibered knot and rkpHzFKpS3, K, gpKq ´
1qq “ 1, then the monodromy of K is either left or right veering.
Remark 3.1.9. There is no analogue of equation (3.2) for the next to top
Alexander graded piece of knot Floer homology of a satellite and its compan-
ion. In general, there is not even an inequality relating them, even for fibered
patterns. For example rkpHzFKpS3, T2,3, 0q “ 1 and rkHzFKpS3, pT2,3q2,1, 1q “ 2
and as Theorem 3.1.10 shows, constructing satellites with certain patterns
can decrease the rank in the next to top Alexander graded piece by an arbi-
trary amount. Note, certain families of patterns do preserve the property of
having one dimensional Floer homology in the next to top Alexander grading,
for example if K is an L-space knot and P is a pattern so that P pKq is also
an L-space knot (for example the pp, qq cable pattern with q ě 2gpKq ´ 1)
p
then by [HW18] both K and P pKq have one dimensional Floer homology in
the next to top Alexander grading.
Recall that the δ-grading on knot Floer homology is define by δ “ m ´
A. We call a knot K Floer thin (or thin) if the δ-grading is constant for all
generators of HzFKpS3, Kq.
Theorem 3.1.10. For each p ą 1, and for any fibered knot K with τpKq “
˘gpKq, or for any fibered thin knot K, we have
rkpHFKpS3, P pp,1qz pKq, gpP pp,1qpKqq ´ 1qq “ 1.
Corollary 3.1.11. For any fibered knot K with τpKq “ ˘gpKq, or for any
fibered thin knot K, the fibered knot P pp,1qpKq has left or right veering mon-
odromy.
Lastly, we use Theorem 3.1.10 to show that for some fibered companion
knots K, the satellite knots P pp,1qpKq are not Floer thin. The main result
we use is [BNS22, Corollary 1.7] which says that a fibered thin knot with
|τpKq| ă gpKq cannot have left or right veering monodromy.
74
Proposition 3.1.12. If K is a non-trivial fibered knot with thin knot Floer
homology such that |τpKq| ă gpKq, then the knot Floer homology of P pp,1qpKq
is not thin.
Since quasialternating knots have thin knot Floer homology by [MO08],
we have the following consequence of Proposition 3.1.12.
Corollary 3.1.13. For any p ą 1 and for any thin fibered knot K with
|τpKq| ă gpKq, the knots P pp,1qpKq are not quasialternating.
Organization
In section 3.2 we introduced the bordered pairing theorem from [LOT18]
and recall the work of [HRW22] reinterpreting the bordered invariants in
terms of immersed curves in the punctured torus. In section 3.3, we recall
Chen’s immersed curve version of the pairing theorem from [Che19]. In sec-
tion 3.4 we prove Theorem 3.1.1. In section 3.5, we prove Theorem 3.1.5, as
well as propositions 3.1.6 and 3.1.7. In section 3.6, we prove Theorem 3.1.10
and Proposition 3.1.12.
3.2 Bordered Floer Homology
In this section we introduced the necessary notation to state and inter-
pret the pairing theorem for bordered Floer homology of [LOT18]. Bordered
Floer homology is an invariant that is used to study Heegaard Floer homol-
ogy of three manifolds that have been decomposed along essential embedded
surfaces. In our case, studying satellite operators, we are interested in de-
composing the ambient three manifold, S3 together with a knot K, along an
essential torus. Then one can compute certain algebraic invariants of both
sides and the Floer homology of the ambient three manifold (together with
the knot filtration) can be computed by suitably combining these invariants.
In [LOT18], Lipshitz, Oszváth and Thurston associate, to a three man-
ifold with parameterized torus boundary, a type A and D structure over
75
the torus algebra A. We now briefly describe these concepts. The torus al-
gebra A is defined as follows. Over F it has a basis consisting of two mu-
tually orthogonal idempotents ι0 and ι1 and six other nontrivial elements
ρ1, ρ2, ρ3, ρ12, ρ23, ρ123. The non-zero products in the algebra are given as fol-
lows:
ρ1ρ2 “ ρ12 ρ2ρ3 “ ρ23 ρ1ρ23 “ ρ12ρ3 “ ρ123
ρ1 “ ι0ρ1ι1 ρ2 “ ι1ρ2ι0 ρ3 “ ι0ρ3ι1
ρ12 “ ι0ρ12ι0 ρ23 “ ι1ρ23ι1 ρ123 “ ι0ρ123ι1
If we let I Ă A denote the subring of idempotents, then a type D structure
over A is a unital left I module N together with an I linear map δ : N Ñ
AbI N such that
pµb Iq ˝ pIb δq ˝ δ “ 0
A type A structure is a right unital I module M with a collection of
maps m ii`1 : M bA ÑM , for i ě 0 such that
n n´2
ÿ ÿ
0 “ mn´ipmipxba1b¨ ¨ ¨bai´1qb¨ ¨ ¨ban´1q` mn´1pxb¨ ¨ ¨baiai`1b¨ ¨ ¨banq
i“1 i“1
(3.3)
and so that
m2px, 1q “ x
mipx, ¨ ¨ ¨ , 1, ¨ ¨ ¨ q “ 0
Given a type A structure M and a type D structure N , we can form a
chain complex, called as the box tensor product and denoted M b N . The
underlying vector space is the tensor product M bI N , and the differential is
defined by
76
8
ÿ
Bbpxb yq “ pmi`1 b Iqpxb δipyqq (3.4)
i“0
In the case that the type D structure is bounded, as defined in in [LOT18,
Section 2], then the above sum is finite and the box tensor complex is well
defined.
In what follows, we are interested in the following version of the bor-
dered pairing theorem.
Theorem 3.2.1. [LOT18, Theorem 11.19] Suppose Y is a closed 3-manifold
decomposed as Y “ Y1 Y Y2 with BY1 – ´BY 22 – T . Suppose further that
K Ă Y1 is a knot which becomes null homologous in Y . Then up to homo-
topy equivalence of chain complexes
gCzFKpY,Kq » CzFApY1, Kqb CzFDpY2q
We will give the immersed curve interpretation of this pairing theorem
due to [Che19] for p1, 1q patterns in section 3.3. First, we will describe in
more detail how to compute and interpret CzFDpS3zνpKqq and CzFApS1 ˆ
D2, P q as immeresed curves in the punctured torus in the next two sections.
CzFDpS3zνpKqq from CFK´pKq
In this section, we recall the algorithm from [LOT18, Section 11.5] for
computing CzFDpS3zνpKqq from CFK´pKq. For the definitions of reduced,
filtered basis, we refer the reader to the original source (see also [HW18]).
We call a filtered reduced basis over FrU s vertically simplified if for each ba-
sis element xi exactly one of the following conditions is satisfied
• There is a unique incoming vertical arrow, and no outgoing vertical
arrow, or
• There is a unique outgoing vertical arrow and no incoming vertical ar-
row, or
77
• There are no vertical arrows.
A horizontally simplified basis is defined similarly, replacing vertical
by horizontal in the above. Given a knot K and a framing n, there exists
a pair of bases η̃ “ tη̃1, . . . , η̃ ´2ku and ξ̃ “ tξ̃1, . . . , ξ̃2ku for CFK pKq that
are horizontally and vertically simplified respectively. They are indexed so
that for every pair η̃2i´1 and η̃2i there is a horizontal arrow of length li ě 1
connecting them and similarly, there is a vertical arrow of length ki ě 1
connecting ξ̃2i´1 to ξ̃2i. There are corresponding bases ξ “ tξ0, . . . , ξ2ku
ř
and η “ tη0, . . . , η2ku for ι0CzFDpXk, nq so that if ξ̃ “ 2kj i“0 aij η̃i and
ř
η̃ 2kj “ i“0 bij ξ̃i, then the corresponding change of bases formulas hold with
the coefficients restricted to U “ 0. The summand ι z1CFD has basis
k k
ď ď
tκi1, . . . , κik u Y tλi1, . . . , λil u Y tµ1, . . . , µ ui i |2τpKq´n|
i“1 i“1
There are non-zero coefficient maps induced from the horizontal and
vertical arrows in the complex for CFK´ as follows. A length ki vertical ar-
row from ξ2i´1 to ξ2i induces type D operations, sometimes called coefficient
maps:
ÝρÑ1 i ÐρÝ2Ý3 i ÐρÝ2Ý3 i Ðρξ2i´1 κ1 κ2 . . . κk Ý
12Ý3 ξ
i 2i
Similarly, for each length li horizontal arrow from η2i´1 to η2i, we get
coefficient maps
ρ
η ÝÑ3 ρ2i´1 λi1 ÝÝ
2Ñ3 λi ÝρÝ232 Ñ . . . Ý
ρÝ2Ñ3 ρλi ÝÑ2l ηi 2i
Additionally, there are coefficient maps from ξ0 to η0 depending on the
framing and the value of the invariant τpKq.
• Ýρξ Ý1Ñ20 η0 if n “ 2τpKq
• ρξ0 ÝÑ1
ρ
µ 23
ρ ρ
1 ÐÝÝ . . .ÐÝ2Ý3 µ ÐÝ3m η0 if n ă 2τpKq m “ 2τpKq ´ n
• ÝρÝ12Ñ3 Ýρξ0 µ1 Ý2Ñ3
ρ ρ
. . . ÝÝ2Ñ3 µm ÝÑ2 η0 if n ą 2τpKq, m “ n´ 2τpKq
78
ρ2 ρ3
ξ0 λ ξ1
ρ1
µ1 ρ1
ρ23
µ2 κ
ρ123
ρ3
ξ2
Figure 3.1. Type D structure for 0-framed right handed trefoil complement
For example, for the knot K “ T2,3, the right-handed trefoil, CFK´pT2,3q
has a simultaneously vertically and horizontally simplified FrU s basis tξ̃0, ξ̃1, ξ̃2u
with differential given by Bpξ̃1q “ Uξ̃0` ξ̃2. Applying the above algorithm, we
get the type D structure shown in Figure 3.1.
For any knot K in S3, there is always a vertically distinguished element
of a horizontally simplified basis, which is an element in a horizontally sim-
plified basis with no incoming or outgoing vertical arrows. Similarly, there
is a horizontally distinguished element of a vertically simplified basis. In
[Hom14, Lemma 3.2], it is shown that it is always possible to find a hor-
izontally simplified basis for CFK8pKq so that one of the horizontal ba-
sis elements ξ0 is the vertically distinguished generator of some vertically
simplified basis. Note that the concordance invariant ϵpKq can be defined
in terms of the generator ξ0: If ξ0 occurs at the end of a horizontal arrow,
then ϵpKq “ 1, if ξ0 occurs at the beginning of a horizontal arrow, then
ϵpKq “ ´1. If there is no horizontal arrow to or from ξ0, then ϵpKq “ 0.
Immersed Curves for knot complements
Given a type D structure over the torus algebra, like CzFDpS3zνpKq, nq,
the work in [HRW22] shows how we can represent it as an immersed mul-
ticurve with local systems in the torus, which we now describe. The first
79
step is to construct a decorated graph from the type D structure. Let N be
a type D structure over the torus algebra, and let Ni “ ιiN . This gives a
decomposition N “ N0 ‘ N1. Given bases Bi of Ni, for i “ 0, 1, we con-
struct a decorated graph Γ as follows. The vertices of Γ are in correspon-
dence with the basis elements and are labelled or depending on if the ver-
tex corresponds to a basis element in B0 or B1 respectively. Suppose now
that we have two vertices corresponding to basis elements x and y such that
δpxq “ ρI b y ` ¨ ¨ ¨ , for I P tH, 1, 2, 3, 12, 23, 123u. In this case we put an
edge labelled ρI from x to y. A decorated graph is called reduced if no edge
labelled by ρH appears. The next step is to take a decorated graph and turn
it into an immersed train track in the punctured torus. Let T 2 “ R2{Z2 and
let w “ p1´ ϵ, 1´ ϵq be a basepoint. Let µ and λ be the images of the x and
y axes respectively and embed the vertices of Γ into T 2 so that the vertices
lie on λ in the interval t0u ˆ r1 , 3s and the vertices lie on λ in the interval
4 4
r1 , 3s ˆ t0u. Then we embed the edges into the torus according to the rules
4 4
shown in [HRW22, Figure 19] (see also Figure 3.2). In general this train
track is not necessarily an immersed curve, but work in [HRW22] shows that
for type D structures that arise from 3-manifolds with torus boundary one
can always choose a nice basis so that the train track is an immersed curve
(possibly with local systems). For example, we construct the immersed curve
associated to the trefoil complement in Figure 3.2, where for example the
arc from ξ1 to κ indicates the presence of a ρ1 edge from ξ1 to κ in the deco-
rated graph. We will denote this immersed curve by αpKq.
Properties of Immersed Multicurves for Knot Complements
In this section we recall how the immersed curve αpKq encodes the con-
cordance invariants τpKq and ϵpKq as well as the genus of the knot gpKq.
In order to do this, we fix a representative of the lift of the immersed curve
to the universal cover, called the peg-board representative of the immersed
curve. This is discussed in [HRW22, Section 4.2]. In brief, we assume that
we have chosen a minimal length representative of the immersed multicurve.
80
µ2 µ1 λ κ
ρ3 ρ123
ξ2 ξ2
ξ0 ξ0
ξ1 ξ1
ρ2 ρ1
µ2 µ1 λ κ
Figure 3.2. The immersed curve associated to the 0 framed trefoil
complement
Given a peg-board representative of αpKq, the genus of the knot is half the
maximal number of pegs between the minimum and maximum height at-
tained by the immersed curve. The invariants τpKq and ϵpKq are related to
the essential component γ0 of the immersed curve, see [HRW22] and [HW19,
Proposition 2]. The essential component γ0 is the unique non-vertical seg-
ment of the immersed curve, in the sense that all other components are sup-
ported in a neighborhood of the meridian, and the component γ0 wraps once
around the cylinder (in the covering of the torus corresponding to the longi-
tudinal subgroup). As mentioned in [HRW17, Remark 50] this component
does not carry any non-trivial local system as only one curve component
can wrap around the cylinder (since otherwise the meridional filling would
have rank ě 2). This observation, together with the discussion surrounding
[Hom14, Lemma 3.2] in Section 3.2 implies the following lemma concerning
the shape of the essential component of αpKq lifted to the universal cover.
Lemma 3.2.2. Suppose that K is a knot in S3 and that γ0 is the essential
81
curve component of αpKq lifted to the universal cover.
• If ϵpKq “ 1 and τpKq ě 0 γ0 slopes upwards for 2τpKq rows and turns
down at the top and up at the bottom
• If ϵpKq “ ´1 and τpKq ě 0, then γ0 slopes upwards for 2τpKq rows
and turns up at the top and down at the bottom
• If ϵpKq “ 1 and τpKq ď 0 then γ0 slopes downwards for 2τpKq rows
and turns down at the bottom at up at the top
• If ϵpKq “ ´1 and τpKq ď 0 then γ0 slopes downwards for 2τpKq rows
and turns up at the bottom and down at the top.
• If ϵpKq “ 0, then τpKq “ 0 and γ0 is horizontal at height 0.
In each case the remaining portion of the essential component of the im-
mersed curve and any other component of the immersed curve are contained
in a neighborhood of the meridian.
Proof. We will show that the immersed curve has the claimed form in the
case that τpKq ą 0 and ϵpKq “ 1. The rest of the cases are similar. As
mentioned above, in [Hom14], Hom constructs a horizontally simplified ba-
sis tξ0, η0, ¨ ¨ ¨ , ηNu so that ξ0 is the distinguished element in a vertically
simplified basis with no incoming or outgoing vertical arrows. In the case
ϵpKq “ 1, this generator appears at the end of a horizontal arrow. Suppose
that η1 ÑÝ ξ0 is a length l arrow from η1 to ξ0. In this case, the portion of
CFDpS3z zνpKqq has the following form: From the length l horizontal arrow
from η1 to ξ0, the algorithm in [LOT18] produces a sequence of type D oper-
ations
ρ3 1 ρ23 1 ρ23 1 ρη 21 ÝÑ λ1 ÝÝÑ λ2 ÝÝÑ . . . λl ÝÑ ξ0
(Note that this part of the type D operations or immersed curve is what
changes when ϵpKq changes sign)
Since τpKq ą 0, the unstable chain takes the form
82
Ýρξ Ñ1 ρ0 µ1 ÐÝ2Ý3 ¨ ¨ ¨ Ð
ρÝ2Ý3 ρµ 32τpKq ÐÝ η0
(Note that this part of the type D operations or immersed curve is what
changes when τpKq changes sign)
Using the procedure described in [HRW17] and the previous section,
this decorated graph becomes the portion of the immersed curve shown in
Figure 3.3. As claimed, the immersed curve slopes upwards for 2τpKq rows,
turns down at the top (from the ρ2 from λ1l to ξ0) and turns up at the bot-
tom by the symmetry of the immersed curve under the elliptic involution.
The remaining bullet points follow similarly.
The fact that the remaining portion of the immersed curve is contained
in a neighborhood of the meridian follows since the meridional filling of any
knot complement has rank one. If any other component wrapped around the
longitude, this would imply that the meridional filling has rank ě 2.
CzFApS1 ˆD2, P q for p1, 1q-patterns P Ă S1 ˆD2
As we saw in the previous section, the type D structure from the pair-
ing theorem can be obtained algorithmically from knowledge of CFK´pKq.
For the type A side, there is no such algorithm for determining CzFApS1 ˆ
D2, P q in terms of CFK´pP pUqq. However, when the pattern pS1 ˆ D2, P q
admits a particular type of Heegaard diagram, called a genus-1 doubly-pointed
bordered Heegaard diagram, we can compute CzFApS1 ˆD2, P q directly. We
now describe how to do this. First, we introduce the notation of a genus 1
doubly-pointed bordered Heegaard diagram.
Definition 3.2.3. A genus-1 doubly-pointed bordered Heegaard diagram is
a five tuple pΣ, αa, β, w, zq. Here Σ is a compact oriented surface of genus 1
with a single boundary component. The alpha arcs αa “ pαa, αa1 2q are a pair
of properly embedded, disjoint arcs in Σ with a fixed order to the intersec-
tions αa X BΣ. The basepoint w lies on the boundary of Σ in the complement
of the endpoints of the α arcs; i.e. w Ă BΣzBαa. The resulting subdivision
83
w w
ξ0
ρ ρ1 2
µ1 w λ
1
l w
ρ23
µ2
w w
ρ23 ...
w µm´1w
ρ23
µm
w w
ρ123 η0 ρ3
Figure 3.3. The essential component of the immersed curve for a knot K
with τpKq ą 0 and ϵpKq “ 1. The curve crosses at heights ´τpKq and τpKq.
