EQUIVARIANT KHOVANOV HOMOTOPY TYPE AND PERIODIC LINKS
by
JEFFREY MUSYT
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2019
DISSERTATION APPROVAL PAGE
Student: Jeffrey Musyt
Title: Equivariant Khovanov Homotopy Type and Periodic Links
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of
Mathematics by:
Robert Lipshitz Chair
Nicholas Proudfoot Core Member
Dan Dugger Core Member
Dev Sinha Core Member
David Evans Institutional Representative
and
Janet Woodruff-Borden Vice Provost & Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2019
ii
© 2019 Jeffrey Musyt
All rights reserved.
iii
DISSERTATION ABSTRACT
Jeffrey Musyt
Doctor of Philosophy
Department of Mathematics
June 2019
Title: Equivariant Khovanov Homotopy Type and Periodic Links
In this dissertation, we give two equivalent definitions for a group G acting
on a strictly-unitary-lax-2-functor D : 2n → B from the cube category to the
Burnside category. We then show that the natural Z/pZ action on a p-periodic
link L induces such an action on Lipshitz and Sarkar’s Khovanov functor FKh(L) :
2n → B which makes the Khovanov homotopy type X (L) into an equivariant knot
invariant. That is, if a link L′ is equivariantly isotopic to L, then X (L′) is Borel
homotopy equivalent to X (L).
iv
CURRICULUM VITAE
NAME OF AUTHOR: Jeffrey Musyt
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR, USA
University of Scranton, Scranton, PA, USA
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2019, University of Oregon
Master of Science, Mathematics, 2017, University of Oregon
Bachelor of Science, Neuroscience, 2009, University of Scranton
AREAS OF SPECIAL INTEREST:
Low Dimensional Topology
Knot Theory
Khovanov Homology
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, Fall 2013 - Spring 2019
GRANTS, AWARDS AND HONORS:
Frank W. Anderson Graduate Teaching Award, University of Oregon, 2018
v
ACKNOWLEDGEMENTS
Thanks to Robert for being an excellent advisor, for his help and
encouragement, for his guidance and humor, and for generally being for it.
Thanks to my family for all the love they have shown me during the past
six years. To my Mom and Dad for all the supportive phone calls, to Jon and Jenn
for their continued support and encouragement, to Jordan and Peter for always
brightening my day, and to Grandma Chickie for all her thoughts and prayers.
Thanks to the following long list of friends all of whom share a large part
in my success: My classmates - Keegan, Ryan, Ben, Janelle, Clair, Paul, and
Christophe; My fellow departmental friends - Katie, Helen, Sarah, Andrew, Rob,
3T, Bradley, Mike, Christy, Joe, Maya, Fill, Martin, Nate, Kelly, Ross, Dana,
Nathan, Jake, Eli, and Marissa; My non-departmental friends - Corin, Alicia,
Ashley, and Wendell; My fellow cyclist - Matt; My math circle crew - Maria,
Natalie, Sean, and Thomas; My fellow Tracktown Swing members - Nick, Nika,
Marissa, Sophie, Rachel, Curtis, Dodi, Shane, Shanoah, Bjorn, Matthew, Barb,
Cameron, and Min Yi; and My editor - Gerard.
Thanks to the whole University of Oregon Math Department: Mike for
his advice about teaching; Jessica, Mary, Sherilyn, and Elise, for their help in the
office; and all the professors who helped me become a better mathematician.
vi
To my family.
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1
II. BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 The Cube Category . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Thickened Cube Category . . . . . . . . . . . . . . . . 8
2.3 The Burnside Category . . . . . . . . . . . . . . . . . . . . 9
2.4 Functors from the Cube to the Burnside Category . . . . . . . . 10
2.5 The Khovanov Functor . . . . . . . . . . . . . . . . . . . . 13
2.6 Homotopy Colimits . . . . . . . . . . . . . . . . . . . . . 17
2.7 The Khovanov Homotopy Type . . . . . . . . . . . . . . . . 18
III. GROUP ACTIONS ON CATEGORIES AND FUNCTORS . . . . . . 21
IV. A Z/pZ-ACTION ON THE KHOVANOV HOMOTOPY TYPE . . . . 27
V. PROOF OF INVARIANCE . . . . . . . . . . . . . . . . . . . . 39
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . 56
viii
LIST OF FIGURES
Figure Page
1. Examples of Knot Diagrams . . . . . . . . . . . . . . . . . . . . 4
2. Examples of Periodic Knot Diagrams . . . . . . . . . . . . . . . . 6
3. Examples of Equivariant Reidemeister Moves . . . . . . . . . . . . 7
4. The Ladybug Configuration . . . . . . . . . . . . . . . . . . . . 15
5. Resolutions of Crossings . . . . . . . . . . . . . . . . . . . . . . 41
6. Subcomplexes and Quotient Complexes for the Proof of
Invariance under the Equivariant Reidemeister II Move . . . . . . . . 47
7. Additional Subcomplexes and Quotient Complexes for the Proof
of Invariance under the Equivariant Reidemeister II Move . . . . . . . 48
8. A Subcomplex of C1 from the Proof of Invariance under the
Equivariant Reidemeister III Move . . . . . . . . . . . . . . . . . 52
9. A Subcomplex of C3 from the Proof of Invariance under the
Equivariant Reidemeister III Move . . . . . . . . . . . . . . . . . 53
ix
CHAPTER I
INTRODUCTION
In 1985, Jones described a new polynomial knot invariant satisfying a
skein relation [Jon85]. Two years later, Kauffman gave a state model definition
for the Jones polynomial by defining what is now called the Kauffman Bracket
[Kau87]. The Jones Polynomial is celebrated not only because it is relatively
good at distinguishing knots, but also because it was used by Kauffman [Kau87],
Murasugi [Mur87], Thistlethwaite [Thi87] [Thi88], and Menasco and Thistlethwaite
[MT93] to prove the Tait Conjectures from 1898 [Tai98]. In 2000, based on an
idea of Crane and Frenkle [CF94], Khovanov categorified the Jones polynomial by
assigning a bigraded abelian group to an oriented link [Kho00]. This is a refinement
of the work of Jones in the sense that the graded Euler characteristic of Khovanov
Homology is the unnormalized Jones polynomial. As a further refinement of the
Jones polynomial and Khovanov homology, Lipshitz and Sarkar [LS14] constructed
the Khovanov stable homotopy type XKh(L) by using the notion of flow categories
described by Cohen, Jones, and Segal in [CJS95]. More precisely, for each oriented
link diag∨ram L, Lipshitz and Sarkar constructed a family of suspension spectra
X (L) = jj XKh(L) such that
(1) The reduced cohomology of X jKh(L) is the same as the Khovanov homology
Kh∗,j(L) : ( )
H̃ i X j (L) = Khi,jKh (L)
(2) The homotopy type of X jKh(L) is determined by the isotopy class of the link
L.