The lighter portion of the curve indicates that γ0 is potentially immersed in
the punctured torus, but is contained in a small neighborhood of the
meridian, along with all the other components of the immersed multicurve
αpKq
of BΣ results in the data of a pointed matched circle. The β-curve is an em-
bedded closed loop in Σ so that β is transverse to the α-arcs and the comple-
ment Σzβ is connected. Furthermore, we place a basepoint z in the interior
of Σ without the α-arcs and β-circles, so that if we forget the z basepoint,
the β curve is isotopic to αa2.
This data specifies a three manifold with torus boundary together with
a knot. The three manifold and knot can be recovered by the following recipe.
Attach a two-handle to Σ ˆ r0, 1s along β ˆ t1u. The knot is specified by
connecting the z basepoint to the w basepoint in the complement of β and
pushing the arc into the handlebody compressed by the β curve and con-
necting w to z in the complement of αa in Σ. Note that the α-arcs are the
cores of the 1-handles of the boundary torus. In our case, we have αa1 “ λ
84
λ “ αa1
y0
y1
y2
y3
y4
µ “ αa2 z
y5
y6
y7
w
x0 x1 x2
Figure 3.4. The genus 1 doubly pointed Heegaard diagram for the pattern
P p3,1q
and αa2 “ µ the longitude and meridian of the torus boundary BpS1 ˆ D2q.
See Figure 3.4 for an example of a genus 1 doubly pointed bordered Heeg-
gard diagram. Note that by definition we have β ¨ µ “ 0 and β ¨ λ “ 1 since
if we forget the z basepoint the β curve is isotopic to the meridian. We ori-
ent the meridian as shown in Figure 3.4 and the β curve inherits an induced
orientation from the meridian.
Now we describe how to obain CFApS1 ˆ D2z , P q from a given genus 1
doubly pointed bordered Heegaard diagram. As an F vector space CzFApS1 ˆ
D2, P q is generated by elements of the set
G “ tx|x P β X αau.
For each x P G, we have the following right action of the idempotent subalge-
bra I: x ¨ ι0 “ x if x P αa1 X β and x ¨ ι0 “ 0 otherwise. Similarly, x ¨ ι1 “ x if
x P αa2 X β and x ¨ ι1 “ 0 otherwise.
85
Now, regard the surface-with-boundary Σ as T 2zD2. Let R2 Ñ T 2 de-
note the universal cover of the torus, and set Σ̃ to be the covering space ob-
tained from R2 by removing the lifts of D2. Using this covering space, we
define the maps
mn`1 : M bAbn ÑM
for n ě 0 as follows.
ÿ
mn`1px, ρI1 , ¨ ¨ ¨ , ρInq “ #Mpx, yqy
yPG
where #Mpx, yq is the mod 2 count of index 1 immersed disks in Σ̃ such
that, when we traverse the boundary of the disk we start from a lift of x and
walk along an arc of some lift of αa then along the arc ρI1 on some lift of
BD2, . . . , then walk along some the arc ρIn and then along some lift of αa to
y and finally along a lift of β from y to x.
For example, consider the doubly pointed genus 1 Heegaard diagram
shown in Figure 3.4. The generators of CzFApP q in idempotent ι0 (intersec-
tion of β with αa1) are labelled x0, x1, x2 from left to right and the generators
in idempotent ι1 (intersections of β with αa2) are labelled y0, ¨ ¨ ¨ , y7 from top
to bottom. We draw the lift to the cover Σ̃ in Figure 3.6 and indicate a few
of the type A operations given by the disks shown. The gray disk gives a
m3px0, ρ12, ρ1q “ y3, the green disk gives m2px1, ρ1q “ y1 and the pink disk
gives m py , ρ , ρ q “ y . The full type A module CFApS1 ˆ D2, P p3,1qz3 1 2 1 4 q is
shown in Figure 3.5. In that figure, an arrow labelled ρI1 , ρI2 , . . . , ρIn from x
to y means there is an A8 operation mn`1px, ρI1 , . . . , ρInq “ y.
3.3 The pairing theorem for p1, 1q patterns
The main result in [Che19] is a reinterpretation of the pairing theorem
from [LOT18, Theorem 11.19] in terms of immersed curves when CFApS1z ˆ
D2, P q comes from a p1, 1q pattern P . In this section we recall this theorem.
Let βpP q denote the β curve in the data of a genus one doubly pointed
Heegaard diagram and let αpKq denote the immersed curve for S3zνpKq as
86
ρ1 ρ1 ρ1
x0 y0 x1 y1 x2 y2
ρ2ρ1 ρ2,ρ1 ρ2,ρρ 112,ρ1 ρ12,ρ1 ρ3 ρ12,ρ1
y3 y4 y7 y5 y6
Figure 3.5. CzFApHq where H is the doubly pointed bordered Heegaard
diagram shown in Figure 3.4
described in section 3.2. Chen’s theorem says that to compute HzFKpS3, P pKqq
we can compute the intersection Floer homology of αpKq and βpP q, denoted
CzFKpα, βq, in the torus as follows. Let T 2 “ r0, 1s2{ „ and divide the square
into four quadrants. Include the immersed curve αpKq into the first quad-
rant r1 , 1s ˆ r1 , 1s) and include pβpP q, w, zq into the third quadrant. Then
2 2
extend both curves horizontally and vertically, so that α and β intersect in
the second and fourth quadrants only. In this set up intersections in the sec-
ond quadrant correspond to generators of CFApS3z , P qb CzFDpS3zνpKqq that
come from pairing generators in idempotent ι0 and intersection points in the
fourth quadrant correspond to generators of CzFApS3, P q b CzFDpS3zνpKqq
that come from pairing generators in the ι1 idempotent. The main work
in [Che19] is constructing from a differential in the Lagrangian Floer chain
complex, CzFKpα, βq, a type A operation in CzFApS1 ˆ D2, P q and a corre-
sponding type D operation in CzFDpS3zνpKqq so that these pair in the box
tensor product to produce the given differential.
The data of the torus divided into quadrants, with the curves αpKq and
βpP q included as described, or this same picture lifted to the universal cover,
will be referred to as a pairing diagram for the knot Floer homology of the
satellite P pKq. For an example of a pairing diagram for the knot Floer ho-
moloy of the satellite knot P p3,1qpT2,3q see Figure 3.7. From the picture we
can see that CzFKpS3, P p3,1qpT2,3qq has 41 generators. In that figure, we also
indicate two differentials, in light and dark grey, that contribute to Bb. The
dark grey disk gives a differential in CzFKpS3, P p3,1qpT2,3qq connecting x0 b ξ1
87
x0
y3
x1
y1 y1
y4
Figure 3.6. Lift of the pattern P p3,1q to the cover Σ̃ a single connected lift of
β is shown in bold
88
x0ξ0
x0ξ1
y0µ1 y0κ
Figure 3.7. pairing diagram showing the trefoil pattern P p3,1q paired with 0
framed trefoil companion
to y0 b κ. This arises from pairing the type A operation m2px0, ρ1q “ y0
and the Type D operation δpξ1q “ ρ1 b κ. The light grey disk represents a
differential from x0 b ξ0 to y0 b µ1 given by pairing the type A operation
m2px0, ρ1q “ y0 and the type D operation δpξ0q “ ρ1 b µ1.
For convenience we will usually draw pictures of single lifts of αpKq and
βpP q to the universal cover π : R2 Ñ T 2 of the torus. Here we choose
a single lift of βpP q, call it β̃, and a lift of αpKq, call it α̃, so that α̃ is in
pegboard position with respect to a peg at the midpoint of the arc δw,z of
large enough radius to contain both basepoints w and z. We also require
that α̃ and β̃ intersect transversely and there are no pairs of intersections
that are connected by a Whitney disk that does not cross any basepoint.
This is allowed, since intersection Floer homology is an isotopy invariant (a
topic we come back to in the next section). These conditions ensure that
CzFKpα̃, β̃q » HzFKpS3, P pKqq, where CzFKpα̃, β̃q denotes the intersection
Floer homology of the two curves in R2ztπ´1pwq, π´1pzqu. See Figure 3.8 for
an example computing HFKpS3, P p3,1qz pT2,3qq from a lifted pairing diagram.
89
This figure shows that rkpHFKpS3, P p3,1qz pT2,3qqq “ 21.
The last bit of information we want to extract from the pairing diagram
is the Alexander grading on HzFKpS3, P pKqq. This is achieved by the follow-
ing lemma.
Lemma 3.3.1. [Che19, Lemma 4.1]
Let x and y be two intersection points between α and β. Let ℓ be an arc
on β from x to y, and let δw,z be a straight arc connecting w to z. Then
Apyq ´ Apxq “ ℓ ¨ δw,z
For example, consider the intersection points labelled x and y in Figure
3.8. These intersection points are connected by an arc of the β curve that
is shown in bold in the figure. When we traverse this arc, from x to y along
the orientation of β, we cross five δw,z arcs positively. Then Lemma 3.3.1
implies Apyq ´ Apxq “ 5.
Computing τpP pKqq from a pairing diagram
In this section, we recall from [Che19] the precedure for computing τ
from the pairing diagram for CzFKpP pKqq when P is a p1, 1q pattern. Recall
that the Alexander filtration on CzFKpKq produces a spectral sequence con-
verging to H 3xFpS q. The τ invariant is the minimal Alexander grading of the
cycle that survives to the E8 page. In what follows we give a way of com-
puting this spectral sequence in the pairing diagram for CFKpS3z , P pKqq for
p1, 1q patterns P . First, we recall the following well known lemma that gives
us a way of thinking about passing from one page of the spectral sequence
to the next as cancelling differentials that decrease filtration by the minimal
amount, see [BHL19; Zha18].
Lemma 3.3.2. [BHL19, Lemma 2.4] Suppose pC, dq is a chain complex over
F2 freely generated by elements txiu. Let dpxi, xjq be the coefficient of xj in
dpx q and suppose dpx , x q “ 1. Then the complex pC 1, d1i k l q wih generators
txi|i ‰ k, lu and differential
90
d1pxiq “ dpxiq ` dpxi, xlqdpxkq
is chain homotopy equivalent to pC, dq
Now, suppose C is a filtered chain complex. The above lemma tells us
how to compute the spectral sequence associated to the filtration in stages.
À
The E0 term of the spectral sequence is the associated graded C “ Ci.
Then, we pass from the E0 to the E1 page by cancelling the components of d
that do not shift the grading, and arrive at a chain complex pE1, d1q, where
the d1 differential is defined as in lemma 3.3.2. Continuing in this way, we
pass from the E1 page to the E2 page by cancelling the components of the
differential d1 that shift grading by one, etc. In this way, the spectral se-
quence collapses when we have reached a chain complex filtered chain homo-
topy equivalent to the original one but whose differential is zero. For more
details, see the discussion after Remark 2.5 in [BHL19].
In the spectral sequence induced by the Alexander filtration on CzFKpα̃, β̃q,
the previous discussion shows that passing from one page to the next in this
spectral sequence amounts to cancelling differentials that connect elements
of minimal Alexander filtration difference. We now give an way to see that
cancellation geometrically in the complex CzFKpα̃, β̃q. In the pairing dia-
gram, differentials are given by Whitney disks that connect two intersection
points and cross the z basepoint, but not the w basepoint and the filtration
difference is the number of z basepoints enclosed. To cancel two generators
connected by such a Whitney disk, we perform an isotopy of the β curve
over the disk to a new curve β1 thus cancelling those two intersection points
in the diagram, together with possible more if the Whitney disk wasn’t in-
nermost, i.e. it contains arcs of the α curve in its interior. In any case, all
the intersection points cancelled by isotoping away this Whitney disk will
all have the same filtration difference, so it doesn’t matter if we cancel pairs
of generators of minimal filtration difference one at a time or in bulk. Once
this isotopy is performed, we arrive at a new complex, with fewer genera-
tors. To remember the filtration difference after the cancellation, following
91
Chen we place small arrows on the β1 curve, called A-bouys, that remember
that an isotopy of a Whitney disk crossing some number of z basepoints was
performed. Then, when we compute filtration differences of the remaining
intersection points in the α and β1 complex, we count both intersections of
the β1 curve with the δw,z arcs and the A-bouys.
It remains to observe that when we cancel two intersection points by
isotoping the curve β to β1, the differential d1 on the Lagrangian Floer chain
complex CzFKpα̃, β̃1q, which is given by counting holomorphic disks with
boundary conditions on α̃ and β̃1, is given by the formula d1paq “ dpaq `
dpa, yqdpxq, where dpa, yq is, as above, the coefficient of y in dpaq. To see
this, recall that for a generator x of the Lagrangian Floer chain complex,
the differential is given by
ÿ
dpxq “ npx, yqy
y
where npx, yq counts Maslov index 1 holomorophic disks connecting x to y in
the α, β complex. Now, suppose that we isotope the curve β to a new curve
β1 where β1 results from isotoping β over a Whitney disk that crosses the z
basepoint and cancels the intersection points x and y of minimal filtration
difference. Then by [SRS14][equation 59], the new holomorphic disk count in
the α and β1 complex is given by
n1pa, bq “ npa, bq ` npa, yqnpx, bq
Where a, b P α X β1. This implies that d1paq “ dpaq ` dpa, yqdpxq for
a P CzFKpα̃, β̃1q. Indeed, we have
ÿ ÿ ÿ
d1paq “ n1pa, bqb “ npa, bqb` npa, yq npx, bqb “ dpaq ` dpa, yqdpxq.
b b b
This gives a diagramatic way to run the Alexander filtration spectral se-
quence in a pairing diagram. For example, consider Figures 3.8-3.11. In that
sequence of figures we first see the pairing diagram for HFKpS3, P p3,1qz pT2,3qq
in Figure 3.8. In Figure 3.9, we have indicated all of the Whitney disks that
92
z z z z
w δw,z w δw,z δw,z δw,z
w w
z z z z
w δw,z w w w
x
z z z z
w δw,z w w
w y
z z z z
w δw,z w w w
Figure 3.8. Pairing Figure 3.9. The disks
diagram for shown represent all the
HzFKpS3, P p3,1qpT2,3qq. differentials that lower
Intersection points filtration degree by
labelled x and y one. Cancelling the
satisfy disks by an isotopy, we
Apyq ´ Apxq “ 5 and end up with Figure
Apxq “ 0. 3.10
connect two intersection points of filtration difference one. When we can-
cel these disks by isotoping the β curve over these disks, we arrive at Figure
3.10. In that figure, we have indicated the disks that connect intersection
points of minimal filtration difference. Cancelling these, we arrive at Figure
3.11, where we see three intersection points, two of which are connected by a
Whitney disk, shown in the figure in purple. If we cancel these two genera-
tors we arrive at a pairing diagram with one intersection point. The Alexan-
der grading of this intersection point is then τpP p3,1qpT2,3qq by the discussion
above.
A convenient way to package the entire spectral sequence is shown in
Figure 3.12. Here we see all of the disks we cancelled in the spectral se-
quence, and the A bouys that keep track of the Alexander filtration from the
original complex in all of the subsequent pages. Note that if we draw it like
93
Figure 3.10. The
result of cancelling the
disks in Figure 3.9. Figure 3.11. The
There are two disks result of isotoping β1
that connect in Figure 3.10, we
generators of arrive at a complex
CzFKpα̃, β̃1q of minimal with three generators
filtration difference. and one differential
Cancelling these disks connecting two
we arrive at Figure generators of minimal
3.11 filtration difference
this, we have to cancel all intersection points with filtration difference one
before cancelling any with filtration difference two, etc. We can find the ab-
solute Alexander grading of the generator labelled a (the intersection point
we found to survive the z-basepoint spectral sequence) as follows. By the
symmetry under the elliptic involution, it is easy to see that Apxq “ 0. Then
using Lemma 3.3.1 we have Apaq ´ Apxq “ Apaq “ 5, so τpP p3,1qpT2,3qq “ 5
3.4 Trefoil patterns
In this section we will compute τ of satellite knots with arbitrary com-
panion knot K and pattern P from a family of trefoil patterns that we will
94
c
x
b
a
Figure 3.12. Cancelling all intersection points with filtration difference one
(disk in yellow) and intersection points with filtration difference two (disks
in pink) There are three intersection points remaining.
95
now describe.
Introducing the patterns
The p1, 1q-patterns studied in this paper are constructed by isotoping
the β curve on the doubly pointed bordered Heegaard diagram for the un-
knotted pp, 1q-cable pattern so that we introduce only two extra intersec-
tion points between β and λ. To describe the isotopy, consider first the case
p “ 3. The unknotted p3, 1q-cable pattern is shown in Figure 3.13. Isotope
the β curve by taking the bottom-most horizontal strand and pushing it
once across the longitude of the solid torus. Once we isotope β over the lon-
gitude, we follow the pattern around the meridian until we end up inside the
bigon that contains the z basepoint, without crossing the longitude again.
A intermediate stage of this isotopy is shown in Figure 3.14. If we push the
β curve over the z basepoint, we arrive at the pattern shown in Figure 3.15,
which we will denote by P p3,1q.
In general, we take the bottom most horizontal strand of the β curve in
the genus-1 doubly-pointed bordered Heegaard diagram for the pp, 1q-cable
pattern, push it once over the longitude, and then follow the pattern around
the meridian until we end up inside the bygon that contains the z basepoint.
If we push the β curve over the basepoint, we arrive at a p1, 1q diagram for a
pattern that we denote P pp,1q. The lift of the pattern P pp,1q is shown in Fig-
ure 3.19, where we see that it looks like the lift of the pp, 1q -cable pattern
with one extra arm. By construction, since we only crossed the longitude λ
once in our isotopy, we increased the number of intersections with the lon-
gitude by two. Therefore rkpH 3zFKpS , P pp,1qpUqqq “ 3 for all p ą 1. Alter-
natively, pairing this pattern with C 3zFDpS zUq (whose immersed curve is a
horizontal line) results in three intersection points and no differentials. As
the rank of knot Floer homology detects the trefoil knot [HW18, Corollary
8], we know that P pp,1qpUq has the knot type of the trefoil in S3. As men-
tioned in the introduction, we will call such a pattern P Ă S1 ˆ D2 a trefoil
pattern. In the next section we will use the procedure described in section
96
Figure 3.13. Doubly
pointed Heegaard
diagram for p3, 1q Figure 3.14. Midway
cable pattern through the isotopy
λ
µ z
w
Figure 3.15. Doubly
pointed bordered
Heegaard diagram for Figure 3.16. The
the trefoil pattern trefoil pattern P p3,1q in
P p3,1q the solid torus
3.3 to prove Theorem 3.1.1.