1
Shortly afterwards, a similar spectrum invariant was described by Hu, Kriz, and
Kriz utilizing different techniques [HKK12]. In [LLS15a], Lawson, Lipshitz, and
Sarkar gave an equivalent construction of the Lipshitz-Sarkar Khovanov homotopy
type by defining a strictly-unitary-lax-2-functor F nKh(L) : 2 → B from the cube
category to the Burnside category, and by using this reformulation they showed
that the Lisphitz-Sarkar and Hu-Kriz-Kriz invariants were homotopy equivalent.
In 1961, Fox first suggested studying classes of knots with various forms of
symmetries. One such class of links is periodic links, which are links that possess
a diagram with a rotation symmetry [Fox61]. In 1988, Murasugi showed that
there is a relationship between the Jones polynomials of a periodic link and its
quotient link, creating an obstruction for when links can be periodic [Mur88].
In 2007, Chbili defined a G-equivariant Khovanov homology when G is a cyclic
group of odd order [Chb07]. In 2015, Politarczyk defined another equivariant
version of Khovanov homology for periodic links that is an analogue to Borel
equivariant cohomology [Pol15]. In 2018, Borodicz, Politarczyk, and Silvero
extended Politarczyk’s work by utilizing equivariant cubical flow categories to
define an equivariant Khovanov homotopy type [BP17]. They also related the
Borel equivariant homology of the homotopy type to Politarczyk’s equivariant
Khovanov homology. In 2018, Stoffregen and Zhang also constructed a Khovanov
homotopy type for periodic links by constructing an equivariant version of Lawson,
Lipshitz, and Sarkar’s Khovanov functor [SZ18]. It is currently unknown if these
two constructions result in equivalent equivariant homotopy types.
In the following chapters, we will give a third construction of an equivariant
Khovanov homotopy type XKh(L) for periodic links. More precisely, we will prove
the following
2
Theorem 1.1. For a p-periodic link L, the natural action of Z/pZ on L induces a
Z/pZ action on XKh(L) which makes XKh(L) a naive Z/pZ-spectrum.
Theorem 1.2. If L and L′ are equivariantly isotopic p-periodic links, then XKh(L′)
is Borel homotopy equivalent to XKh(L).
By Borel homotopy equivalent, we mean that we can find collection of naive Z/pZ-
spectra Xi and Yi−1 such that we get a composition of roofs
X1 X2 Xi
XKh(L) Y . . . ′1 Yi−1 XKh(L )
where the downward maps are equivariant and induce homotopy equivalences but
the inverse maps Yj → Xi need not be equivariant.
To prove theorems 1.1 and 1.2 we will do the following: we will give two
equivalent definitions for what we mean for a group G to act on a strictly-unitary-
lax-2-functor (chapter 3); we will then define how Z/pZ acts on the Khovanov
functor FKh(L) : 2
n → B and how this group action can be extended to an action
on XKh(L) (chapter 4); and finally we will show that Xkh(L) is an equivariant knot
invariant (chapter 5).
3
CHAPTER II
BACKGROUND
In this chapter we will recall definitions and set notation so that we can
describe both the Khovanov functor F nKh : 2 → B and the construction of
the Khovanov homotopy type XKh(L). We begin with our main object of study,
periodic links.
Definition 2.1. A link L of m components is a piecewise linear embedding of m
disjoint copies of S1 in S3. A knot is a link with only one component.
It is often easier to represent a knot by using a knot diagram, which is the
projection of the knot onto a plane with small breaks to indicate where one strand
crosses over another strand. The convention is that the projection of the over-
strand remains intact while the projection of the under-strand is broken. Figure
1 contains a few examples of knot diagrams.
Slight changes to the embedding of the circles that make up a link L do not affect
how “knotted” or “linked” the components of L are in S3, and so links are only
considered up to isotopy.
Figure 1. Diagrams for the Unknot, Trefoil, and Figure-8 knot.
4
Definition 2.2. Let f, g : qmi=1S1 → S3 be two piecewise linear embeddings of
qm 1 3i=1S in S . An isotopy from f to g is a piecewise linear continuous map H :
qmi=1S1× [0, 1]→ S3 such that H(−, 0) = f , H(−, 1) = g, and H(−, t) is a piecewise
linear embedding of qm 1i=1S in S3 for all t ∈ [0, 1].
Using this definition, we can now describe an equivalence relation for links.
Definition 2.3. Two links L1 and L2 are equivalent if there exists an isotopy
between them.
Explicitly describing the isotopy between two links can often be quite difficult, and
so it is often helpful to use the following theorem of Reidemeister to determine
when two knot diagrams represent the same equivalence class of links.
Theorem 2.4. [Rei74] Two links L1 and L2 are equivalent if and only if a diagram
D1 representing L1 can be transformed into a diagram D2 representing L2 by a
sequence of the following three types of moves
The main focus of this paper will be a specific class of links called periodic
links.
Definition 2.5. A link L is called p-periodic if it possesses a knot diagram with a
2π rotational symmetry about a point not in the image of L.
p
In figure 2, rotating each knot diagram about the marked center point shows that
the Hopf Link is 2-periodic and that the Trefoil is 3-periodic.
5
Figure 2. A 2-periodic diagram for the Hopf Link and a 3-periodic diagram for
the Trefoil
Two p-periodic links L1 and L2 are equivariantly isotopic if there exists an
isotopy between L1 and L2 that respects the
2π rotation symmetry of the two links.
p
As with the non-equivariant case, it is often easier to think of this equivariant
isotopy in terms of diagrams, so we will now define the concept of equivariant
Reidemeister moves.
Definition 2.6. Given a p-periodic knot diagram D for a link L, an equivariant
Reidemeister move of type I (resp. II or III) is the result of performing a regular
Reidemeister move of type I (resp. II or III) and the p − 1 images of that move
under the rotational action on D.
Examples of the first and second equivariant Reidemeister moves can be seen in
figure 3. We now give a proof of the following proposition, which can be thought of
as an equivariant verison of Reidemeister’s theorem.
Proposition 2.7 ([BPS18] - Proposition 2.7). Let L1 and L2 be two p-periodic links
and let D1 and D2 be two p-periodic diagrams representing L1 and L2, respectively.
Every equivariant isotopy from L1 to L2 can be realized by a sequence of equivariant
Reidemeister moves from D1 to D2.
Proof. Quotienting D1 and D2 by the rotation action will result in two isotopic
diagrams Dq1 and D
q
2 representing the quotients of the links L1 and L2. Since D
q
1
6
Figure 3. Examples of equivariant Reidemeister moves.
and Dq2 are isotopic, there is a sequence of regular Reidemeister moves transforming
one into the other. This sequence of moves between Dq and Dq1 2 lifts to a sequence
of equivariant Reidemeister moves from D1 to D2.
Another important fact about periodic links is that Z/pZ acts naturally on
any p-periodic link L. More precisely, i ∈ Z/pZ acts on L by rotating the link 2iπ
p
radians. This is the natural action that will be used to induce an action on XKh(L).
2.1 The Cube Category
The objects of the n-dimensional cube categ(ory) 2n are elements of the
product {0, 1}n. There is a partial ordering on Ob 2n with (u1, . . . , un) ≥
(v1, . . . , vn) wh(ene)ver ui ≥ vi for all 1 ≤ i ≤ n. This partial ordering also induces a
grading on Ob 2n given by the L1-norm ∑n
|u| = ui.
i=1
It will occasionally be useful to know the difference in grading between u and v,
and so we will sometimes write u >k v when u ≥ k and |u| − |v| = k. Additionally,
we will occasionally write u •v when u >1 v in order to emphasize that u and
7
v are joined by a single edge in the cube. The partial ordering on the objects also
induces the morphism structure in 2n with there being a unique morphism ϕu,v
between u and v whenever u ≥ v and no morphism otherwise. (i.e Hom2n(u, v) =
{ϕu,v} when u ≥ v and Hom2n(u, v) = ∅ otherwise.) We will view 2n as a strict 2-
category that contains no non-identity 2-morphisms. It will be helpful later to have
the following sign function
Definition 2.8. For u = {u1, u2, . . . } >1 v = {v1, v2, . . . }, let k be the unique
element of {1, 2, . . . , n} such that uk >∑vk. Definek−1
sgnu,v = ui (mod 2).
i=1
2.2 The Thickened Cube Category
The objects of the thickened cube category 2̂n are composable pairs of
−ϕ−u→,v −ϕ−v,wmorphisms u v → w for any u, v, w ∈ Ob(2n). A morphism between
ϕu,v ϕv,w ′ ϕu′,v′ ′ ϕ ′ ′u −−→ v −−→ w and u −−−→ v ,wv −−−→ w′ is a commutative diagram of the form
ϕu,v ϕv,w
u v w
ϕu,u′ ϕv′,v ϕw,w′
u′ v′ϕ w
′.
u′,v′ ϕv′,w′
Note the upward direction of the middle vertical map. We will occasionally
refer to a morphism by the triple (ϕu,u′ , ϕv′,v, ϕw,w′) of its vertical maps. The
composition of two morphisms (ϕu,u′ , ϕv′,v, ϕw,w′) and (ϕu′,u′′ , ϕv′′,v′ , ϕw′,w′′) is just
formed by vertically stacking the two commutative diagrams, or more succinctly
(ϕu′,u′′ , ϕv′′,v′ , ϕw′,w′′) ◦ (ϕu,u′ , ϕv′,v, ϕw,w′) = (ϕu′,u′′ ◦ϕu,u′ , ϕv′,v ◦ϕv′′,v′ , ϕw,w′ ◦ϕw′,w′′).