τ of 0-Framed Satellites With Arbitrary Companions
In the previous section we constructed, for each p ą 1, a trefoil pattern
in the solid torus. It follows from [Che19, Lemma 6.3] that wpP pp,1qq “ p` 1.
The pattern P p3,1q is shown in the solid torus in Figure 3.16. In this section
we show how to compute τpP pp,1qpKqq for K an arbitrary knot in S3. As we
97
will see, the answer only depends on the values of τpKq and ϵpKq.
Proof of Theorem 3.1.1. We begin with a discussion of how to determine the
absolute Alexander grading of intersection points representing generators of
HzFKpS3, P pp,1qpKqq in the pairing diagrams in Figures 3.19 and 3.20. For
example, in Figure 3.19 we see a lift of the β curve to the universal cover.
The dotted portions of the β curve represents that the β curve crosses p ´ 3
columns that are not drawn, and the β curve is completely horizontal. If we
focus in on one row, for example the row highlighted in Figure 3.19, we can
determine the relative Alexander grading of all the intersection points by
Lemma 3.3.1. We then determine the relative Alexander grading of all the
other generators by noting that by [Che19, Lemma 6.3], if x and x1 are inter-
section points that occur on arcs of the β curve that differ by a meridional
deck transformation (shifting the picture in the universal cover down a row),
then their Alexander grading difference is wpP q, where wpP q denotes the
winding number of the pattern. For example in Figure 3.19 the intersection
points x and x1 lie on arcs of the β curve that are related by a meridional
deck transformation. It is easy to see that Apxq ´ Apx1q “ p ` 1 “ wpP q.
Now, to determine the absolute Alexander grading, note that the conju-
gation symmetry of knot Floer homology is witnessed in the pairing dia-
gram by the hyperelliptic involution. That is, if we rotate the entire picture
by π, and exchange the z and w basepoints, we will get the same complex.
Therefore, if any intersection is fixed under this involution then it must have
Alexander grading zero. In particular, we can see that the intersection that
is fixed will occur along the arc of the β curve in Figure 3.19 that contains
the point labelled x. Since all intersections along this arc will have Alexan-
der grading zero by Lemma 3.3.1 it is enough to compute Alexander grading
relative to any intersection between αpKq and β that lies on this arc. From
now on, we assume that this has been done and the Alexander gradings that
appear in Figure 3.19 are absolute and not relative.
Now, we turn to discuss how we determine which intersection point sur-
vives the z basepoint Alexander grading spectral sequence. By an isotopy
of the β curve only crossing z basepoints, we can isotope β to the light blue
98
curve in Figures 3.19 and 3.20 in the far right of the diagram. From this ob-
servation, we see that there is a choice of cancelling disks in the pairing dia-
gram so that when we run the spectral sequence, the last remaining intersec-
tion point lies in the far right column of the pairing diagram. This choice of
cancelling disks echos the choice made in the example shown in Figure 3.12
above. Next, recall that the immersed curve αpKq, for a general knot K,
consists of two kinds of components. There is the essential curve component
γ0 with no non-trivial local systems which wraps around the longitude of the
torus and there are (potentially) other components that are immersed with
local systems which all lie in a neighborhood of the meridian. Since β can be
isotoped away from a neighborhood of the meridian by crossing only z base-
points, the intersection point that survives the Alexander filtration spectral
sequence is an intersection between the essential component γ0 and βpP q.
Therefore, since the essential curve component has the form described in
Lemma 3.2.2 and depends only on the values of τpKq and ϵpKq, it remains
to analyse the following cases to determine the absolute Alexander grading
of the generator that survives.
τpKq ą 0, ϵpKq “ 1: In this case, the part of the essential component of the
immersed curve for K coming from the unstable chain slopes upward for
2τpKq rows and turns down at the top and up at the bottom. See Figure
3.17 for an example when p “ 3 and K “ T2,3 and Figure 3.19 for the
general case, where in that figure, we pay attention to the piece of the es-
sential component that is dotted and we only draw the portion of the essen-
tial component of αpKq that carries the intersecion that survives the spec-
tral sequence. In this case we see that the surviving intersection point is the
one labelled a in Figure 3.19. To compute what this Alexander grading is,
we use Lemma 3.3.1. When we follow the β curve from the generator with
Alexander grading 0, labelled x in the figure, we travel down τpKq rows and
then cross one extra δw,z arc. The τpKq rows results in a change in Alexan-
der filtration by wpP qτpKq “ pp ` 1qτpKq, and crossing one more δw,z arc
gives the result:
99
z z z z
w δw,z w δw,z δw,z δw,z
w w
z z z z
w δw,z w w w
x
z z z z
w δw,z w w y
w y
z z z z x
w δw,z w w w
Figure 3.17. The lift
of the trefoil pattern Figure 3.18. The lift
P p3,1q shown in Figure of the trefoil pattern
3.15 paired with the P3 shown in Figure
right handed trefoil. 3.15 paired with the
We have left handed trefoil. We
τpP p3,1qpT2,3qq “ find τpP p3,1qpT2,´3qq “
Apyq “ 5. Apyq “ 0.
τpP pp,1qpKqq “ pp` 1qτpKq ` 1.
For example, in Figure 3.17, we saw earlier that τpP p3,1qpT2,3qq “ 5.
τpKq ą 0, ϵpKq “ ´1: In this case, that part of the essential component of
the immersed curve for K slopes upward for 2τpKq rows, but it turns down
at the bottom and up at the top. See Figure 3.19, where we pay attention
to the solid portion of the essential component of the immersed curve in the
bottom right that turns down and contains the intersection point labelled b.
Since b is the only intersection point remaining after isotoping β to the light
blue curve, we see that Apbq “ τpP pp,1qpKqq. Using Figure 3.19, we see that
this intersection point occurs exactly τpKq`1 rows below the generator with
Alexander grading zero. Using Lemma 3.3.1, we see that
100
z z z z
w w w w
´pp` 1qτpKq
z z z z
w w w w
´pp` 1q
x1
z z z z
w w w w
A “ ´1
A “ 0
x A “ 1
z z
2 p´
z z
1
w w w w
pp` 1qpτpKq ´ 1q
z z z z
w w w w
pp` 1qτpKq a
z z z z
b
w w w w
Figure 3.19. The general case with τpKq ą 0 and ϵpKq “ ˘1. ϵpKq “ 1 is
shown as a dotted arc, and ϵpKq “ ´1 is shown as a solid arc
101
τppP pp,1qqpKqq “ pp` 1qpτpKq ` 1q
τpKq ă 0, ϵpKq “ 1: In this case, the essential component of the immersed
curve α slopes downward for 2τpKq rows, and turns up at the top and down
at the bottom, see Figure 3.20 where in the case ϵpKq “ 1, we focus on the
solid portion of the essential component of αpKq in the upper right, which
contains the intersection point labeled a. Just as in the previous cases we see
that a survives the z basepoint spectral sequence and we find the Alexander
grading of a by counting how may rows above the generator with Alexander
grading zero this intersection point lies. From Figure 3.20 we see that the
intersection point y lives exactly τpKq rows above the intersection point x
with Apxq “ 0. Therefore, by Lemma 3.3.1 Apyq “ wpP qτpKq. Then, we see
that Apaq ´ Apyq “ 1, so we have
Apaq “ τpP pp,1qpKqq “ pp` 1qτpKq ` 1.
τpKq ă 0, ϵpKq “ ´1: This case is similar to the previous cases. Here the
relevant portion of the α immersed curve slopes downward and turns down
at the top and up at the bottom, see Figure 3.20 paying attention to the
dotted portion of the curve in the upper right. The intersection point la-
belled b is the one that survives the z-basepoint spectral sequence. We count
the number of rows above the central row that this intersection point occurs
to compute Apbq. The result is
τpP pp,1qpKqq “ pp` 1qpτpKq ` 1q.
For an example, consider Figure 3.18. We see that the intersection point
that survives the z basepoint spectral sequence lies on both the bold portion
of the α curve and the bold portion of the β curve. It is easy to see from the
picture that τpP p3,1qpT2,´3qq “ 0, since travelling along the bold potion of the
β curve, we do not cross any δw,z arcs.
ϵpKq “ 0: In this case, we also have τpKq “ 0. Hence the essential curve
component is horizontal. Therefore, the intersection point that survives the
102
z z z z
w w w w
A “ pp` 1qτpKq a
y
z z z z
w w w b w
A “ ´pp` 1q
z z z z
w w w w
Ax“ 0
z z z z
w w w w
A “ pp` 1qp|τpKq| ´ 1q
z z z z
w w w w
A “ pp` 1q|τpKq|
z z z z
w w w w
Figure 3.20. The general case with τpKq ă 0 and ϵpKq “ ˘1. ϵpKq “ ´1 is
shown as a dotted arc and ϵpKq “ 1 is shown as a solid arc
103
z basepoint spectral sequence has Alexander grading 1, which is the same as
τpT2,3q as expected.
3.5 Three Genus and Fiberedness
In this section we will prove theorem 3.1.5 from the introduction. Recall
that the knot Floer homology detects both the three-genus and the fibered-
ness of a knot K Ă S3 in the following sense. The genus of a knot is the
largest Alexander grading supporting non-zero Floer homology by [OS04a].
Further, the knot is fibered if and only if the knot Floer homology is one di-
mensional in this top Alexander grading by [Juh08c].
Recall from [HMS08] that, for a non-trivial companion knot, the satel-
lite knot P pKq is fibered if and only in the companion knot K is fibered in
S3 and the pattern is fibered in the solid torus. Therefore, to prove theorem
3.1.5, it is enough to show that P pp,1qpT2,3q is fibered. Furthermore, for a non
trivial knot K, we have the classical genus of a satellite formula
gpP pKqq “ |wpP q|gpKq ` gpP q. (3.5)
So, to compute gpP q it is enough to compute gpP pT2,3qq.
Proof of Lemma 3.1.3. We will make use of the pairing diagram in Figure
3.21 which computes HFKpS3, P pp,1qz pT2,3qq. In the diagram, we see that the
generator a has the largest Alexander grading of any intersection point, and
we compute using Lemma 3.3.1 that Apaq “ p ` 2. Hence gpP pp,1qpT2,3qq “
p` 2. Then using equation 3.5 we have
p` 2 “ gpP pp,1qpT2,3qq “ pp` 1qgpT2,3q ` gpP q “ p` 1` gpP q,
which implies that gpP q “ 1.
With Lemma 3.1.3 in hand, we can prove that the for all p ą 1, pat-
terns P pp,1q are fibered.
104
Proof of Theorem 3.1.5. Since knot Floer homology detects fibered knots,
and a satellite knot is fibered if and only if the pattern and comanion are
fibered, to show that the pattern P pp,1q is fibered it is enough to show that
rkpH 3zFKpS , P pp,1qpT2,3q, p ` 2qq “ 1 for all p ą 1. To this end, consider
the pairing diagram for HzFKpS3, P pp,1qpT2,3qq. By the proof of Lemma 3.1.3,
we know that a has the largest Alexander grading of any intersection point.
To show that P pp,1qpKq is fibered, we will show that for any other intersec-
tion point x in the pairing diagram, we have Apxq ă Apaq. To this end,
note that the Alexander grading is weakly decreasing as we travel up the
pairing diagram on the β curve. It follows from lemma 3.3.1 that Apaq ´
Apbq “ 1 and that Apxq ď Apbq for any other intersection point x. Therefore
HzFKpS3, P pp,1qpT2,3q, p ` 2q is one dimensional for all p ą 1, and so the satel-
lite knot P pp,1qpT2,3q is fibered for all p ą 1. Since a satellite knot with non
trivial companion is fibered if and only if both the pattern is fibered and the
companion is fibered [HMS08], it follows that P is a fibered pattern.
Recall from the introduction that fibered knots have unique minimal
genus Seifert surfaces. Hence, for a fibered pattern P and a fibered knot
K, the satellite knot P pKq also has a unique minimal genus Seifert sur-
face. Given this, one might wonder when the operation of taking a satel-
lite of a non-trivial knot can increase or decrease the number of non-isotopic
Seifert surfaces in the knot complement. In this direction, we prove Proposi-
tions 3.1.6 and 3.1.7 from the introduction, which imply that for knots with
small rank knot Floer homology in the top Alexander grading the process
of taking a satellite with a fibered pattern preserves the property of having
a unique minimal genus Seifert surface as well as the property of having a
depth at most one codimension one taut folitation of the complement.
Proof of Proposition 3.1.6. Suppose K is a knot with rkpH 3zFKpS ,K, gpKqqq ă
4 and P is a fibered pattern. Then by [Juh08a, Theorem 2.3] it follows that
K has a unique minimal genus Seifert surface up to isotopy. By equation
3.2 we have rkpHFKpS3z , P pKq, gpP pKqqqq ă 4. Hence P pKq also has a
unique minimal genus Seifert surface by [Juh08a][Theorem 2.3]. Repeating
105
A “ 0
A “ 1
A “ 2 A “ p´ 1
b
A “ p
A “ p` 1 a
A “ p` 2
Figure 3.21. The pairing diagram computing HzFKpS3, P pp,1qpT2,3qq
106
the above argument, we see that P ipKq has a unique minimal genus Seifert
surface up to isotopy for all i ě 1.
Proof of Proposition 3.1.7. Suppose K is a knot with rkpHzFKpS3, K, gpKqqq “
3. So agpKq, the coefficient of ptg`t´gq in the symmetrised Alexander polyno-
mial for K, is equal to χpHFKpS3z , K, gpKqqq so is non zero. It follows from
[Juh08c, Theorem 1.8] that S3zνpKq has a depth ď 1 taut foliation trans-
verse to BpνpKqq. Then, equation 3.2 implies that rkpHzFKpS3, P pKq, gpP pKqqqq “
3, and so a p p qq “ χpH 3zg P K FKpS , P pKq, gpP pKqqqq ‰ 0, where agpP pKqq is the
analogous coefficient of the Alexnader polynomial for P pKq. So S3zP pKq
has a depth ď 1 taut foliation transverse to BνpKq again by [Juh08c, Theo-
rem 1.8]. This argument can be repeated to show that S3zP ipKq also has a
depth ď 1 taut foliation transverse to BνpP ipKqq for all i ě 1.
3.6 Next to top Alexander grading
In this section we prove Theorem 3.1.10 from the introduction. First,
we recall the notion of right and left veering monodromy following [BNS22].
Suppose that Σ is a surface with non-empty boundary and a and b are two
properly embedded arcs in Σ. We say that a is to the right of b at p, de-
noted a ěp b if p is a common endpoint of both arcs and either a is isotopic
to b rel boundary, or after isotoping a rel boundary so that it intersects b
minimally, a is to the right of b in a neighborhood of p. Now, suppose that
ϕ : Σ Ñ Σ is a homeomorphism of Σ which restricts to the identity on a
boundary component B of Σ. Then we say that ϕ is right veering at B if
ϕpaq ěp a
for every properly embedded arc a Ă Σ and every p P Ba X B. A map ϕ is
called right veering if it is right veering at every boundary component of Σ.
We call a map ϕ left veering if its inverse is right veering.
Recall from Theorem 3.1.8 that, for a fibered knot, we can detect when
the monodromy of a fibration is right or left veering by computing the next
107
b
b
a a
Figure 3.22. P pp,1q Figure 3.23. P pp,1q
paired with a fibered paired with a fibered
knot with thin knot K with
τpKq “ gpKq |τpKq| ă gpKq
to top Alexander graded piece of the knot floer homology to be one dimen-
sional.
Lemma 3.6.1. If K is a fibered knot with τpKq “ ˘gpKq, then
rkpHFKpS3, P pp,1qpKq, gpP pp,1qz pKqq ´ 1qq “ 1.
Proof. Since K is a fibered knot with τpKq “ ˘gpKq, we must have ϵpKq “
sgnpτpKqq and the essential curve component has the form described in
Lemma 3.2.2. Since the knot is fibered, there are no other components of the
immersed curve that pass through at height ´gpKq, so the red arcs shown
in Figure 3.22 are representative of what the immersed curve of a general
fibered knot with τpKq “ ˘gpKq looks like near the bottom row of the
lifted pairing diagram. As we showed in the proof of Theorem 3.1.1, the
knot P pp,1qpKq has one dimensional Floer homology in the top most Alexan-
der grading, and the intersection point labelled a carries this Alexander
grading. Continuing with this reasoning we have that Apbq “ gpP pp,1qpKqq´1
by Lemma 3.3.1, since starting from a, we encounter one δw,z arc before we
reach the intersection point labelled b. Now, no other intersections between
the β curve and αpKq occur before we reach another δw,z arc while travers-
ing β up the diagram. Since the Alexander grading is weakly decreasing as
we travel up the β curve, it follows that b is the unique intersection point
108
with Alexander grading gpP pp,1qpKqq´1, hence HFKpS3, P pp,1qpKq, gpP pp,1qz pKqq´
1q is one dimensional, as desired.
Lemma 3.6.2. Suppose K is a fibered thin knot with |τpKq| ă gpKq. Then
rkpHFKpS3, P pp,1qpKq, gpP pp,1qz pKqq ´ 1qq “ 1
Proof. First, recall from [Pet13] that CFK8 has a simultaneously vertically
and horizontally simplified basis with repect to which it decomposes as a
direct sum of a staircase summand and boxes, and all horizontal and ver-
tical differentials have length one. In this case the immersed curve αpKq
consists of the essential component together with figure eight components,
as shown in figure 3.23. Since |τpKq| ă gpKq, the essential component
of the immersed curve doesn’t pass through at height ˘gpKq and the por-
tion that does consists of a single figure eight component, as shown in Fig-
ure 3.23. We know that the intersection point a represents the sole gener-
ator with Alexander grading gpP pp,1qpKqq, and the generator b has Alexan-
der grading gpP pp,1qpKqq ´ 1. Similar to the proof of Lemma 3.6.1, we see
from the diagram that any other intersection point has Alexander grading
ă gpP pp,1qpKqq ´ 1. Therefore b is the sole intersection point with Alexan-
der grading gpP pp,1qpKqq ´ 1 so HFKpS3, P pp,1qpKq, gpP pp,1qz pKqq ´ 1q is one
dimensional, as desired. .
Proof of Theorem 3.1.10. If K is any fibered knot such that τpKq “ ˘gpKq,
then Lemma 3.6.1 implies that P pp,1qpKq has left or right veering monodromy.
If K is any fibered thin knot such that |τpKq| ă gpKq, then Lemma 3.6.2
implies that P pp,1qpKq has left or right veering monodromy.
Finally, we prove Proposition 3.1.12 from the introduction.
Proof of Proposition 3.1.12. Suppose K is a non-trivial fibered thin knot
with |τpKq| ă gpKq. By Theorem 3.1.10, the fibered knot P pp,1qpKq has
right or left veering monodromy. Therefore, by [BNS22, Corollary 1.7] to
show that P pp,1qpKq is not thin it is enough to show for each p ą 1 that
|τpP pp,1qpKq| ă gpP pp,1qpKqq. By Corollary 3.1.4 we know that gpP pp,1qpKqq “
109
pp` 1qgpKq ` 1. In the case ϵpKq “ 0, Theorem 3.1.1 implies τpP pp,1qpKqq “
1 ă pp ` 1qgpKq ` 1 “ gpP pp,1qpKqq, since gpKq ą 0. In the case ϵpKq “ 1,
Theorem 3.1.1 implies |τpP pp,1qpKqq| ď pp ` 1q|τpKq| ` 1 ă pp ` 1qgpKq `
1 “ gpP pp,1qpKqq, where the strict inequality is by assumption. In the case
ϵpKq “ ´1 we have that ´g ă τpKq ă g. Then |τpP pp,1qpKqq| “ pp `
1q|τpKq ` 1| ă pp` 1qgpKq ă pp` 1qgpKq ` 1 “ gpP pp,1qpKqq.
110
3.7 n-Twisted Satellites with Generalized Mazur Patterns
In this section, we study the immersed curve pairing theorem in the
case that the knot complement has framing n, or equivalently when we add
full twists around the meridian to the pattern knots. We compute the genus
and determine the fiberedness and the Heegaard Floer concordance invari-
ants τ and ϵ of satellite knots with arbitrary companions K and patterns
from a family of knots in the solid torus, which we denote Qi,jn , shown in
Figure 3.24. Here j P Zą0 is the winding number of the pattern, n P Z is
the number of full twists around the meridian, and i P Zě0 denotes the num-
ber of full twists added to the clasp region in the box labelled i in Figure
3.24. We refer to the patterns Qi,jn as n-twisted generalized Mazur patterns,
since Q0,10 is the Mazur pattern and Q
i,1
0 is a generalized Mazur pattern in
analogy with the generalized Whitehead doubles of [Tru16] (See recent work
of [PX24] for a similar family of patterns also called generalized Mazur pat-
terns). Given a knot K, the satellite knot with n-twisted generalized Mazur
pattern Qi,jn pKq can either be viewed as a 0-twisted satellite with pattern
Qi,jn or as an n-twisted satellite with pattern Q
i,j
0 . In this paper, we mostly
adopt the latter perspective.
In [Lev16], Levine computed τ and ϵ of 0-twisted satellites with Mazur
pattern and arbitrary companions by explicitly determining the bordered bi-
module C{FDApXQq associated to the complement of the Mazur pattern in
the solid torus and using the bordered pairing theorem of [LOT18]. Levine
used this to compute τ and ϵ of 0-twisted satellites with Mazur pattern.
More recently, in [CH23], Chen and Hanselman showed that the UV “ 0
quotient of the full knot Floer complex of satellite knots with p1, 1q-patterns
can be computed using the immersed curve pairing theorem. They then re-
covered, in a more direct way, Levine’s computation of τ and ϵ of 0-twisted
satellites with Mazur pattern [CH23, Theorem 6.9].
One consequence of Levine’s computation of ϵ of satellites with Mazur
pattern is that the Mazur pattern does not act surjectively on the smooth
concordance group. Levine then used this to construct a knot in the bound-
111
n
i
Figure 3.24. The pattern Qi,j. In the box labelled i, there are i full twists on
two strands as shown in the box on the bottom left. In the box labelled n
insert n full twists on j ` 2 strands
ary of a contractible 4-manifold that does not bound a PL disk there or
in any other contractible 4-manifold with the same boundary, answering a
question of Kirby and Akbulut [Lev16, Theorem 1.2].
In this work, we extend these computations to determine τ and ϵ of n-
twisted satellites with patterns Qi,j. As a special case of our work, we show
that τ of an n-twisted satellite knot with Mazur pattern and companion K
depends only on the value of n relative to 2τpKq, which echos the computa-
tions of τ of n-twisted Whitehead doubles [Hed07]. Interestingly this is not
the case for τ of satellites with patterns Qi,j with winding number j ą 1,
where we show that the value of τ depends linearly on n and quadratically
on j. Further, we show that for any companion knot K, ϵpQi,jn pKqq ‰ ´1.
This shows that for all i ě 0, j ą 0 and n P Z the patterns Qi,jn do not act
surjectively on the smooth concordance group. See [PX24] for another family
of patterns that have a similar property.
In another direction, we extend recent computations of Petkova and
Wong in [PW21], where they showed that the genus and fiberedness of the
112
n-twisted Mazur pattern in the solid torus can be determined from the bor-
dered type A structure CFApS1 ˆ D2, Q0,1z q, using the bordered pairing the-
orem and classical results about the genus and fiberedness of satellites knots
[Sch53; HMS08]. We expand on these computations and give closed formu-
las for the genus of n-twisted satellite knots with patterns Qi,j and arbitrary
companions, and we determine for which i, j and n the pattern knots Qi,jn
are fibered in the solid torus. We also show that for any non-trivial compan-
ion K the satellite knot Qi,jn pKq is not Floer thin.
Statement of Results
Recall that for the n-twisted satellite knot PnpKq with non-trivial com-
panion knot K, we have [Sch53]
gpPnpKqq “ |wpP q|gpKq ` gpPnq, (3.6)
where wpPnq “ pPn X ptptu ˆD2qq is the winding number of the pattern and
gpPnq is the genus of a relative Seifert surface for Pn. A consequence of this
formula is that to determine gpPnpKqq for an arbitrary non-trivial compan-
ion knot K, it is enough to determine gpPnpT2,3qq. We use this observation
together with the fact that knot Floer homology detects the genus of knots
in S3 to prove the following:
Theorem 3.7.1. For K be a non-trivial knot in S3, j P Zą0, i P Zě0 and
n P Z
$
jpj ` 1q
’
&jgpKq ` n` 1 n ě 0
gpQi,j 2n pKqq “ jpj ` 1q
’
%jgpKq ` |n| ` p1´ jq n ă 0
2
Equation 3.6, and so the proof of Theorem 3.7.1, requires the compan-
ion knot to be non-trivial. However, a similar computation gives gpQi,jn pUqq:
113
Theorem 3.7.2. For j P Zą0, i P Zě0 and n P Z
$
jpj ` 1q
’
’ n` 1 n ą 0
’
’
& 2
gpQi,jn pUqq “ 0 n “ 0
’
’
’jpj ` 1q
’
% |n| ` 1´ j n ă 0
2
Note that when j “ 1 and i “ 0 Theorem 3.7.1 and Theorem 3.7.2
recover [PW21, Theorem 1.0.5]
Recall from [HMS08] that a satellite knot PnpKq is fibered if and only
if the companion knot K is fibered in S3 and the pattern knot Pn is fibered
in S1 ˆD2. This implies that to show that a satellite knot PnpKq is fibered,
it is enough to show that the satellite knot PnpT2,3q is fibered. Since a knot
K Ă S3 with gpKq “ g is fibered in S3 if and only if rankpHzFKpS3, K, gqq “
1 [Ni07; Juh08b], we see that to determine if a pattern Pn is fibered it is
enough to compute the top Alexander graded piece of the knot Floer ho-
mology of P i,jnpT2,3q. For P “ Q , in Lemma 3.9.2 we compute the rank of
the top Alexander graded piece of the knot Floer homology of Qi,jn pT2,3q and
show
Theorem 3.7.3. Let K be a non-trivial fibered knot in S3. Then the satel-
lite knot Qi,jn pKq is fibered if and only if either j ě 2, i “ 0 and n ‰ 0 or
j “ 1, i “ 0 and n ‰ ´1, 0.
Note that the case j “ 1 and i “ 0 of Theorem 3.7.3 recovers [PW21,
Theorem 1.0.6]. The proof of Theorem 3.7.3 actually shows that for any
companion knot K, the rank of H 3 i,j i,jzFKpS ,Qn pKq, gpQn pKqqq is greater than
or equal to i` 1.
Recall that a knot is called Floer thin if all the generators of the knot
Floer homology are supported in the same δ grading, where δpxq “ Mpxq ´
Apxq. We show
Theorem 3.7.4. For any non-trivial companion knot K, the satellite knots
Qi,jn pKq are not Floer thin.
We also consider the case when the companion knot is trivial.
114
Theorem 3.7.5. For K “ U the satellite knots Qi,jn pUq are Floer thin if and
only if j “ 1 and n “ ´1.
Note that Theorems 3.7.4 and 3.7.5, in the case i “ 0 and j “ 1, recover
[PW21, Theorem 1.01].
In [OS03b] and [Hom14] two smooth concordance invariants of knots derived
from the UV “ 0 quotient of the full knot Floer complex CFK8 are intro-
duced, called τpKq and ϵpKq. These invariants have proved fruitful in the
study of the knot concordance group [Hed07; Hom14; Lev16]. We give an
explicit computation of τ and ϵ of satellite knots with arbitrary companion
knots K and patterns Qi,jn .
Theorem 3.7.6. If K is a knot in S3 with ϵpKq “ ´1, then for all i ě 0,
j ě 1 and n P Z
τpQi,jn pKqq “
jpj ´ 1q
jpτpKq ` 1q ` n.
2
If K is a knot in S3 with ϵpKq “ 1, then for all i ě 0, j ě 1 and n P Z
$
jpj ´ 1q
’
&jτpKq ` n` 1 n ă 2τpKq
τpQi,jn pKqq “ 2
p q ` jpj ´ 1q’%jτ K n n ě 2τpKq
2
If K is a knot in S3 with ϵpKq “ 0, then for all i ě 0, j ě 1 and n P Z
$
jpj ´ 1q
’
& n n ě 0
τpQi,jn pKqq “ 2jpj ´ 1q
’
% n` j n ă 0
2
Theorem 3.7.7. For any knot K and for any i ě 0, j ě 1 and n P Z, we
have ϵpQi,jn pKqq P t0, 1u.
The invariant ϵ is a concordance invariant, and takes values in t0, 1,´1u.
If we let CQ denote the rational homology knot concordance group (for the
definition see [Lev16]) then an immediate Corollary of Theorem 3.7.7 is
Corollary 3.7.8. F or all i ě 0, j ě 1 and n P Z, the satellite operators
Qi,jn : CQ Ñ CQ are not surjective.
115
As mentioned above, this shows that when we add full twists to the
clasp region of the Mazur pattern (by increasing the parameter i) and when
we add meridian twists to the Mazur pattern (by changing the framing n)
we get a bi-infinite family of winding number 1 patterns that do not act sur-
jectively on the smooth (or Q-homology) concordance group and thus gives
infinitely many examples of knots in homology spheres that do not bound
PL disks in any contractible 4-manifold. See the recent work of [PX24] for
another infinite family of winding number 1 unknot patterns with the same
property. Our construction also gives many patterns of arbitrarily large
winding number and various knot types in S3 that also are not surjective
satellite operators, and in particular shows that for these patterns, the im-
age of the concordance invariant ϵ is not sensitive to twisting the pattern or
changing the framing of the pattern knot complement.
3.8 Background
In this section we review some concepts from the immersed curve refor-
mulation of bordered Floer homology and the bordered pairing theorem for
p1, 1q-patterns. We assume the reader is familiar with the various flavors of
knot Floer homology and the work of [LOT18]. We quickly review the neces-
sary background to state the immersed curve reformulation of the bordered
invariants and bordered pairing theorem from [Che19; CH23; HRW22]. In
Section 3.8 we introduce some notation and prove a structure theorem for
the immersed curve associated to an n framed knot complement. Then in
Section 3.8 we discuss p1, 1q-patterns and the work of [Che19] with an eye
towards extracting the UV “ 0 quotient of the knot Floer complex from the
pairing diagram as in [CH23], and then in Section 3.8 we discuss the specific
family of p1, 1q-patterns that gives rise to the patterns knots Qi,j.
Immersed Curves for n-Framed Knot Complements
Note that the pair pS3, PnpKqq can be obtained by gluing S3 ´ νpKq
with framing n to pS1 ˆ D2, P q or by gluing S3 ´ νpKq with framing 0 to
116
λ ρ ξ3 0
ρ2
ξ0 λ
κ η0
ρ123 ρ1
η0 κ
Figure 3.25. Type D Figure 3.26. Type D
structure for structure for
complement of knot K complement of knot K
with τpKq ą 0 and with τpKq ą 0 and
ϵpKq “ 1, where we ϵpKq “ ´1, where we
replace the dotted replace the dotted
arrow from ξ0 to η0 by arrow from ξ0 to η0 by
the appropriate the appropriate
unstable chain unstable chain
the pair pS1 ˆ D2, Pnq. We want to study the pairing CzFDpS3 ´ νpKq, nq b
CzFApS1 ˆ D2, P q which computes CzFKpS3, PnpKqq from the perspective of
immersed curves. With this goal in mind, we want to understand the essen-
tial component of the immersed curve associated to an n-framed knot com-
plement. This lemma is a generalization of Lemma 3.2.2 to the case when
the framing of the knot complement is arbitrary.
Definition 3.8.1 ([HRW22; HRW17; HW19]). Given a knot K Ă S3, let
αpK,nq denote the immersed multi-curve representing the type D structure
CzFDpS3 ´ νpKq, nq.
As in [HW19, Proposition 2] we single out a special component of the
immersed multi-curve αpK,nq, denoted γ0 and called the essential compo-
nent of the immersed curve (See also [HRW22, pp 43-44]). As mentioned
there, if we lift the curve to R2 and consider the vertical axes Z ˆ R, then
the essential component is the only component of the immersed curve that
117
crosses from tiu ˆ R to ti ` 1u ˆ R and in the case when the framing n “ 0,
this potion of the essential component of the immersed curve component has
slope 2τpKq (That is is spans 2τpKq rows) and it either turns up, down or
continues straight after passing through ti ` 1u ˆ R if ϵpKq “ ´1, 1 or 0
respectively. We extend these observations to a structure theorem for a por-
tion of γ0 of n framed knot complements.
Lemma 3.8.2. Suppose τpKq ě 0 and ϵpKq “ 1. If n ă 2τpKq, then the es-
sential component of the immersed curve has slope 2τpKq´n and turns down
immediately after passing through ti` 1uˆR, see Figure 3.27. If n ě 2τpKq,
then the essential component of the immersed curve has slope 2τpKq ´ n and
turns down immediately after crossing through ti` 1u ˆ R, see Figure 3.28.
Proof. When ϵpKq “ 1, by [Hom14] there is a reduced horizontally sim-
plified basis so that the vertically distinguished generator ξ0 of CFK´pKq
is an element of this horizontally simplified basis and occurs at the end of
a horizontal arrow (symmetrically the horizontally distinguished generator
η0 occurs at the end of a vertical arrow). If τpKq ě 0 then the algorithm
from [LOT18, Theorem 11.26] shows that the type D structure contains the
portion shown in Figure 3.25, where the dotted arrow is replaced by the ap-
propriate unstable chain.
Then the algorithm in [HRW17, Sections 2.3-2.4] shows that the essen-
tial component of the immersed curve lifted to the cover R2zπ´1pzq has the
form shown. In Figure 3.27 and 3.28 we see the resulting curves for n ă
2τpKq and n ě 2τpKq respectively and indicate how the curves are built
from the type D structure. Intersections with the vertical lines in the fig-
ure correspond to generators of ι 3z0CFDpS ´ νpKq, nq and the intersections
with the horizontal lines correspond to generators of ι1CzFDpS3´ νpKq, nq. If
δpxq “ ρI b y ` ¨ ¨ ¨ , then there is an arc ρI from x to y, as described in the
figures.
Similarly, we can show
118
ξ0
κ
ξ
ρ 0µ1 1 ρ2
λ η0ρ123 ρ2
µm
ρ23
ρ23
ρ123 ξ0
ρ23 ρ2
λ
κ µm
ρ ρ123 3η0
κ
η0
tiu ˆ R ti` 1u ˆ R tiu ˆ R ti` 1u ˆ R
Figure 3.27. The Figure 3.28. The
unstable portion of unstable portion of
αpK,nq with τpKq ě 0 αpK,nq with τpKq ě 0
and ϵpKq “ 1 and and ϵpKq “ 1 and
2τpKq ą n n ě 2τpKq
119
2τpKq ´ n
2τpKq n
n´ 2τpKq
2τpKq
ρ23
ρ λ
ρ 3123
ξ0 ρ3
ρ1
ξ
µ 01
ρ23
µi
ρ1 η0
ρ3 ρ23
η0
µj κ
ρ23
ρ23
µm
ρ1 η0 ρ3
tiu ˆ R ti` 1u ˆ R tiu ˆ R ti` 1u ˆ R
Figure 3.29. The Figure 3.30. The
unstable portion of unstable portion of
αpK,nq with τpKq ě 0 αpK,nq with τpKq ě 0
and ϵpKq “ ´1 and and ϵpKq “ ´1 and
n ě 2τpKq n ď 0 ď 2τpKq
120
Lemma 3.8.3. Suppose τpKq ě 0 and ϵpKq “ ´1. If n ě 2τpKq, then the
essential component of the immersed curve has slope 2τpKq ´ n and turns up
after crossing through ti ` 1u ˆ R, see Figure 3.29. If n ă 2τpKq, then the
essential component of the immersed curve has slope 2τpKq ´ n and turns up
after crossing through ti` 1u ˆ R, see Figure 3.30.
The statements about the form of the essential component of the im-
mersed curve in the case τpKq ď 0 are similar. In summary, the essential
component of the immersed curve has slope 2τpKq ´ n and turns up, down
or continues straight depending on whether ϵpKq “ ´1, 1 or 0.
(1,1)-Unknot Patterns
In this section, we review some notation and results about p1, 1q un-
knot patterns. In the case that the p1, 1q pattern knot P is an unknot pat-
tern, meaning that P pUq „ U , Chen showed that the β curve for the genus
1 doubly-pointed Heegaard diagram for P can be encoded by two integers
pr, sq, where gcdp2r´1, s`1q “ 1 [Che19, Theorem 5.1]. In this parametriza-
tion, r denotes the number of rainbows and s denotes the number of stripes
(see [Che19, Figure 15]). The pattern described by the pair pr, sq corresponds
to the two bridge link bp2|s| ` 4|r|, ϵprqp2|r| ´ 1qq [Che19, Theorem 5.4].