This category is the result of a general thickening process applied to the cube
category. A similar process can be applied to any small category.
8
2.3 The Burnside Category
The objects of the Burnside category, B, are finite sets. A morphism
between X and Y in B is a triple (A, s, t) where A is a finite set, s : A → X is
a set map, and t : A → Y is a set map. (s and t are often called the source map
and target map respectively.) The triple (A, s, t) is often called a correspondence
(or span) between X and Y and is usually depicted as
A
s t
X Y .
Given two correspondences (A, sA, tA) ∈ HomB(X, Y ) and (B, sB, tB) ∈
HomB(Y, Z) the composition of these two morphisms is given by (C, sC , tC) where
C = A×Y B = {(a, b) ∈ A×B | tA(a) = sB(b)}
sC(a, b) = sA(a) tC(a, b) = tB(b).
Diagrammatically this looks like
A×y B
A B
sA tA sB tB
X Y Z.
Additionally, given two correspondences (A, sA, tA) and (B, sB, tB) from X to Y , a
morphism of correspondences is a bijection between A and B such that following
triangles commute
A
B
X Y .
9
∼=
The Burnside category is a weak 2-category with the morphisms of correspondences
acting as the 2-morphisms. See, for instance, [LLS15b, Section 3] for more details.
In order to define the Khovanov functor in section 2.5, it will help to define
a functor A : B → Ab from B to the category of abelian groups.
Definition 2.9. For X ∈ Ob(B), define A(X) = Z〈X〉, the free abelian group
with basis X. For a correspondence (A, s, t) ∈ HomB(X, Y ) define the map A(A) :
A(X)→ A(Y ) on each of the basis elements x ∈ X as
∑
A(A)(x) = |s−1(x) ∩ t−1(y)|y.
y∈Y
If we consider the objects of B to be isomorphism classes of correspondences
instead of just correspondences, then we can view B as a regular category. In this
case, A identifies B as a full subcategory of Ab whose objects are finitely generated
free abelian groups and whose morphisms are matrices with non-negative entries.
2.4 Functors from the Cube to the Burnside Category
We will now give two equivalent definitions for a functor D : 2n → B from
the cube category to the Burnside category. As mentioned in sections 2.1 and 2.3
we can view 2n as a strict 2-category and B as a weak 2-category. This means that
when we refer to a functor from the cube category to the Burnside category, we
really mean a strictly-unitary-lax-2-functor. A more detailed explanation for these
types of functors can be found in [Béa67], where they are referred to as strictly
unitary homomorphisms.
Definition 2.10. A strictly-unitary-lax-2-functor D from the cube category 2n to
the Burnside category B consists of
(i) A finite set D(u) ∈ B for every u ∈ Ob (2n)
10
( )
(ii) A finite correspondence D(ϕu,v) ∈ HomB D(u), D(v) for every u ≥ v
(iii) A 2-isomorphism Du,v,w : D(ϕv,w)×D(v) D(ϕu,v)→ D(ϕu,w) for all u > v > w
such that for all u > v > w > z the following diagram commutes
Id×Du,v,w
D(ϕw,z)×D(w) D(ϕv,w)×D(v) D(ϕu,v) D(ϕw,z)×D(w) D(ϕu,w)
Dv,w,z × Id Du,w,z
D(ϕv,z)×D(v) D(ϕu,v) D(ϕu,z)
Du,v,z .
I(t should be noted that D) (ϕw,z)×D(w) D(ϕv,w)×D(v) D(ϕu,v) denotes D(ϕw,z)×D(w)
(D(ϕv,w) ×D(v) D(ϕu,v)) when going across the top of the diagram, and
D(ϕw,z) ×D(w) D(ϕv,w) ×D(v) D(ϕu,v) when going down the left side. These are
not the same but they are canonically identified.
Notice that condition (ii) in the above definition requires a finite
correspondence be given for every morphism in 2n. In some sense this is the
complete set of data for the morphisms, and so we will often use the phrase
“complete definition” when we want to specifically refer to definition 2.10.
The second definition is composed of a similar set of data. However, instead
of giving a correspondence D(ϕu,v) for every morphism ϕ
n
u,v in 2 , this definition
only gives a correspondence for each edge of the cube. With this in mind (and to
contrast the above definition), we will often refer to definition 2.11 as the “edge
definition” for D.
Definition 2.11. A strictly-unitary-lax-2-functor D from the cube category 2n to
the Burnside category B consists of
(e.i) A finite set D(u) ∈ B for every u ∈ 2n
11
( )
(e.ii) A finite correspondence D(ϕu,v) ∈ HomB D(u), D(v) for every edge u •v
(e.iii) An isomorphism Du,v,v′,w : D(ϕv,w) ×D(v) D(ϕu,v) → D(ϕv′,w) ×D(v′) D(ϕu,v′)
•v •
for every face u w• .
v′
•
Such that the following two conditions are satisfied
•v •
(C1) For every face u •w• , D
−1
u,v′,v,w = D
′ u,v,v
′,w
v
•v •w
′′
•
•
(C2) For every three dimensional face u •v′ w′ •••z the following square•
•
v′′ •
•
w
commutes
Dv,w′′,w′,z × Id
D(ϕw′′,z)×D(w′′) D(ϕv,w′′)×D(v) D(ϕu,v)
Id×Du,v,v′,w′′ D(ϕw′,z)×D(w′) D(ϕv,w′)×D(v) D(ϕu,v)
D(ϕw′′,z)×D(w′′) D(ϕv′,w′′)×D(v′) D(ϕu,v′) Id×Du,v,v′′,w′
Dv′,w′′,w,z × Id D(ϕw′,z)×D(w′) D(ϕv′′,w′)×D(v′′) D(ϕu,v′′)
D(ϕw,z)×D(w) D(ϕv′,w)×D(v′) D(ϕu,v′) Dv′′,w′,w,z × Id
D(ϕw,z)×D(w) D(ϕv′′,w)×D(v′′) D(ϕu,v′′)
Id×Du,v′,v′′,w
Given the data from the complete definition, it is easy to produce the
required data for the edge definition by simply setting D −1u,v,v′,w = Du,v′,w ◦ Du,v,w.
Showing that the data from the edge definition is sufficient to produce the data for
the total definition requires selecting maximal chains between any two vertices. A
maximal chain mu,v between vertices u >k v is a choice of edges
12
u = zu,v •0 ... • z
u,v • • u,v
i .... zk = v.
connecting u and v. By using these maximal chains and the face isomorphisms
from (e.iii), it is possible to recover the complete definition of D. For a more
detailed explanation, see [LLS15a, Prop. 4.3].
2.5 The Khovanov Functor
Following section 4 of [LLS15a], we now define the Khovanov functor
F (L) : 2nKh → B which is a specific strictly-unitary-lax-2-functor from the
cube category to the Burnside category that captures the information from the
Khovanov chain complex. To do this, we will first define a functor F : (2n)opKh,Ab →
Ab and then refine it to produce FKh(L).
For an oriented link diagram, L, with n crossings and a fixed ordering on the
crossings, we can construct a cube of resolutions for L by replacing each crossing
with one of the following two resolutions
0 1
which are referred to as the the 0-resolution of 1-resolution respectively. Given
a vertex u = {u1, . . . un} ∈ Ob (2n), performing the ui-resolution on the ith
crossing of the link results in a collection of embedded circles in S2, denoted Lu.
Additionally, for an edge u •v , Lu can be changed into Lv by either merging two
circles together or splitting apart a circle into two circles. Let V = Z〈x+, x−〉 be a
free rank-2 Z module with the following multiplication and comultiplication
13
m(x+ ⊗ x+) = x+ ∆(x+) = x+ ⊗ x− + x− ⊗ x+
m(x+ ⊗ x−) = x− ∆(x−) = x− ⊗ x−
m(x− ⊗ x+) = x−
m(x+ ⊗ x+) = 0
W⊗e can define a function F n opKh,Ab : (2 ) → Ab as follows. For u ∈ 2n, FKh,Ab(u) =
S∈π (L ) V . For the morphism ϕu,v corresponding to the edge u •v , there are0 u
two cases. When Lu is obtained from Lv by merging two circles, F
op
Kh,Ab(ϕu,v)
applies the multiplication map to the corresponding factors of FKh,Ab(u) and the
identity map to the remaining factors. In the other case, Lu is obtained from Lv
by splitting apart a single circle, and so F opK ,Ab(ϕu,v) applies the comultiplicationh
map to corresponding factor of FKh,Ab(u) and the identity map to the remaining
factors. The total complex of the cube, which can be formed by multiplying the
edge map u •v by (−1)sgnu,v and summing over the vertices of each grading, is
the Khovanov complex.
To refine F : (2n)opKh,Ab → Ab into a strictly-unitary-lax-2-functor
F nKh : 2 → B, it suffices to describe a set of data for the vertices, edges, and
faces of the cube that satisfy condition (C1) and (C2) from definition 2.13. For
each u, FKh(u) = {x : πo(Lu) → {x+, x−}} is the preferred basis of the Khovanov
generators. For the∑edge morphism u •v and for each y ∈ FKh(v), notice that
F (ϕopKh,Ab u,v)(y) = x∈F (u) x,yx where x,y is a matrix whose entries are 0 and 1.Kh
This means, we can define FKh(ϕu,v) = {(y, x) ∈ FKh(v)× FKh(u)|x,y = 1}.
To define the isomorphism FKh u,v,v′,w : FKh(ϕv,w) ×F F (ϕKh(v) Kh u,v) →
•v •
FKh(ϕv′,w)×F (v′) FKh(ϕu,v′) for the face u •w• , we first note that since FKh Kh,Ab
v′
is a commutative diagram there is a 2-isomorphism between FKh(ϕv,w) ×FKh(v)
14
Figure 4. (a) An example of a link L that would result in a ladybug configuration.
(b) An isotopy of the 00-resultion of L which gives the ladybug configuration its
name. (c) The resolution of the link from (a) with the right arc pairs 1 and 2
labeled.