For example, see Figure 3.31 where we have drawn the doubly pointed
Bordered Heegaard diagram for the unknot pattern described by the pair
pr, sq “ p4, 2q, and Figure 3.32 where we have drawn the same genus 1
bordered Heegaard diagram with the pattern knot that it determines. In
general, the knot determined by the p1, 1q unknot pattern given by the pair
pr, sq has a presentation with r ´ 1 rainbow arcs and s` 1 stripes, see Figure
3.38.
As above, let CzFKpα, β, z, wq denote the intersection Floer homology
of the two curves α and β in T 2ztz, wu as described in [Che19, Theorem
1.2]. The generators of CzFKpα, β, z, wq are the intersection points of the two
curves, and the differential counts embedded bigons with left boundary on
the β curve and right boundary on the α curve. As proved in [CH23], we
121
z
w
Figure 3.32. The knot
in green in S1 ˆD2
Figure 3.31. The p1, 1q determined by the
pattern determined by p1, 1q pattern with β
the pair pr, sq “ p4, 2q curve the blue curve
can recover the UV “ 0 quotient of the full knot Floer complex by consid-
ering disks that contain either z or w basepoints (but not both) and label
them by V and U respectively. The component of the differential induced by
counting bigons crossing the z basepoint will be called vertical differentials
and denoted Bv, and those crossing the w basepoint horizontal differentials
and denoted Bh.
Now, if π : R2 Ñ T 2 denotes the universal cover of the torus, let β̃ be
a connected component of π´1pβq in R2zpπ´1tz, wuq and let α̃pK,nq be a lift
of αpK,nq to R2, as in Figures 3.27-3.30. Then by [Che19, Proof of Theorem
1.2] C ´1 ´1zFKpα̃, β̃, π pzq, π pwqq – CzFKpα, β, z, wq. Indeed, it is easy to see
that there is a correspondence at the level of generators, and it is similarly
straightforward to see that differentials on both sides agree. See Figures
3.33, 3.35, and 3.36. Throughout we work with the lifted pairing diagram.
We assume that the intersection between the two curves is reduced, mean-
ing that the only bigons contributing to the differential are the bigons that
cross either the z or the w basepoint, this is easily obtained by an isotopy of
122
α̃pT2,3, 0q β̃
A “ 3
f
z z z z a z
j
w w b w w w
e d
c
g A “ 0
z z z z z
i
w w w w w
h
A “ ´3
Figure 3.33. The lifted pairing diagram for CzFKpα̃pT2,3, 0q, β̃, w, zq
αpK,nq across the Whitney disks that don’t contain a basepoint. With these
conventions, the following is proved in [Che19, Theorem 1.2 and Lemma 4.1]
and [CH23, Theorem 6.1]:
Theorem 3.8.4. For P a p1, 1q pattern, HzFKpS3, P znpKqq “ CFKpα̃pK,nq, β̃pP qq
and moreover CFK 3FrU,V s{UV pS , PnpKqq – pCzFKpα̃pK,nq, β̃, z, wq, Bv, Bhq.
Furthermore, given two intersection points x and y between α̃pK,nq and
β̃pP q, Apyq ´ Apxq “ ℓx,y ¨ δw,z, where ℓx,y is an arc on the β curve that
goes from x to y and A denotes the Alexander grading of generators of the
knot Floer homology.
See Figure 3.33 for an example, where we have drawn the lifted pair-
ing diagram for the satellite knot Q0,30 pT2,3q. In that figure, we have labelled
some intersection points, and drawn the δw,z arcs. Theorem 3.8.4 implies
that the intersection points are in bijection with the generators of the knot
Floer homology HzFKpS3, Q0,30 pT2,3qq. Moreover, by taking an arc along the
β curve from c to a, for example, we see that Apaq ´ Apcq “ ´1. The knot
Floer homology has a symmetry given by HzFKpS3, K,Aq – HzFKpS3, K,´Aq,
and we can see this symmetry in the pairing diagram by rotating the whole
picture by π and exchanging the w and z basepoints. It follows that Apcq “
0 and we can always upgrade the relative Alexander grading given by Theo-
123
d U j b
V V
k
V 3
h U i
V V
g l
U
Figure 3.34. The piece of the complex CFKFrU,V s{UV pS3, Q0,30 pT2,3qq that
contains the intersection point d with Apdq “ τpQ0,30 pT2,3qq and d` h
generates HxFpS3q.
rem 3.8.4 to an absolute Alexander grading. In Figure 3.33 we find Apbq “ 3,
Apeq “ Apdq “ 4 and Apfq “ 3.
Another consequence of Theorem 3.8.4 is that since we can recover the
UV “ 0 quotient of the full knot Floer complex, we can compute both τ
and ϵ of satellite knots with p1, 1q-patterns. We return to this in Section 3.11
later, but we remark here that by counting disks that cross only the z base-
point in Figure 3.33, the intersection points d, g, and h form a subcomplex
of CzFKpQ0,30 pT2,3qq such that the cycle d ` h generates HxFpS3q (obtained by
setting V “ 1 in the above subcomplex). This cycle can be extended to a
vertically simplified basis of CFK´pQ0,30 pT2,3qq in the sense of [Hom14, Sec-
tion 2]. Moreover, the intersection points i and j satisfy Bhpi ` jq “ d ` h,
so the distinguished element of the vertically simplified basis is in the im-
age of the horizontal differential and this implies [Hom14, Section 3] that
ϵpQ0,30 pT2,3qq “ 1. Further, it is easy to see that the intersection point d
satisfies Apdq “ τpQ0,30 pKqq. See Figure 3.34, where we have indicated a
portion of the complex over FrU, V s{UV . Note that the above argument
only involved intersection points between the unstable portion of the curve
αpT2,3, 0q in the first column and the β curve. We return to this observation
in section 3.11, where we see that this holds in general for the patterns given
124
α̃pT2,3, 1q A “ 9 A “ 9 α̃pT2,3,´1q
A “ 6 A “ 6
A “ 3 A “ 3
Figure 3.35. The Figure 3.36. The
pairing diagram for pairing diagram for
CzFKpαpT2,3, 1q, βpQ0,3qq CzFKpαpT2,3,´1q, βpQ0,3qq
by the β curve βpi, jq.
The pairing diagrams and their lifts become more complicated when we
consider knots with non-zero framing since the unstable chain gets longer
for most values of n, which we need for computing the knot Floer homology
of satellites with n-twisted patterns. For example, see Figures 3.35 and 3.36
where we have the pairing diagram for Q0,3´1pT2,3q and Q
0,3
1 pT2,3q. In those
figures, the intersection point c satisfies Apcq “ 0 and we have indicated
some of the Alexander gradings of intersection points.
The Curves βpi, jq
In this section we introduce the specific p1, 1q-patterns that give rise to
the pattern knots Qi,j shown in Figure 3.24.
Definition 3.8.5. Let βpi, jq denote the β curve for the p1, 1q pattern which
in the parameterization of [Che19] is given by pr, sq “ p2` j ` 2pj ` 1qi, jq.
The doubly pointed bordered Heegaard diagram associated with βpi, jq
125
2` j ` 2pj ` 1qi j ` 2pj ` 1qi` 1
¨ ¨ ¨ ¨
¨
¨
i
1q
`
2pj
`
1 ¨ ¨ ¨
z
¨ ¨ ¨
w
2` j ` 2pj ` 1qi` j 1` 2pj ` 1qi
Figure 3.37. The p1, 1q pattern that determines the pattern knot Qi,j.
Figure 3.31 shows the case i “ 0 and j “ 2
is shown in Figure 3.37, and from that description it is easy to see that knot
determined by the p1, 1q pattern with β curve βpi, jq is shown in Figure 3.38.
In that figure there are r ´ 1 “ 1 ` j ` 2ip1 ` jq “ p2i ` 1qp1 ` jq rainbows
and s ` 1 “ j ` 1 stripes. Each pair of strands represents j ` 1 parallel
strands, as indicated, and there are 2ipj ` 1q of them. If we pull the p2i `
1qp1 ` jq rainbows from the left side of the figure around the orange arc, we
end up with Figure 3.39. In that figure the bold line represents j consecutive
strands. We isotope the j strands by pulling i the bold piece of the knot,
and end up at Figure 3.40. Here there are j strands winding around the hole
of the torus and 2i ` 1 rainbows. It is straightforward to verify that this is
the knot Qi,j0 shown in Figure 3.24.
126
¨ ¨ ¨
2` 2j j
1
j `
` 1
j ` j
1
j ` 1
j j ` 1`
1 j ` 1
Figure 3.38. The knot in S1 ˆD2 determined by the p1, 1q pattern with
β “ βpi, jq
In order to understand the pairing diagram for CzFKpα̃pK,nq, β̃pi, jqq we
make some observations about the lifted β curve β̃pi, jq. When i “ 0 the
curve βp0, jq is determined by the pair pr, sq “ p2 ` j, jq. In this case, it is
easy to see that the lift β̃p0, jq has the form shown in Figure 3.41 top row.
Indeed, each “wave” contributes one to the count of rainbows, and there are
j ` 1 “waves”, and there is one extra rainbow at the left end. Said another
way, the lifted β curve β̃p0, jq is obtained from β̃p0, j ´ 1q by the finger move
shown in Figure 3.41 and this isotopy introduces one more rainbow and one
more stripe to β̃p0, j ´ 1q.
Next we claim that the transition from β̃p0, jq to β̃p1, jq corresponds to
“twisting up” each wave, which is shown in Figure 3.43. Indeed, here we see
127
Figure 3.40. Isotope the j
consecutive strands that
Figure 3.39. The knot are bold in Figure 3.39 to
from Figure 3.38 after an obtain this knot, which is
isotopy Qi,j0
z z z z z z z z
w w w w w w w w
z z z z z z z z
w w w w w w w w
z z z z z z z z
w w w w w w w w
j
1 2 ` 1 j j ` 1
2
Figure 3.41. The isotopy that produces βp0, j ` 1q from βp0, jq.
128
A´ 1 ¨ ¨ ¨
z z z
A´ 1 ¨ ¨ ¨ A´ 1A´ A2 ´ 3 δw,z
z z z A
A w w w
A ´´ 2 A´ 1
w 1w w A ¨ ¨ ¨A´
A 2 ¨ ¨ ¨
Figure 3.43. twist up
Figure 3.42. The curve the curve β̃p0, jq to get
β̃p0, jq for the knot the curve β̃p1, jq for
Q0,j the knot Q1,j
A´ 1 ¨ ¨ ¨
A´ 1 ¨ ¨ ¨ A´ 2A´ 1
A´ 2
´ ´ A´ 1A 2 A 3 A
A´ 1 A´ 2 A´ 1A ¨ ¨ ¨
A A´ 1 A´ 2
¨ ¨ ¨
Figure 3.45. The
Figure 3.44. The collapsed β̃p1, jq curve
collapsed β̃p0, jq curve for the knot Q1,j
for the knot Q0,j
that twisting up adds an extra 2 rainbows for each wave region, and thus
2pj ` 1q new rainbows in total. In general, β̃pi, jq is obtained from β̃p0, jq
by twisting up each wave region i times, and we see that this corresponds to
adding 2pj ` 1qi new rainbows, and no new stripes, to the lifted β curve.
For convenience we label the arcs of the β curves between lifts of the
δw,z arcs by relative Alexander gradings that an intersection between αpK,nq
and βpi, jq on that arc would carry if there were intersections on that arc. It
is straightforward to see that these Alexander grading labels increase as we
move from right to left and bottom to top along the lift β̃pi, jq. Moreover,
from the description of twisting up and [Che19, Lemma 4.1] the following
lemma is immediate (see Figures 3.42-3.43).
129
´ 1
A
Lemma 3.8.6. For any knot K and for any i ą 0, we have
tA : HFKpS3, Qi,jz n pKq, Aq ‰ 0u “ tA : HzFKpS3, Q0,jn pKq, Aq ‰ 0u
In order to simplify arguments and pictures in the next section, we in-
troduce a modified version of the lifted β curve, called the collapsed β curve.
Definition 3.8.7. Let Bpi, jq denote the curve β̃pi, jq after collapsing the
lifts of the arcs δw,z to a single point
See Figure 3.44 and 3.45 where we draw Bp0, jq and Bp1, jq together
with the Alexander gradings of arcs. The following lemma is immediate.
Lemma 3.8.8. As an F-vector space, pairing with the collapsed β curve is
the same as pairing with the β curve: CzFKpα̃pK,nq, Bpi, jqq – CzFKpα̃pK,nq, β̃pi, jqq
and moreover, we can recover the Alexander grading of any intersection
point in the collapsed pairing diagram.
Although twisting up does not change the set of Alexander gradings la-
belling arcs of the β curves by Lemma 3.8.6, twisting up does change the
number of arcs of the collapsed β curve that are labelled with a fixed Alexan-
der grading. We will return to this observation in section 3.9 (see Lemma
3.9.2).
3.9 Three-Dimensional Invariants
In this section we compute the genus of the patterns Qi,jn , determine the
set of triples pi, j, nq so that the pattern Qi,jn is fibered in the solid torus, and
show that whenever K is a non-trivial companion the satellite knots Qi,jn pKq
are not Floer thin.
Three-Genus and n-twisted Satellites
In this section we use Theorem 3.8.4 and the collapsed pairing diagram
for n-framed satellite knots to prove Theorems 3.7.1 and 3.7.2 from the in-
troduction. Recall that our strategy is to determine gpQi,jn pT2,3qq directly
130
from the pairing diagram and deduce the forumla for a general non-trivial
companion from Equation 3.6. An immediate Corollary of Theorem 3.7.1 is
a computation of the genus of the n-twisted pattern knot Qi,jn in S1 ˆD2.
Corollary 3.9.1. For n P Z, i P Zě0 and j P Zą0, the pattern knot Qi,jn in
S1 ˆD2 has genus
$
&1 n ě 0
gpQi,jn q “
jpj ` 1q |n| `
2 %1´ j n ă 0
Proof. Equation 3.6 shows that
gpQi,jn q “ gpQi,jn pT2,3qq ´ jgpT2,3q “ gpQi,jn pT2,3qq ´ j.
To prove Theorems 3.7.1 and 3.7.2, we will make use of the collapsed
pairing diagram. Note first that since gpKq “ maxtA : HFKpS3z , K,Aq ‰
0u, Lemma 3.8.6 implies that gpQ0,jn pKqq “ gpQi,jn pKqq, so it is enough to
consider the case i “ 0.
In Figures 3.46-3.48, we see the top half of the lifted pairing diagram
CzFKpα̃pT2,3, nq, β̃p0, jqq. The other half is determined by the symmetry of
the pairing diagram coming from the symmetry of knot Floer homology. We
work with the collapsed pairing diagram to simplify the pictures, since we
are not interested in any of the differentials and only in the Alexander grad-
ings in this section. Note that by [Che19, Lemma 6.3], the Alexander grad-
ing of intersection points of αpT2,3q and βp0, jq increase by ´wpQ0,jn q “ j
as we go up one row in the pairing diagram, so to determine the largest
Alexander grading of an intersection point in the pairing diagram, it is enough
to determine the number of rows between the central intersection point c
(with Apcq “ 0) and the top of the pairing diagram.
Proof of Theorem 3.7.1. As mentioned, by Lemma 3.8.6, it is enough to
determine the genus in the case i “ 0, and by Equation 3.6 it is enough
to compute gpQ0,jn pT2,3qq. To this end, consider first the case n ě 0. It is
131
αpK,nq b
αpK,´nq
a
a1
jpj ´ 1q
j ` n
2
j
´ ´1j ´ 1 j 2 0
j j ´ 1 1 0
j ` 1 j 2 1
0 c
Figure 3.46. The pairing diagram for Q0,jn when j is odd and n ą 0
132
n
ˆ ˙
1 j ´ 1 n
2
easy to see that the intersection point labelled a in Figures 3.46-3.48 has
the largest Alexander grading. Indeed, the Alexander gradings increase by
j for each row we go up in the pairing diagram and the Alexander gradings
labelling each arc in the collapsed pairing diagram increase by one for each
column we go over from right to left in the pairing diagram. To determine
Apaq, note that there are a total of 2pj ` 2q ` pn ´ 2qpj ` 1q lifts of the
curve βp0, jq needed to account for all the intersections between αpT2,3, nq
and βp0, jq. Indeed there are pj ` 2q lifts of the CFK8pT2,3q region (which
occupies 2 rows) and there are pj ` 1q lifts of the unstable region, which
spans n ´ 2 rows. There are then three cases to distinguish. If j is odd, then
there are an even number of rows and the central intersection point occurs
between these rows. Moreover, since there are an odd number of CFK8pT2,3q
regions, by symmetry of the pairing diagram the central intersection point
occurs in the middle of the central CFK8 region of the curve. See Figure
3.46. If j is even, then there are an odd number of lifts of the unstable re-
gion and so the central intersection point occurs somewhere along the un-
stable region. If n is even or odd, then the number of rows is either even or
odd. If n is even, we are in the situation pictured in Figure 3.48 and if n
is odd, we are in the situation pictured in Figure 3.47. In any case, to de-
termine Apaq it is enough to count the number of rows between the central
intersection point c and the intersection point labelled b. In all the cases the
number of rows are indicated in the figure. We describe the case j even and
n odd in detail and leave the rest to the reader.
In Figure 3.47 the central intersection point c occurs on the central lift
n´ 1
of the unstable chain. There are rows between that intersection point
2
j ´ 2
and the intersection point labelled d in Figure 3.47. Then, there are n
2
rows between the interesction point d and the intersection point e, and n` 1
rows between e and b. Therefore,
ˆ ˙
p q ´ p q “ n´ 1 ` j ´ 2A b A c j n` n` 1 .
2 2
j
Then it is straightforward to verify that Apaq ´ Apbq “ ` 1. Simplifying,
2
133
αpK,nq αpK,´nq
a b a1
e
d
c
Figure 3.47. The general pairing diagram for j even and n odd
we see that when n ě 0
p i,jp qq “ p q “ ` jpj ` 1qg Qn T2,3 A a j n` 12
This finished the proof in the case n ě 0. When n ă 0, Figures 3.46-3.48
show both the curve αpK,nq and the curve αpK,´nq and we can see that
the difference gpQi,j pT i,j´n 2,3qq ´ gpQn pT2,3qq “ j. For example in Figure 3.47,
the intersection point with the largest Alexander grading in the pairing with
αpK,´nq is labelled a1: Apa1q “ gpQ0,j 1´npKqq. Then Apaq´Apa q “ ℓa1,a ¨δw,z “
jpj ` 1q
j. Therefore when n ă 0, we have gpQi,jn pT2,3qq “ j ` |n| ` 1´ j2
134
n´ 1 j ´ 2
n
2 2
n 1
αpK,nq αpK,´nq
a1
a
Figure 3.48. The general pairing diagram for j even and n even
Proof of Theorem 3.7.2. The computation of the genus when the compan-
ion knot is the unknot is similar to the proof of Theorem 3.7.1 and left to
the reader (see Case 0 in the proof of Theorem 3.7.6 for the relevant pairing
diagram).
Fiberedness
In this section we prove Theorem 3.7.3. By [HMS08] a necessary con-
dition for a satellite to be fibered is for the companion to be fibered and to
135
n
ˆ ˙
n´ 2 j ´ 2
n
2 2
a
g ´ 2
g ´ 1
a g1 a2
0
Figure 3.50. The
pairing diagram for
Q0,jn pT2,3q when
n ă ´1. The
Figure 3.49. The lifted Alexander grading
pairing diagram for labels of the β arcs are
H 0,jzFKpQ0 pT2,3qq as in Figure 3.51
determine the fiberedness of any satellite Qi,jn pKq with fibered companion, it
is enough to determine if the satellite knot Qi,jn pT2,3q is fibered.
Now, recall that a knot K in S3 is fibered if and only if HzFKpS3, K, gpKqq
has rank one [Ni07; Juh08b]. In the previous section we determined that
largest Alexander grading, so the genus, of any satellite knot with pattern
Qi,jn and companion K. In this section, we will determine the rank of the
knot Floer homology of Qi,jn pT2,3q in Alexander grading gpQi,jn pT2,3qq. Us-
ing this, we will show when this has rank one. In the following, let g “
gpQi,jn pT2,3qq
Lemma 3.9.2.
$
’
’2pi` 1q if n “ 0 and j ě 1
’
&
dimpHzFKpS3, Qi,jn pT2,3q, gqq “ 2pi` 1q if n “ ´1 and j “ 1
’
’
’
%pi` 1q else
Proof. We will first determine the rank of HFKpS3, Q0,jz n pT2,3q, gq then we will
see how the rank changes when we increase i by twisting up the β curve.
136
g ´ 2
g ´ 1
a
g
g ´ 1
g ´ 1´ j g ´ 2´ j
g ´ j g ´ 1´ j
g ´ j ` 1
b g ´ j
g ´ 1´ j
g ´ 1´ 2j g ´ 2´ 2j
g ´ 2j g ´ 1´ 2j
g ´ 2j ` 1 g ´ 2j
Figure 3.51. The general pairing diagram showing intersection points with
largest possible Alexander grading when i “ 0 and n “ ´1. For each
increase in i, there is one more arc in the top right with Alexander grading
label g, and one more arc in the second to top row and second to right
column with label g ´ 1` j
137
Suppose first that n “ 0, In this case there are two intersection points in
the top row of the pairing diagram that contribute to HzFKpS3, Q0,j0 pT2,3q, gq,
shown in Figure 3.49 and labelled a1 and a2. Direct inspection shows that
there are no other intersection points in this Alexander grading. Therefore
dimpHFKpS3, Q0,jz 0 pT2,3qqq “ 2.
Suppose now that n ă 0. We first deal with the case n ă ´1. The
pairing diagram for this case is shown in Figure 3.50. In that figure we see
that there is one intersection point with Alexander grading gpQ0,jn pT2,3qq
labelled a in the top row of that figure. Inspection of the pairing diagram
shows that all the intersection points in the lower rows of the pairing dia-
gram all carry Alexander gradings ă g regardless of the value of j. Hence
dimpH 3zFKpS ,Q0,jn pT2,3q, gqq “ 1 when n ă ´1 and j ě 1.
The pairing diagram for the case n “ ´1 is shown in 3.51. In that fig-
ure, we see that there is one intersection point in Alexander grading g in the
top row of the pairing diagram, labelled a. All other arcs of the β curve in
this row (and thus all other intersection points in this row) carry an Alexan-
der grading label ă g. Consider the next to top row of the pairing diagram.
The largest possible Alexander grading is the Alexander grading of the inter-
section point labelled b, which is g ´ j ` 1. This is always strictly less than
g unless j “ 1. Further, regardless of the value of j, all other intersection
points carry an Alexander grading ď g ´ 1. So in the case that n “ ´1, we
see that dimpHFKpS3, Q0,1z ´1pT2,3q, gqq “ 2 and dimpHFKpS3, Q
0,j
z
´1pT2,3q, gqq “ 1
when j ą 1.
The case that n ě 1 is similar. In that case we see that for all j ě 1 and
n ě 1, rkpHFKpS3, Q0,jz n pT2,3q, gqq “ 1.
This proves the theorem in the case i “ 0. To deal with the cases i ą 0,
recall that the lifted curve β̃pi, jq is obtained from the lifted curve β̃pi ´ 1, jq
by twisting up, as shown in Figure 3.43. We see in Figures 3.52 and 3.53
that for each intersection point of α̃pT2,3, nq with β̃pi ´ 1, jq in Alexander
grading g, there is one more intersection point of α̃pT2,3, nq with β̃pi, jq in
that same Alexander grading. The theorem follows.
With Lemma 3.9.2 in hand, we can prove Theorem 3.7.3 from the intro-
138
Figure 3.53. The top
Figure 3.52. The top left of the pairing
left of the pairing diagram when n ą 0
diagram when n ą 0 and i “ 3. The
and i “ 2. The intersection points
intersection points connected by a spiral
connected by a spiral are in the same
are in Alexander Alexander grading
grading g “ gpQi,jn pKqq g “ gpQi,jn pKqq
duction.
Proof of Theorem 3.7.3. By [HMS08], the pattern knot Qi,jn is fibered in
S1 ˆ D2 if and only if the satellite knot Qi,jn pT2,3q is fibered in S3. By the
computation in lemma 3.9.2 and the fact that a knot in S3 is fibered if and
only if rankpHzFKpS3, K, gpKqqq “ 1 [Ni07], we see that the pattern knot Qi,jn
is fibered for j ě 2 if and only if i “ 0 and n ‰ 0 and when j “ 1 Qi,1n is
fibred if and only if i “ 0 and n ‰ 0,´1.
3.10 Thickness and unknotting number of generalized Mazur satellites with
non-trivial companions
In this section we give lower bounds on the thickness and torsion order
for n-twisted satellites with patterns Qi,j and arbitrary non-trivial compan-
ions. Recall that a knot K is called Floer thin if for all pairs of generators
x and y of HzFKpS3, Kq Mpxq ´ Apxq “ Mpyq ´ Apyq. Equivalently, if we
define the δ-grading as δpxq “ Mpxq ´ Apxq a knot is thin if the δ grading
139
x
y
Figure 3.54. Illustration of two intersection points in the pairing diagram
with a length j ` 1 vertical differential between them. The red arc is a
portion of αpK,nq that exhibits the genus detection of knot Floer homology
is constant for all generators. This concept was introduced in [MO08], where
they showed that all quasi-alternating knots have thin knot Floer homology.
Suppose that there is a length k vertical arrow between two distinct
generators x and y of the knot Floer homology. Then Apyq “ Apxq ´ k and
Mpyq “ Mpxq ´ 1. In this case, if we consider the collapsed δ grading we see
that δpxq “ Mpxq ´ Apxq and δpyq “ Mpxq ´ 1 ´ pApxq ´ kq “ δpxq ` k ´ 1.
So if k ą 1, these two generators are supported in distinct δ gradings, and so
the knot K is not Floer homologically thin.
Theorem 3.10.1. Suppose that K is a non-trivial companion knot. Then
the satellite knots Qi,jn pKq are not thin.
The proof of Theorem 3.10.1 relies on the observation that, since knot
Floer homology detects the genus of knots, if a knot K is non-trivial there
is always a portion of the immersed curve in each column that exhibits this.
We are only interested in the portion of the immersed curve in the second
column of the pairing diagram which is shown in figure 3.54.
Proof. Suppose K is a non-trivial companion knot. Then the curve αpK,nq
contains a portion as shown in Figure 3.54 by the genus detection of knot
Floer homology. We see that there are two intersection points, denoted x
140
x
y
Figure 3.55. The pairing CzFKpαpU, nq, βpi, jqq when n ă ´1
x
y
Figure 3.56. The pairing CzFKpαpU,´1q, βpi, jqq
and y, that are connected by a length j ` 1 vertical differential. Hence the
knot Qi,jn pKq is not Floer thin.
Next, we investigate what happens when the companion knot K is the
141
unknot. In that case, since Qi,j0 pUq „ U , it is clear that the 0-twisted satel-
lite is Floer thin. In all other cases, we prove the following
Theorem 3.10.2. The satellite knot Qi,jn pUq is not Floer thin unless n “ ´1
and j “ 1.
Note that Theorems 3.10.1 and 3.10.2 in the case j “ 1 recover [PW21,
Theorem 1.01].
Proof. Since K “ U , the pairing diagram CzFKpαpU, nq, βpi, jqq has the form
shown in Figures 3.55 and 3.56. Figure 3.55 shows the case when n ă ´1
and Figure 3.56 shows the case when n “ ´1. The case when n ą 0 is sim-
ilar and left ot the reader. In the case n ă ´1 inspecting Figure 3.55 we
see that there is a length j ` 1 vertical differential between the intersection
points labelled x and y. In the case n “ ´1, Figure 3.56 shows that there is
a length j vertical differential between the intersection points labelled x and
y. Inspection of the pairing diagram shows that these are the longest pos-
sible vertical differentials in the complex C 3zFKFrU,V s{UV pS ,Qi,jn pUqq. There-
fore, when n ă ´1, the satellite knot Qi,jn pUq is never thin and when n “ ´1,
the satellite knot Qi,jn pUq is thin if and only if j “ 1.
The V -torsion order of a knot, OrdV pKq is the smallest integer k with
the property that V kpTorspHFK´FrV spS3, Kqqq “ 0. The proofs of Theo-
rem 3.10.1 and 3.10.2, in addition to determining when the satellite knots
Qi,jn pKq are not thin, also gives a lower bound on the torsion order of Qi,jn pKq:
Corollary 3.10.3. When K is non-trivial, or when K “ U and n ‰ ´1
Ord i,jV pQn pKqq ě j ` 1. When K “ U and n “ ´1 then OrdV pQ
i,j
´1pUqq ě j.
Proof. The proof of theorem 3.10.1 and 3.10.2 shows that the chain omplex
gCFK´pQi,jn pKqq has a length j ` 1 vertical differential in the case K is non-
trivial or K “ U and n ‰ ´1, or a length j vertical differential in the case
K “ U and n “ ´1.
Since the torsion order is a lower bound for the unknotting number of a
knot [AE20] the following Corollary is immediate. This verifies a conjecture
142
of Hom, Lidman and Park in the case that the pattern knot is an n-twisted
generalized Mazur pattern [HLP22, Conjecture 1.10].
Corollary 3.10.4. The satellite knots Qi,jn pKq with non-trivial companions
have unknotting number at least j ` 1 “ wpQi,jn q ` 1.
3.11 Heegaard Floer Concordance Invariants and Twisting
In this section we determine the dependence of the invariants τ and ϵ on
the parameters i, j and the twisting parameter n. First, we will determine
the invariants τpQ0,jn pKqq and ϵpQ0,jn pKqq in terms of τpKq, ϵpKq, j and n,
and then we will show that τpQi,jn pKqq and ϵpQi,jn pKqq are independent of
i P Zě0.
Recall that by Theorem 3.8.4, the complex CFK 3 0,jFrU,V s{UV pS ,Qn pKqq
can be extracted from the pairing diagram by considering disks that cover
either the z or w basepoint and do not cover both. Let CFK 3 0,jFrV spS ,Qn pKqq
denote the complex obtained by only counting disks that cross the z-basepoint
(so the U “ 0 quotient of CFK 3 0,jFrU,V s{UV pS ,Qn pKqq). Theorem 3.8.4 shows
that this complex is isomorphic to gCFK´pS3, Q0,jn pKqq and so has homol-
ogy isomorphic to HFK´pS3, Q0,jn pKqq as an FrV s module. The structure
theorem for HFK´ implies that it has a single free FrV s summand, and the
generator of this summand has Alexander grading τpQ0,jn pKqq by [OST08,
Appendix A]. Therefore, to determine the value of τ of satellites with arbi-
trary companions, arbitrary framings and patterns Q0,j, we will use Theo-
rem 3.8.4 to identify a collection of intersection points, so generators of the
complex CFK pS3 0,jFrU,V s{UV , Qn pKqq, that form a subcomplex with respect
to the vertical z-basepoint differentials (when we set U “ 0) and generate
the FrV s free part of the homology of HFK´pS3, Q0,jn pKqq. Setting V “ 1 in
this complex gives H 3xFpS q and so, said another way, we identify a cycle in
HzFKpS3, Q0,jn pKqq that, in the V -filtration, survives in H 3xFpS q.
We will see in the pairing diagram that the form of this subcomplex is
completely determined by the piece of the essential component of α̃pK,nq
in the first column of the lifted pairing diagram corresponding to Lemmas
143
3.8.2 and 3.8.3. Once we identify the cycle that generates the FrV s-free part
of the homology (so survives in the spectral sequence to H 3xFpS q), it can be
extended to be the distinguished element of some vertically simplified basis,
as in [Hom14, Section2.3]. Then it is possible to determine ϵpQ0,jn pKqq from
the horizontal (w-basepoint crossing) differentials. By [Hom14, Definition 3.4
and Lemma 3.2] ϵpKq “ 1,´1 or 0 depending on whether the distinguished
element of the vertically simplified basis has a horizontal differential into it,
out of it, or neither respectively.
As in Lemmas 3.8.2 and 3.8.3, we distinguish multiples cases for the
essential component of α̃pK,nq depending on τpKq, ϵpKq, and n. In each
case the form of the pairing diagram, and thus the subcomplex carrying the
FrV s free part of the homology, changes. Moreover, the Alexander grading
labels of the arcs of the β curve relative to the central intersection point of
the pairing diagram also change. As in the proof of Theorem 3.7.1, there are
also multiple sub-cases depending on whether j and n are even or odd. We
mostly draw the pairing diagram in the case j is odd, since the pictures are
slightly simpler. We analyze the case j even and n odd in Figure 3.70, and
leave the rest of the cases where j is even to the reader.
Proof of Theorem 3.7.6. The proof is divided into many cases, first by the
value of ϵpKq, then into whether τpKq is positive or negative, and then into
various cases of whether or not n ě 2τpKq or n ă 2τpKq. The pictures look
slightly different when, for example τpKq ě 0 and either n ď 0 ď 2τpKq or
0 ď n ď 2τpKq, so we separately analyze those cases as well.
Case 0 ϵpKq “ 0: In this case it follows that τpKq “ 0 [Hom14], and
the essential component of the immersed curve αpK,nq is the same as the
immersed curve for the n-framed unknot complement, and so τpQ0,jn pKqq “
τpQ0,jn pUqq. The case n “ 0 is clear, since Q
i,j
0 pUq „ U . We indicate the
pairing diagrams for the cases n ă 0 and n ą 0 in Figures 3.57 and 3.58.
In the case that n ă 0, the intersection points labelled ta u2m`1k k“1 form a sub-
complex of CFK 3FrV spS ,Q0,jn pUqq with respect to the vertical differentials
that contains an FrV s free part. Setting V “ 1 in the above subcomplex,
ř
we see that the cycle a generates HFpS3` x2i 1 q. Note that the intersection
144
a2m`1
a2m
a1
b
c
c
b1
a a2 1
b3
a3
Figure 3.57. Figure 3.58.
ϵpKq “ τpKq “ 0 and ϵpKq “ τpKq “ 0 and
n ă 0 n ą 0
ř ř
points b2i`1 satisfy Bhp b2i`1q “ U a2i`1, so that ϵpQ0,jn pUqq “ 1 by
ř
[Hom14, Section 3]. Recall that Ap a2i`1q “ maxtApa2i`1qu, and from
this it is easy to see that τpQ0,jn pUqq “ Apa1q. Then, Apa1q “ ℓc,a1 ¨ δw,z,
where ℓc,a is the arc of the lifted β curve from c to a by [Che19, Lemma
4.1]. Now as remarked the Alexander grading labels of arcs of the β curve
change by ´j for each row we go down in the pairing diagram, so we see
ˆ ˙
j ´ 1
that Apaq “ τpQ0,jn pUqq “ ´j |n| ´ “
jpj ´ 1q
1 n` j
2 2
In the case that n ą 0, The intersection points taku form a subcomplex
of CFK 3 0,jFrV spS ,Qn pUqq that contains an FrV s-free part, and we see that
the cycle a1 generates HxFpS3q. Directly from Figure 3.58 we see that the
intersection point b satisfies Bhpbq “ U2a so ϵpQ0,j1 n pUqq “ 1. Furthermore,
we have that Apa1q “ τpQ0,jp qq “
jpj ´ 1q
n U n.2
Case 1 ϵpKq “ 1: In the case ϵpKq “ 1, we first distinguish between
the cases τpKq positive and negative and then distinguish further between
various sub-cases depending on the value of n relative to τpKq.
145
j ´ 1
n
2
j ´ 1
n
2
a2i`1
a2i`2
c
a1 b
j ´ 1
|n| ´ 2τpKq
2
b1
a1
a2i´2
b2i´1
a2i´1
a2i
b2i`1
a2i`1
a2m
c
b2m`1
a2m`1
Figure 3.59. The Figure 3.60. The
pairing diagram when pairing diagram when
τpKq ě 0, ϵpKq “ 1 τpKq ą 0 ϵpKq “ 1
and n ě 2τpKq and j and n ď 0 ă 2τpKq
odd and j odd
146
ˆ ˙
j ´ 1
n´ 2τpKq
τpKq 2 2τpKq
n´ 2τpKq
2τpKq ´ n
τpKq
b1
a1
a2i
b2i`1
a2i`1
a2m
b2m`1
a2m`1
c
Figure 3.61. Case τpKq ą 0, ϵpKq “ 1 and 0 ď n ă 2τpKq with j odd
147
τp q j ´ 3K n n
2
a U1 b1
V j`1 a U3 b3
¨ ¨ ¨ a2m´2 V
V
a2 V j`1
a2m´3 V j`1 a2m
V
V
a4 . . . a
U
2m`1 b2m`1
a U2m´1 a2m`1 b V
a2m
Figure 3.62.