FKh(ϕu,v) and FKh(ϕv′,w) ×F (v′) FKh(ϕu,v′). Namely, for x ∈ FKh(u) and z ∈Kh
FKh(w), the cardinalities of
A := s−1x,z (z) ∩ t−1(x) ⊆ FKh(ϕv,w)×F (v) FKh(ϕu,v) andKh
A′ −1x,z := s (z) ∩ t−1(x) ⊆ FKh(ϕv′,w)×FKh(v′) FKh(ϕu,v′)
are the (x, z) entries in the matrix FKh,Ab(ϕ
op
u,v) ◦ FKh,Ab(ϕop
op
v,w) and FKh,Ab(ϕu,v′) ◦
F opKh,Ab(ϕv′,w) respectively, and these two matrices are the same. In most cases
the cardinalities of the above sets are either zero or one. In either case, there is a
unique isomorphism F ′Kh u,v,v′,w|Ax,z : Ax,z → Ax,z. The only exceptional case is
when a circle Cw in Lw splits to form two circles C
1
v and C
2
v in Lv and two circles
C1v′ and C
2
v′ in Lv′ ; these two circles merge back to a single circle Cu in Lu; x labels
Cu by x−; and z labels Cw by x+.
15
In this case, we can define the isomorphism by using the ladybug
configuration (See figure 4). To use the ladybug configuration, we first draw
the circle Cw from Lw. We then draw an arc av where we would need to pinch
Cw together to form the two circles Cv1 and Cv2 in Lv. We then draw a second
arc av′ where we would need to pinch the Cw circle together to form the two
circles Cv′ and Cv′ in L
′
v. Using av and av′ we can define the right pair of arcs in1 2
(Cw, ∂av ∪ ∂av′) as the arcs you get by walking along av and av′ and then turning
right. We then can choose to label one of the right pair arcs as 1 and the other
as 2. (This choice of labeling will not affect FKh u,v,v′,w). Each right pair arc is
contained entirely within one of the two circles in both Lv and Lv′ . This means
we can label the two circle of L 1 2v as Cv and Cv based on which right pair arc they
contain. Similarly, we can label the two circles of Lv′ as C
1
v′ and C
2
v′ . With these
identifications we can define the two elements of Ax,z as
α = ((Cw → x+), (C1v , C2v )→ (x−, x+), (Cu → x−)),
β = ((Cw → x+), (C1 2v , Cv )→ (x+, x−), (Cu → x−))
and the two elements of A′x,z as
α′ = ((Cw → x+), (C1v′ , C2v′)→ (x−, x+), (Cu → x−)),
β′ = ((C → x ), (C1w + v′ , C2v′)→ (x+, x−), (Cu → x−)).
With these two two-element sets identified FKh u,v,v′,w can be defined by mapping
α →7 α′ and β 7→ β′. From the definition of FKh u,v,v′,w it is clear that F−1Kh u,v′,v,w
= FKh u,v,v′,w so condition (C1) is satisfied. Checking condition (C2) requires
•v •w
′′
•
•
fixing a three-dimensional face u •v′ w′ •••z and fixing Khovanov generators•
•
v′′ •
•
w
x ∈ FKh(u) and z ∈ FKh(z). There are six correspondences coming from the six
16
paths through the cube which correspond to three bijections FKh,u,v∗,w∗ and three
bijections FKh v∗,w∗,z. One needs to check that these bijections agree (taking into
account the ladybug formation.) The proof that these bijections do in fact agree
follows from lemmas 5.14 and 5.17 in [LS14].
2.6 Homotopy Colimits
In order to define the Khovanov homotopy type, we will end up needing
to take a homotopy colimit, so for completeness sake, we include Vogt’s definition
[Vog73] as described by Lawson, Lipshitz, and Sarkar in [LLS15a, Section 2.9].
Definition 2.12. Let C be a small category and D : C → Top• be a C -diagram in
Top• (or Top). Let
Cn(A,B) = {(fn, . . . , f1) ∈ (Mor(C ))n|fn ◦ · · · ◦ f1 : A→ B is defined in C } n > 0
C0(A,A) = {(idA)} C0(A,B) = ∅ for A 6= B.
The homotopy colim(it of D, hocolim(D), is⊔ ⊔ )∞
Cn(A,B)× In ×D(A) ∪ {∗}/ ∼
A,B∈C n=0
where I is the unit interval and {∗} an extra point, where ∼ is given as follows:(tn, fn, . . . , t2, f2;x) f1 = id(tn, fn, . . . , fi+1, titi−1, fi−1, . . . , t1, f1;x) fi = id, 1 < i(tn, fn, . . . , ti+1, fi+1 ◦ fi, ti−1, , . . . , t1, f1;x) ti = 1, i < n(tn, fn, . . . , t1, f1;x) ∼ (tn−1, fn−1, . . . , t1, f1;x) tn = 1(tn, fn, . . . , fi+1;D((fi ◦ f1)(x)) ti = 0∗ x = base point
with {∗} as base point for a diagram D in Top•. The unbased version is obtained by
deleting {∗} and the last relation.
17
2.7 The Khovanov Homotopy Type
We now have all the required background needed to define the Khovanov
homotopy type XKh(L) given a link diagram L and the Khovanov functor FKh(L) :
2n → B. The construction from [LLS15a, Section 4] begins by defining a family of
ϕu,v ϕv,w ( )
functors F̂ k nKh : 2̂ → Top• for eack k ∈ N. For each u −−→ v −−→ w ∈ Ob 2̂n we
define (
k −ϕ
∨ ∏
−u→,v −ϕ−v,
)
F̂Kh u v →
w
w = Sk.
a∈FKh(ϕu,v) b∈FKh(ϕv,w)
s(b)=t(a)
−ϕ−u→,v ϕv,wTo define the image of a morphism (ϕu,u′ , ϕv′,v, ϕw,w′) between u v −−→ w and
′ ϕ ′ ′ ϕ ′−−u−,→v ′ −−v−,w
′
u v → w′∨ we need∏to define a map ∨ ∏
Sk −→ Sk.
a∈FKh(ϕ ′ ′u,v) b∈FKh(ϕv,w) a ∈FKh(ϕu′,v′ ) b ∈FKh(ϕv′,w′ )
s(b)=t(a) s(b′)=t(a′)
To do this we note that it suffices to construct the map on each piece of the wedge
sum, so fix an a ∈ FKh(ϕu,v). The maps FKh u,u′,v and FKh u,v′,v induce a bijection
F ∼Kh(ϕu,v) = FKh(ϕv′,v)×FKh(v) FKh(ϕu′,v′)×F (u′) FKh(ϕKh u,u′).
This means each a can be identified with a triple (y, a′, x) in the fiber product
above. With this identification, we can define F̂ kKh to send the wedge summand
corresponding to a to the wedge summand corresponding to a′. Similarly, the maps
FKh v′,w,w′ and FKh u,v′,v give an isomorphism
FKh(ϕv′,w′) = FKh(ϕw,w′)×F (w) FKh(ϕv,w)×F (v) FKh(ϕv′Kh Kh ,v)
so for each b ∈ FKh(ϕv′,w′) we get b′ = (z, b̄, ȳ). If we let ∆b represent the diagonal
map then, w∏e can consider the s∏equence of ma∏ ∆ ∏
ps ∏
k −−b−→bS Sk ∼= Sk
b∈FKh(ϕv,w) b∈FKh(ϕv,w) b′=(z,b̄,ȳ)∈F ′Kh(ϕv′,w) b ∈FKh(ϕv′,w)
s(b)=t(a) s(b)=t(a) b̄=b b′=(z,b,y)
ȳ=y s(b)=t(a)
18
We now note that {b′ ∈ F ′Kh(ϕu′,v′)|b = (z, b, y)} is a subset of {b′ ∈
FKh(ϕu′
′
,v′)|t(a ) = s(b′)} since s(b′) = s(y) = t(a′). This means we can extend
the above map to a map
∏ ∏ ∏ ∏ ∏ ∏
Sk b
∆b Sk ∼ Sk k−−−−−→ = → S
b∈FKh(ϕv,w) b∈FKh(ϕv,w) b′=(z,b̄,ȳ)∈F (ϕ ′ ) b′Kh ∈FKh(ϕ ) b
′∈F (ϕ )
v ,w v′,w Kh v′,w′
s(b)=t(a) s(b)=t(a)
b̄=b b′=(z,b,y) s(b′)=t(a′)
ȳ=y s(b)=t(a)
by mapping to the base point in the remaining factors. This is our desired
map. Applying this map to every part of the wedge sum gives the definition of
F̂ kKh((ϕu,u′ , ϕv′,v, ϕw,w′)).
We now note that there are natural transformations Sn ∧ F̂ k (L)→ F̂ k+nKh Kh (L)
between our family of thickenedKhovanov functors F̂ kKh(L) given by
n ∧ ∨ ∏ k ∨ ∏ ∨ ∏S S  ∼ Sn ∧ Sk → Sn+k=
a∈F (ϕ b∈F (ϕ ) b∈ϕ b∈ϕKh u,v) Kh v,w a∈FKh(ϕu,v) v,w a∈FKh(ϕu,v) v,w
s(b)=t(a) s(b)=t(a) s(b)=t(a)
The first part of this map is just given by commuting the smash product with the
wedge sum. The second part of this map is given by applying the following map to
each ∏ ∏
σn : Sn ∧ X ni → S ∧Xi
i i
where we view Sn ∧ X as [0, 1]n × X/(∂[0, 1]n × X ∪ [0, 1]n × {∗}) and where
σn(y, x1, . . . , x
n
n) = ((y, x1) . . . (y, xn). If all of the Xi are (k − 1)-connected this σ
induces isomorphisms on πi for 0 ≤ i ≤ 2k − 2. With these natural transformations,
we can view all of the F̂ kKh’s as being a diagram of spectra F̂Kh : 2̂
n → S .
Finally, we let 2̂n+ be the category obtained from 2̂
n by adding a new object
∗ and new morphisms ((u → v → w) → ∗) from each vertex of 2̂n with w 6=
→ + +
0. Similarly, we will define F̂ k : 2̂n → S by setting F̂ kKh + Kh |2̂n = F̂ kKh and
+ +
F̂ kKh (∗) = {∗}. Taking the homotopy colimit of F̂Kh and formally desuspending
19
by the number of negative crossings n− in L produces XKh(L). That is XKh(L) =
Σ−
+
n−hocolim(F̂Kh ).
We should mention that in definition 2.15 we described the homotopy
colimit for a functor D : C → Top• and not a functor to the category of spectra.