Subcomplex carrying Figure 3.63.
the cycle that Subcomplex carrying
generates HxFpS3q cycle that generates
together with HFpS3x q and horizontal
horizontal differentials differentials from
from Figures 3.70 and Figures 3.60, 3.61, and
3.71 3.64
Case 1.1a τpKq ě 0 and n ě 2τpKq: This case is shown in Figure
3.59. In that figure, the intersection points labelled taku2m`1k“1 form a sub-
complex that contains the FrV s-free part of CFK pS3, Q0,jFrV s n pKqq and it is
easy to see that the intersection point labelled a2m`1 generates HxFpS3q, ob-
tained by setting V “ 1 in the above sub-complex. Then, the intersection
point a2m`1 is a distinguished element of some vertically simplified basis of
CFK´pKq. Since the intersection point labelled b satisfies Bhpbq “ U2a2m`1,
the cycle a2m`1 is a boundary with respect to the horizontal differential, so
it follows from [Hom14, Section 3] that ϵpQ0,jn pKqq “ 1. It remains to deter-
mine τpQ0,jn pKqq “ Apa2m`1q. By symmetry of the pairing diagram, we see
that the intersection point c satisfies Apcq “ 0, and then by [Che19, Lemma
4.1] Apa2m`1q “ Apa2m`1q ´ Apcq “ ℓc,a2m`1 ¨ δw,z. Now, to determine the
quantity ℓc,a2m`1 ¨ δw,z we see in Figure 3.59 that the arc ℓc,a2m`1 traverses
p q ` j ´ 1τ K n rows up the pairing diagram, and the Alexander grading
2
changes by j for each row we go up in the pairing diagram. Therefore
ˆ ˙
p q “ j ´ 1 jpj ´ 1qA a2m`1 j τpKq ` n “ jτpKq ` n.
2 2
Case 1.1b τpKq ą 0 and n ď 0 ă 2τpKq: This case is shown in Figure
148
3.60. In the pairing diagram, we can see that the subcomplex generated by
the intersection points ta 2m`1kuk“1 together with the vertical differentials car-
ries the FrV s-free part of the HFK´pS3, Q0,jn pKqq. This subcomplex is shown
in Figure 3.62 together with the horizontal differentials, and that the cycle
řm
k“0 a2k`1 survives in HxFpS3q. Then this cycle forms the distinguished el-
ement of some vertically simplified basis. Further, we see that for each k,
ř ř
Bhpb2k`1q “ Ua2k`1, so we have Bhp mk“0 b
m
2k`1q “ U k“0 a2k`1. Therefore
ϵpQ0,jn pKqq “ 1. Now, it remains to determine Apa q “ τpQ0,j1 n pKqq. By sym-
metry, Apcq “ 0 and
ˆ ˙
Apa1q “ Apa1q ´ Apcq “ ℓa1,c ¨ δw,z “ ´j τp
j ´ 1
Kq ` |n| ´ 2τpKq ` 1,
2
since the Alexander grading changed by ´j for each row we go down in the
pairing diagram. Simplifying, we see that
τpQ0,jn pKqq “ jτpKq `
jpj ´ 1q
n` 1.
2
Case 1.1c τpKq ą 0 and 0 ď n ă 2τpKq: This case is shown in Figure
3.61. The analysis here is exactly as in the previous case. The subcomplex
taku carries the FrV s-free part of the homology HFK´pS3, Q0,jn pKqq, and the
ř
cycle mk“0 a2k`1 survives in H
3
xFpS q, so can be taken to be the distinguished
ř
elements of a vertically simplified basis. Further, we have Bhp mk“0 b2k`1q “
ř
U m a , so just like in the previous case it follows that ϵpQ0,jk“0 2k`1 n pKqq “ 1
It remains to determine Apa1q: Counting the number of rows between a1 and
c in the pairing diagram, we find that
ˆ ˙
p 0,jp qq “ p q´ p q “ p q ` j ´ 1 ` “ p q` jpj ´ 1qτ Qn K A a1 A c j τ K n 1 jτ K n`12 2
This ends the analysis of the case ϵpKq “ 1 and τpKq non-negative.
Next, we move on to the case ϵpKq “ 1 and τpKq non-positive.
Case 1.2a τpKq ď 0, ϵpKq “ 1 and n ă 2τpKq
In this case, the pairing diagram is shown in Figure 3.64. The intersec-
tion points labelled ta u2m`1k k“1 generate the free part of CFKFrV spQ0,jn pKqq,
ř
the cycle mk“0 a2k`1 is the cycle that survives in HxFpS3q, and τpQ0,jn pKqq “
149
a2m`1
a2m
a2i`1
a2i
a2i´1
c
|τpKq|
a2i´2
a3
a2 a1
b
b1
a
a 12m
b2m`1
a2m`1
Figure 3.64. τpKq ď 0
ϵpKq “ 1 and Figure 3.65. τpKq ď 0
n ď 2τpKq ϵpKq “ 1 and n ě 0
ř ř
Apa1q. The intersection points tb um2k`1 k“0 satisfy Bhp
m
k“0 b2k`1q “ U
m
k“0 a2k`1,
so ϵpQ0,jn pKqq “ 1. Exactly in the previous cases, we find that
ˆ ˙
τpQ0,jn pKqq “ Apa1q “ ´j ´τpKq `
j ´ 1 | | ` “ p q ` jpj ´ 1qn 1 jτ K n` 1
2 2
Case 1.2b τpKq ď 0, ϵpKq “ 1 and n ě 0 ě 2τpKq In this case, the
pairing diagram has the form shown in Figure 3.65. In this case the FrV s
150
j ´ 1
|n|
2τpKq ´ n 2
2|τpKq|
j ´ 1
|τpKq| n´ 2|τpKq| n2
2|τpKq|
¨ ¨ ¨ a4
a U1 b1
V j`1 a2
V
V j a
U
3 b3
a3 V j`1 V
a2 V j`1 ¨ ¨ ¨2
a U1 b
a4
Figure 3.66.
Subcomplex carrying Figure 3.67.
the cycle that Subcomplex carrying
generates HFpS3x q cycle that generates
corresponding to the H 3xFpS q and horizontal
cases in Figures 3.59, differentials from
3.65, and 3.68 Figures 3.69 and 3.72
free part of the homology is generated by the intersection points ta u2m`1k k“1 .
Further, the intersection point a1 generated HxFpS3q. Just as above, the in-
tersection point b satisfies Bhpbq “ U2a1 and hence ϵpQ0,jn pKqq “ 1. Further-
more Apa1q “ τpQ0,jn pKqq. Inspecting the pairing diagram we find that
ˆ ˙
p q ´ p q “ p q ` j ´ 1 “ p q ` jpj ´ 1qA a1 A c j τ K n jτ K n
2 2
Case 1.2c τpKq ď 0, ϵpKq “ 1 and 0 ě n ě 2τpKq The pairing dia-
gram for this case is shown in Figure 3.68. The intersection points ta u2m`1k k“0
generate the free part of the homology, and the intersection point a1 gener-
ates HxFpS3q. In the pairing diagram, the intersection point labelled b satis-
fied Bhpbq “ U2a , so ϵpQ0,j1 n pKqq “ 1. Further, we compute
ˆ ˙
τpQ0,jn pKqq “ p
j ´ 1
A a1q “ ´j |τpKq| ` |n| “ jτp
jpj ´ 1q
Kq ` n
2 2
That finishes the cases where ϵpKq “ 1.
Case 2 ϵpKq “ ´1: As in the case ϵpKq “ 1, we distinguish between
various sub-cases depending on the sign of τpKq and the value of n relative
to 2τpKq
Case 2.1a τpKq ě 0, ϵpKq “ ´1 and n ď 0 ă 2τpKq
151
c
c
b1
a1
a2
b3
a3
a2i´2
b2i´1
a2i´1
a2i
b2i`1
a2m`1
a2i`1
a2m
a2m
a1 b2m`1
b
a2m`1
Figure 3.69. The
Figure 3.68. The pairing diagram when
pairing diagram when τpKq ą 0 ϵpKq “ ´1
τpKq ă 0 ϵpKq “ 1 and n ă 0 ă 2τpKq
and 0 ą n ą 2τpKq and j odd
152
j ´ 1
|n|
2 |τpKq|
j ´ 1
|n| ´ 2τpKq
2 τpKq
1
This case is shown in Figure 3.69. The intersection points taku2m`1k“1 gen-
erate a subcomplex with respect to the vertical differentials and carries the
FrV s-free part of the homology of CFK 3 0,jFrV spS ,Qn pKqq. Setting V “ 1
ř
we see that the cycle m 3xk“0 a2k`1 generates HFpS q. The intersection points
ř ř
tb m h m m 0,j2k`1uk“0 satisfy B p k“0 b2k`1q “ Up k“0 a2k`1q, so ϵpQn pKqq “ 1. We
determine τpQ0,jn pKqq “ Apa1q from the pairing diagram and find
ˆ ˙
p q “ ´ p q ` j ´ 1 | | ´ p q ´ “ p p q ` q ` jpj ´ 1qA a1 j τ K n 2τ K 1 j τ K 1 n
2 2
Note: The case τpKq ą 0, ϵpKq “ ´1 and 0 ă n ă 2τpKq is similar and
left to the reader.
Case 2.1b τpKq ě 0, ϵpKq “ ´1 and n ě 2τpKq
This case is shown in Figure 3.71. In this case subcomplex generated
by the intersection points taku generate the FrV s-free part of the homology.
We see that the intersection point labelled a generates HFpS3x1 q and that
Bhpbq “ Ua1. Therefore ϵpQ0,jn pKqq “ 1. Now, it is easy to see from the
pairing diagram that Apa1q “ Apa3q, and
ˆ ˙
τpQ0,jn pKqq “ Apa1q “ j τpKq `
j ´ 1
n` 1 “ jpτpKq ` q ` jpj ´ 1q1 n
2 2
Case 2.2a τpKq ď 0, ϵpKq “ ´1 and n ě 0 ě 2τpKq This case
is shown in Figure 3.70 where the intersection points taku form a subcom-
plex that carries the FrV s-free part of the homology, and the cycle a2m`1
generates H 3xFpS q. Considering disks that cross the w-basepoint, we see
that Bhpbq “ Ua2m`1 and so ϵpQ0,jn pKqq “ 1. It remains to determine
τpQ0,jn pKqq “ Apa2m`1. This is similar to the previous cases, but we point
out what happens in the case when j is even and n is odd. In this case the
central intersection point c with Apcq “ 0 is shown in Figure 3.70. Since j is
even, the central intersection point occurs along the unstable chain, as in the
proof of Theorem 3.7.1. Just as in the previous cases, we find that
ˆ ˙
p n´ 1 j jA a2m`1q ´ Apcq “ j ´ τpKq ` p ´ 2 n` n` 2τpKq ` 1q `
2 2 2
153
a1
a2
a2i
a2i`1
a2m´1
b
a2m a2m`1
c
Figure 3.70. Case τpKq ă 0, ϵpKq “ ´1 and n ą 0 ą 2τpKq with j even
154
n´ 1 ´ p q p jτ K ´ 2qn
2 2
n` 2τpKq ` 1
Simplifying, we see that
τpQ0,jn pKqq “ p
jpj ´ 1q
A a2m`1q “ jpτpKq ` 1q ` n.
2
Case τpKq ď 0 ϵpKq “ ´1 and 0 ą n ą 2τpKq
This case is similar to the previous cases and is left to the reader.
Case τpKq ď 0 ϵpKq “ ´1 and n ď 2τpKq The pairing diagram for
this case is shown in Figure 3.72. In that figure we see that the intersection
points labelled taku generate the free part of the homology of CFK pS3, Q0,jzFrV s n pKqq
ř
and when we set V “ 1 the cycle mk“0 a generates HFpS3x2k`1 q. The in-
ř ř
tersection points tb2k`1u are such that Bhp mk b2k`1q “ U k“0 a2k`1, so
ϵpQ0,jn pKqq “ 1. It remains to determine τpQ0,jn pKqq “ Apa1q. Inspecting
the pairing diagram, we find that
jpj ´ 1q
τpQ0,jn pKqq “ jpτpKq ` 1q ` n2
Lemma 3.11.1. For any j P Zą0, n P Z and i P Zě0,
τpQi,j 0,jn pKqq “ τpQn pKqq
Proof. Inspection of the pairing diagram shows that the intersection points
that form a subcomplex of CFK 3 i,jFrV spS ,Qn pKqq that generates the FrV s
free part of the homology is independent of i. That is, twisting up the β
curve does not change the subcomplex under consideration and as remarked
before, does not change the Alexander gradings of the previously existing
intersection points. See Figures 3.73 and 3.74. In particular in all cases the
cycle that survives to HxFpS3q and the Alexander grading of that cycle is in-
dependent of i.
Lemma 3.11.2. For any j P Zą0, n P Z and i P Zě0,
ϵpQi,jn pKqq “ ϵpQ0,jn pKqq.
155
a2m`1
a2m
a3
c
b
a a2 1
c
b1
a2 a1
b3
a3
Figure 3.71. The
pairing diagram when
τpKq ą 0 ϵpKq “ ´1 Figure 3.72. τpKq ă 0
and n ě 2τpKq and j ϵpKq “ ´1 and
odd n ď 2τpKq
Proof. There are a few cases depending on the shape of essentail component
of the curve αpK,nq, but the proof is essentially local in nature so we only
indicate the local modification to the complex. Consider the case when the
intersection point with Alexander grading τpQi,jn pKqq and the vertical sub-
complex nearby this intersection point has the form shown in figures 3.75
and 3.76. For example this covers the cases when τpKq ě 0 and ϵpKq “ 1
and n ď 2τpKq and τpKq ď 0, ϵ “ 1 and n ď 2τpKq. When we twist the
β curve up once, notice that there are now two intersection points b and b1
with a horizontal differential to a. However, this does not change the com-
156
j ´ 1
n` 1
τpKq 2
j ´ 1
n´ 1
2 |τpKq|
a a
Figure 3.73. The Figure 3.74. The
subcomplex that subcomplex that
carries the FrV s-free carries the FrV s-free
part of the homology part of the homology
before twisting after twisting
putation of ϵ, since we can perform a change of basis, letting b1 “ b and
b2 “ b ` b1. Then Bhpb1q “ a and Bhpb2q “ 0. We see from figures 3.77 and
3.78 that this pattern continues for each addition twist we add to the lifted
β curve.
157
b1
z z z z
w w w w
b
a a
Figure 3.75. A Figure 3.76. Another
horizontal differential horizontal differential
to the intersection to the intersection
point that survives the point that survives the
spectral sequence to spectral sequence to
HFpS3x q when i “ 1 HxFpS3q when i “ 1
b1
z z z z
w w w w
b
a a
Figure 3.77. A Figure 3.78. Another
horizontal differential horizontal differential
to the intersection to the intersection
point that survives the point that survives the
spectral sequence to spectral sequence to
HFpS3x q when i ą 1 HFpS3x q when i ą 1
158
CHAPTER 4
KHOVANOV STABLE HOMOTOPY TYPE AND RIBBON
CONCORDANCE
4.1 Introduction
This chapter contains previously published material. In 2014, Lipshitz
and Sarkar introduced a stable homotopy refinement of Khovanov homol-
ogy [LS14a]. For each fixed j it takes the form of a suspension spectrum X j.
The cohomology H˚pX jq of this spectrum is isomorphic to the Khovanov ho-
mology Kh˚,j. In subsequent work (e.g. [LS14c]) they used this refinement
to define stable cohomology operations on Khovanov homology. This lead to
a refinement of Rasmussen’s s-invariant for each nontrivial cohomology oper-
ation, and in particular for the Steenrod squares [LS14c]. In this short note
we offer a solution to the following question posed in Lipshitz-Sarkar [LS18,
Question 3]: Are there prime knots with arbitrarily high Steenrod squares on
their Khovanov homology? Explicitly, we prove the following theorem:
Theorem 4.1.1. Given any n, there exists a prime knot Pn so that the op-
eration
n i,j i`n,jSq : KĂh pPnq Ñ KĂh pPnq
is nontrivial for some pi, jq. Here KĂh denotes reduced Khovanov homology.
Corollary 4.1.2. Given any n, there exists a prime knot Pn so that the op-
eration
Sqn : Khi,jpP q Ñ Khi`n,jn pPnq
is nontrivial, on unreduced Khovanov homology, for some pi, jq.
In fact a stronger version of Theorem 4.1.1 is true:
Theorem 4.1.3. Given any n, there exists a hyperbolic knot Hn so that the
operation
n i,j i`n,jSq : KĂh pHnq Ñ KĂh pHnq
is nontrivial for some pi, jq. Here KĂh denotes reduced Khovanov homology.
159
Theorem 4.1.4. Given any n, there exists a prime satellite knot Sn so that
the operation
n i,j i`n,jSq : KĂh pS Ănq Ñ Kh pSnq
is nontrivial for some pi, jq. Here KĂh denotes reduced Khovanov homology.
Our technique for proving all of the above theorems is to find a ribbon con-
cordance from any given knot K to a prime, hyperbolic or satellite knot,
then appeal to the following generalization to reduced Khovanov homology
of a theorem of Wilson and Levine-Zemke (for the original statement see
[Wil12; LZ19] or Theorem 4.2.2 below).
Theorem 4.1.5. Suppose C is a ribbon concordance between knots K and
K 1. Then the induced map FC : KĂhpKq Ñ KĂhpK 1q is injective.
Recall that any prime knot K is either a hyperbolic knot, a satellite
knot, or a torus knot. With this in mind, Theorems 4.1.1—4.1.4 suggest the
following question:
i,j
Question: For any given n, is there a torus knot Tn so that Sqn : KĂh pTnq Ñ
i`n,j
KĂh pTnq is nontrivial for some pi, jq?
The organization of the chapter is the following. In Section 4.2, we review
the results of Wilson and Levine and Zemke [Wil12; LZ19] showing that rib-
bon concordances induce split injections on Khovanov homology. In Section
4.3, we prove the analogue of this theorem for reduced Khovanov homology.
In Section 4.4, we show that any knot is ribbon concordant to a prime knot,
following the arguments in [Lic81; KL79]. In Section 4.5, we collect various
results about the naturality of Steenrod squares with respect to births, Rei-
demeister moves and saddle maps and the behavior of the Khovanov stable
homotopy type under connected sums. In Section 4.6 we show that the non-
triviality of Steenrod squares on composite knots constructed by Lipshitz-
Sarkar [LLS15, Corollary 1.4] and [LS18, Corollary 3.1] propagates to the
nontriviality of Steenrod squares on the Khovanov homology of prime knots.
In Section 4.7 we prove Theorem 4.1.3 using results of Silver and Whitten
[SW05]. In Section 4.8 we prove Theorem 4.1.4 using results of Livingston
[Liv81].
160
Acknowledgements The author would like to thank his advisor Robert
Lipshitz, as well as Danny Ruberman for pointing out Theorem 4.1.3, Chuck
Livingston for pointing out his construction of ribbon concordances to prime
knots that leads to Theorem 4.1.4, and the anonymous referee for their care-
ful reading that improved the exposition.
4.2 Khovanov Homology and Ribbon Concordances
In this section we review the behavior of Khovanov homology under rib-
bon concordances. Unless explicitly stated otherwise, throughout this paper
we write KhpKq to mean KhpK;F2q.
Definition 4.2.1. let K0 and K1 be links in S3. A concordance from K0 to
K1 is a smoothly embedded cylinder in r0, 1s ˆ S3 with boundary ´pt0u ˆ
K0q Y pt1u ˆK1q. A concordance C is said to be ribbon if C has only index
0 and 1 critical points with respect to the projection r0, 1s ˆ S3 Ñ r0, 1s.
Throughout this paper, we will use the notation C to denote the ribbon
concordance C upside-down.
Theorem 4.2.2. [Wil12; LZ19] If C is a ribbon concordance from K0 to
K1, then the induced map
KhpCq : KhpK0q Ñ KhpK1q
is injective, with left inverse KhpCq. In particular, for any bigrading pi, jq
the group Khi,jpK0q is a direct summand of the group Khi,jpK1q.
The proof of this theorem involves decomposing the cobordism D :“
C ˝ C as the disjoint union of the identity cobordism (a cylinder) and sphere
components joined to the cylinder by tubes (formed from the ribbons and
their duals). For details, see [LZ19] or [Wil12]. In the next section, we present
an analogue of Theorem 4.2.2 for reduced Khovanov homology, after review-
ing the necessary definitions.
161
4.3 The Base-point action and Reduced Khovanov Homology
We begin with the definition of the base-point action on Khovanov ho-
mology. For grading conventions, see [Shu14].
Definition 4.3.1. Fix a diagram of the knot K and pick a base-point q P K
not on any of the crossings. Then we make the Khovanov complex CKhpKq
of K into a module over F rXs{X22 as follows. Generators of the chain groups
are complete resolutions of K and a choice of 1 or X for each component of
the complete resolution. Multiplication by X is zero if the generator labels
the circle containing q with an X and if the generator labels the circle con-
taining q by 1 it changes the label of the circle to X. With our grading con-
ventions (see [Shu14]), multiplication by X has bidegree p0,´2q. That is
X : Khi,jpKq Ñ Khi,j´2pKq.
Definition 4.3.2. Let F be the F2rXs{X2 module F2 where X acts trivially.
Then define
CrKhpKq :“ CKhpKq bF2rXs{X2 F.
The homology of the complex CrKhpKq is called reduced Khovanov ho-
mology and denoted KĂhpKq.
Theorem 4.3.3. [Shu14, Corollaries 3.2.B and 3.2.C] The action of X on
CKhpKq commutes with the Khovanov differential, so induces a map (also
called X) on homology. Further,
1. The following sequence is exact:
¨ ¨ ¨ ÝÑX Khi,j`2pKq ÝÑX Khi,jp X XKq ÝÑ Khi,j´2pKq ÝÑ ¨ ¨ ¨
2. The reduced Khovanov homology over F2 is isomorphic to the kernel
of X (which is the image of X by part 1), and we have the direct sum
decomposition
162
i,j i,j´1 i,j`1Kh pKq – KĂh pKq ‘KĂh pKq.
With these preliminaries in mind, we prove Theorem 4.1.5 from the in-
troduction.
Proof of Theorem 4.1.5. By Theorem 4.2.2 we know that the map FC :
KhpK0q Ñ KhpK1q is a split injection with left inverse FC . By Theorem
4.3.3, for a P t0, 1u,
KĂhpKaq – KerpX : KhpKaq Ñ KhpKaqq – ImpX : KhpKaq Ñ KhpKaqq.
Therefore, it is enough to show that the map FC is a F2rXs{X2 module
map. Indeed, then FC |Ker maps KerpX : KhpK0q Ñ KhpK0qq to KerpX :
KhpK1q Ñ KhpK1qq and FC |Ker maps KerpX : KhpK1q Ñ KhpK1qq to
KerpX : KhpK0q Ñ KhpK0qq. Further, FC |Ker ˝ FC |Ker “ id|Ker. Therefore
FC |Ker is a split injection.
Now, any cobordism can be decomposed into births (0-handles) and saddle
moves (1-handle attachments) and deaths (2-handles). So, to show that the
maps induced on Khovanov homology by cobordisms respect the X action,
it suffices to verify the following.
1. Births and deaths respect the module structure with respect to a base-
point not on the circle dying or being born.
2. The isomorphisms of Khovanov homology associated to Reidemeister
moves respect the module structure.
3. The maps associated with saddles respect the module structure.
Item 1 is clear from the definition of the X action, provided we chose
a base-point on the original knot diagram, away from where the births and
deaths occur.
Item 2 follows from Proposition 2.2 of Hedden-Ni [HN13]. Evidently, the ho-
motopy equivalences induced from Reidemeister moves commute with the
163
X action if the Reidemeister moves does not involve a strand moving across
a base-point. Therefore it suffices to show that moving a strand across the
base-point does not change the action of X on homology. This follows by
writing down an explicit chain homotopy between the different base-point
actions associated with choosing two marked points, on the same compo-
nent, on opposite sides of a crossing. These homotopy equivalences appear in
[HN13, Lemma 2.3].
Item 3 reduces to a local calculation in a complete resolution. Either
the saddle cobordism merges two components, or splits one component into
two. In either case, it is easy to check that the maps involved commute with
the X action.
4.4 Knots and Prime Tangles
The main theorem of this section is the following:
Theorem 4.4.1. [Lic81; KL79] Any knot is ribbon concordant to a prime
knot.
The proof of this theorem is standard and is well explained elsewhere
in the literature. We include a review of the techniques used in the proof
for the convenience of the reader and to introduce some notation. We begin
with a definition and a convention [Lic81; KL79; Ble82].
Definition 4.4.2. A (4-ended) tangle with no closed components is an em-
bedding of r0, 1s \ r0, 1s into B3 so that t0, 1u Y t0, 1u map to S2 “ BB3. We
specify a tangle by a diagram, see Figure 4.3. We denote such a tangle by
pB, T q or just T . A tangle pB, T q is prime if both of the following conditions
hold:
1. Any 2-sphere embedded in B that intersects the knot transversely at
two points bounds on one side a three ball A so that A X T is homeo-
morphic to the standard ball arc pair pD2 ˆ r0, 1s, 0ˆ r0, 1sq.
164
T1 T2
T1 T2
Figure 4.1. T1 `p T2 Figure 4.2. pT1 `p T2q ` Cl
2. pB, T q is not a rational tangle. Equivalently, pB, T q does not contain
any separating disks.
One motivation for the name prime tangle and illustration of their use
is indicated by the following:
Theorem 4.4.3. [Lic81, Lemmas 1, 2] The sum of two prime tangles is a
prime knot. The partial sum of two prime tangles is a prime tangle.
For the proof, see [Lic81]. In this paper, we use the notation `p for the
partial sum of two tangles and the notation T1 ` T2 for the sum of two tan-
gles. These operations depend on a choice of which endpoints are identified.
In the present work, the operations `p and ` mean the operations in Fig-
ures 4.1 and 4.2 respectively. For our purposes, we make this explicit as fol-
lows. Let NW, NE, SW, SE denote the northwest, northeast, etc corners of
the diagram of a tangle T . Then T1`pT2 means the tangle formed by joining
the NE and SE corners of T1 to the NW and SW corners of T2 respectively
by unknotted arcs. Further, T1 ` T2 means the tangle formed from T1 and T2
by joining the NE corner of T1 with the SE corner of T2, the SE corner of T1
with the NE corner of T2, etc. See Figure 4.2, which shows pT1 `p T2q ` Cl.
Note that, even though we use the ` sign to denote the tangle sum opera-
tion, it is usually not commutative.
Lemma 4.4.4 (See [KL79], [Ble82]). For any nontrivial knot K in S3 there
is an embedded S2 meeting K transversely in four points separating S3 into
two three balls A and B so that
165
1. pA,AXKq is a trivial two-stranded tangle (so homeomorphic, as pairs,
to pD2 ˆ I, tp´1{2, 0qu ˆ I Y tp1{2, 0qu ˆ Iqq, and
2. pB,B XKq is a prime tangle.
Lemma 4.4.5. The clasp tangle Cl is a prime tangle.
Proof. Since each of the individual strings that compose the clasp tangle
are unknotted, condition 1 in the definition of a prime tangle is automati-
cally satisfied. We just need to verify that the clasp is not a rational tangle.
Suppose for the sake of contradiction that it is. Recall that a knot built out
of two rational tangles is a two-bridge knot. It is a classical fact (originally
proved by Schubert, see J. Schultens [Sch03] for a modern proof) that the
bridge number of a knot, bpKq, satisfies bpK#K 1q “ bpKq ` bpK 1q ´ 1.
Further, the only knot with bridge number 1 is the unknot. These two facts
together imply that two-bridge knots are prime. However, the numerator
closure of the clasp tangle is clearly a connected sum 31#mp31q.
Figure 4.3. The clasp tangle Figure 4.4. The numerator closure ofCl the clasp tangle
Proof of Theorem 4.4.1. Since any knot can be decomposed as a connected
sum of prime knots, and connected sum is compatible with concordance, it
suffices to prove the result for a knot K “ K1#K2 where Ki are prime. By
Lemma 4.4.4, we can find two disjoint three balls B1 and B2 so that Ti “
Bi X Ki is a prime tangle and pS3zBiq X Ki is an untangle. Now, consider
T1 `p T2. This tangle is prime by Theorem 4.4.3. The denominator closure
166
of the resulting tangle is K1#K2. The tangle sum pT1 `p T2q ` Cl is then
a prime knot by Theorem 4.4.3. The ribbon concordance, shown in Figure
4.7—Figure 4.10, between K1#K2 and pT1 `p T2q ` Cl establishes the result.
T2 T2
Figure 4.5. Denominator closure of Figure 4.6. Denominator closure of
the clasp tangle T2 `p T2
4.5 Steenrod Operations and Stable Homotopy Type
In this section we review, in bare bones fashion, the necessary facts
about Khovanov stable homotopy type needed in establishing Theorem 4.1.1.
We begin with a theorem, which explains how the Khovanov stable ho-
motopy type behaves under the operation of connected sum. Throughout
this section, let L denote a link of one or more components.
Theorem 4.5.1. [LLS15, Theorem 2]
ł
X jrKhpL1#L2q » X
j1
r pL1q ^ X j2r pL2q.
j1`j2“j
Next, we recall the precise naturality statement enjoyed by stable coho-
mology operations.
Theorem 4.5.2. [LS14b, Theorem 4] Let S be a smooth cobordism in r0, 1sˆ
S3 from L1 to L2, and let FS : Kh˚,˚pL1q Ñ Kh˚,˚`χpSqpL2q be the map asso-
ciated to S. Let α : H˚r p¨;Fq Ñ H˚`nr p¨;Fq be a stable cohomology operation.
Then the following diagram commutes up to sign:
167
Khi,jpL1;Fq α Khi`n,jpL1;Fq
FS FS
Khi,j`χpSqpL ;Fq α Khi`n,j`χpSq2 pL2;Fq.
Then, for α a stable cohomology operation, the following diagram com-
mutes:
KhpU \K;F α2q KhpU \K;F2q
– –
F rXs{X22 bKhp
Idbα
K;F 22q F2rXs{X bKhpK;F2q
m m
KhpK;F α2q KhpK;F2q.
The bottom square commutes by Theorem 9, since the X action on Kh can
also be viewed as induced from a merge cobordism U \ K Ñ K where the
unknot is placed near the basepoint. The top square commutes since the
Khovanov spectrum of the unknot is homotopy equivalent to a wedge of two
S0’s in grading ´1 and 1. This homotopy equivalence induces the map in co-
homology that identifies F2rXs{X2 bKhpK;F2q with KhpK;F2q ‘KhpK;F2q
with appropriate grading shifts.
Commutativity of the above diagram is the statement that any stable
cohomology operation is a map of F2rXs{X2 modules. It follows that the
analogous diagram to the one in Theorem 4.5.2, with Khovanov homology
replaced by reduced Khovanov homology commutes, commutes.
Lemma 4.5.3. [LLS15, Corollary 1.4] For any n there is a knot Kn so that
the operations
n i,j i`n,jSq : KĂh pKnq Ñ KĂh pKnq
and
Sqn : Khi,jpKnq Ñ Khi`n,jpKnq
168
are nontrivial, for some pi, jq.
For the proof, see [LLS15, Proof of Corollary 1.4, Page 67]. They find,
´1,0
for the knot K “ 15n41127, a class α P KĂh pK;F2q so that Sq1pαq ‰ 0 P
0,0
KĂh pK;F2q and Sqipαq “ 0 for i ą 1. Then, letting Kn “ K#K# ¨ ¨ ¨#K,
the Cartan formula and Theorem 4.5.1 give the result.
Since the knot Kn in the above theorem is the knot K connect summed with
itself n times, we can view Kn as the denominator closure of the partial tan-
gle sum K `p ¨ ¨ ¨ `p K (see Figure 4.7).
T T T1 T2 1 2
Figure 4.7. K1#K2 \ Unknot
Figure 4.8. K1#K2 \ Unknot after
isotopy
T1 T2
T1 T2
Figure 4.10. The result of adding a
band; the final stage of the ribbon
Figure 4.9. Another isotopy. concordance between K1#K2 and the
prime knot pT1 `p T2q ` Cl .
169
4.6 Proof of Theorem 1
In this section, we collect the results from the previous sections together
to construct a proof of Theorem 4.1.1.
Proof of Theorem 1. By Lemma 4.4.4, the knot K from the proof of Lemma
4.5.3 can be decomposed as a prime tangle T2 and an untangle T1, so that
the denominator closure of T2 is K (this is “K with ears"; see [Ble82]). Then,
the knot Kn “ K# ¨ ¨ ¨#K is the denominator closure of the (prime) tan-
gle T2 `p T2 `p ¨ ¨ ¨ `p T2, where recall `p denotes the partial sum of tan-
gles. Consider the ribbon concordance C given in Theorem 4.4.1, from Kn to
Pn :“ pT2 `p T2 `p ¨ ¨ ¨ `p T2q ` Cl. This is illustrated, for n “ 2, in Figure
4.7—Figure 4.10 by replacing T1 by T2 in the figures. By Theorem 4.4.3 and
Lemma 4.4.5, Pn is a prime knot.
By Theorem 4.1.5, the map
F : KĂC hpKnq Ñ KĂhpPnq
is injective with left inverse given by FC where C is the concordance C upside-
down. Therefore KĂhpPnq “ KĂhpKnq ‘ G for some complement G. Theorem
4.5.2 implies that the following commutes (note that the Euler characteristic
of any concordance is 0):x
´n,0 Sqn 0,0
KĂh pKnq KĂh pKnq
FC FC
´n,0 Sqn 0,0
KĂh pP Ănq Kh pPnq.
This immediately implies Theorem 4.1.1, since the vertical maps are
injective.
Next, we prove Corollary 4.1.2 from the introduction.
Proof of Corollary 1. This proof follows closely the proof of [LLS15, Corol-
lary 1.4]. There is a long exact sequence in Khovanov homology induced
from the cofiber sequence:
170
X j´1r pPnq Ñ X jpP q Ñ X j`1rn pPnq.
The long exact sequence takes the form:
i,j`1 π i,j´1 i`1,j`1¨ ¨ ¨ Ñ KĂh pPnq Ñ Khi,jpPnq ÝÑ KĂh pP Ănq Ñ Kh pPnq Ñ ¨ ¨ ¨
Since over the field F :“ Z{2Z the Khovanov homology of any knot K
i,j`1 i,j´1
is isomorphic to the direct sum KĂh pK;Fq ‘ KĂh pK;Fq of the shifted
reduced homology, the map π above is surjective. So, there is a class γ P
Kh´n,1pPnq so that πpγq “ β, where the class β is as in the proof of Theorem
4.1.1. Naturality of the Steenrod squares establishes the result.
Remark 1 : The above proof applies to any stable homotopy refinement of
Khovanov homology that satisfies the analogue of Theorems 4.5.1 and 4.5.2.
The idea of the proof also offers an obstruction to ribbon concordance be-
tween two knots. If P and Q are knots with a ribbon concordance between
them, the Khovanov homology of P is a summand of the Khovanov homol-
ogy of Q with the same stable cohomology operations as the Khovanov ho-
mology of Q.
As an illustration of the above remark, we have the following lemma. To
state it, we recall that in [See12], Seed constructed pairs of links L and L1 so
that KhpL;Zq – KhpL1;Zq but the invariants XKhpLq and XKhpL1q are not
stably homotopy equivalent. Then, the following lemma is immediate.
Lemma 4.6.1. For each of Seed’s pairs of knots, there is no ribbon concor-
dance between them.
4.7 Hyperbolic Knots and Invertible Concordances
In this section we prove that there are hyperbolic knots with arbitrarily
high Steenrod operations on their reduced Khovanov homology. The main
theorem of this section, Theorem 4.1.3 from the introduction, is a direct con-
sequence of the following that appears in [SW05, Theorem 2.2(iv)]:
171
Theorem 4.7.1. Given any knot K Ă S3 there is a hyperbolic knot H and a
ribbon concordance from K to H.
The proof of Theorem 4.1.3 now is the same as the proof of Theorem
4.1.1 in Section 4.6. Following the notation of Section 4.6, and letting C de-
note the composite ribbon concordance from Kn to Hn, the following com-
mutes:
´n,0 n 0,0
KĂh pKnq
Sq
KĂh pKnq
FC FC
´n,0 n 0,0
KĂh p SqHnq KĂh pHnq.
Remark 2 : It was pointed out to us by Danny Ruberman that there is a
stronger result possible. In Kawauchi [Kaw89], it is shown that for any knot
K there is an invertible concordance from K to a hyperbolic knot. This al-
lows the propagation of Steenrod Squares without the injectivity results of
Wilson or Levine-Zemke. See also [Kim00].
4.8 Satellite Knots
In this section, we show how results from [Liv81] imply Theorem 4.1.4.
Proof of Theorem 4.1.4. The reader is referred to [Liv81] for details of Liv-
ingston’s construction. Glancing at Figure 2 of [Liv81] shows that there is a
ribbon concordance from the unknot to a non-trivial knot K 1 contained in
the solid torus S1 ˆ D2. Now, consider the knot Kn discussed in section 4.6
and the satellite Sn formed from K 1 as pattern and Kn as companion. Liv-
ingston shows that Sn is prime. The ribbon concordance from the unknot to
K 1 gives a ribbon concordance from Kn to Sn. The remainder of the proof
goes through exactly as in the proof of Theorems 4.1.1 and 4.1.3.
172
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