+
To resolve this discrepancy, we note that the homotopy colimits of the the F̂ kKh ’s
+
together with the structure maps σn form a classical spectrum (hocolim(F̂ kKh ), σ
n)
+ + +
and that (hocolim(F̂Kh )) k k
n
k = hocolim(F̂Kh ), and so (hocolim(F̂Kh ), σ ) =
XKh(L).
We should also mention that for the remainder of our discussion, we will
suppress the formal grading shift as it is clear in the proofs of invariance that any
grading shifts will agree with the above.
20
CHAPTER III
GROUP ACTIONS ON CATEGORIES AND FUNCTORS
The main goal of this chapter is to give two equivalent definitions for what
it means for a group G to act on a strictly-unitary-lax two functor D : 2n → B
from the cube category to the Burnside category. We begin by giving a definition
for what it means for a group G to act on a category C.
Definition 3.1. For a group G, the group action of G on a category C is the
following collection of data:
(1) an autoequivalence Gg : C → C for each g ∈ G
(2) an isomorphism of functors ηg,h : GhG ∼g = Ghg for each pair g, h ∈ G
such that for all g, h, i ∈ G the following diagram of functors is commutative:
GiGhGg GiGhg
GihGg Gihg.
Given g ∈ G, A ∈ Ob(C), and f a morphism in C, we will usually write gA to
mean Gg(A) and gf to mean Gg(F ). We now define what it means for G to act on a
functor between two 1-categories.
Definition 3.2. Let F : C → C ′ be a functor between categories C and C ′ and
let G be a group that acts on C. A group action of G on F is a collection of maps
Rg,A : F(A) → F(gA) such that for all A,B ∈ Ob(C), f ∈ Hom(A,B), and for all
g, h ∈ G, the following hold:
(1) For the identity element e ∈ G, Re,A is the identity morphism
21
(2) Rh,gA ◦Rg,A = Rhg,A
(3) The following diagram commutes
F(f)
F(A) F(B)
Rg,A Rg,B
F(gA) F(gB).
(F ◦ Gg)(f)
Unfortunately, the above definition does not immediately apply to a strictly-
unitary-lax-2-functor D : 2n → B because such functors only preserve compositions
of morphisms up to an isomorphism. Instead, we describe the following larger set of
data to explain precisely how G acts on D.
Definition 3.3. Let D : 2n → B be a strictly-unitary-lax-2-functor from the
cube category to the Burnside category described by the “complete set of data”
(definition 2.10) and let G be a group that acts on 2n. A group action of G on D
is a collection of maps
Rg,u : D(u)→ D(gu) and Sg,ϕu,v : D(ϕu,v)→ D(ϕgu,gv)
for any u, v, w ∈ Ob(2n) with u >k v >l w and for any g, h ∈ G, such that
(c.i) For the identity element e ∈ G, Re,u : D(u) → D(eu) and Se,ϕu,v : D(ϕu,v) →
D(ϕeu,ev) are the identity maps
(c.ii) Rh,gu ◦Rg,u = Rhg,u
(c.iii) Sh,ϕu,v ◦ Sg,ϕu,v = Shg,ϕu,v
(c.iv) Rg,u ◦ s = s ◦ Sg,ϕu,v and Rg,u ◦ t = t ◦ Sg,ϕu,v
22
(c.v) Sg,ϕu,w ◦Du,v,w = Dgu,gv,gw ◦ (Sg,ϕv,w × Sg,ϕv,w).
We have chosen to label the conditions (c.i) through (c.v) in order to emphasize
that this definition is for a group action on D where D is described by the complete
set of data. The following definition is a similar collection data but for D described
by the edge set of data. To distinguish between the two definitions, we will include
the superscript e on the maps and we will label the conditions (e.i) - (e.v).
Definition 3.4. Let D : 2n → B be a strictly-unitary-lax-2-functor from the cube
category to the Burnside category described by the edge set of data (definition 2.11)
and let G be a group that acts on 2n. A group action of G on D is a collection of
maps
Reg,u : D(u)→ D(gu) and Seg,ϕ : D(ϕu,v)→ D(ϕgu,gv)u,v
•v •
for any u, v, v′, w ∈ Ob(2n) with u • •w and for any g, h ∈ G, such that
v′
(e.i) For the identity element e ∈ G, Re,u : D(u) → D(eu) and Se,ϕu,v : D(ϕu,v) →
D(ϕeu,ev) are the identity maps
(e.ii) Reh,gu ◦Re eg,u = Rhg,u
(e.iii) Seh,ϕ ◦ Se eu,v g,ϕ = Su,v hg,ϕu,v
(e.iv) Re ◦ s = s ◦ Se and Re ◦ t = t ◦ Seg,u g,ϕu,v g,u g,ϕu,v ( )
(e.v) (Se eg,ϕ ′ ×Re ′ Sg,ϕ ′ ) ◦D e eu,v,v′,w = Dgu,gv,gv′,gw ◦ Sv ,w g,v u,v g,ϕ ×Re Sv,w g,v g,ϕu,v
An advantage of the second definition is that it contains much less data since the
maps Seg,ϕ need only be defined for the edges of the cube. Additionally, we onlyu,v
need to check that certain conditions hold for each face of the cube. We will now
show the promised equivalence between definition 3.3 and definition 3.4.
23
Proposition 3.5. The complete set of data for the action of G on D : 2n → B
can be used to construct the edge set of data for the group action in such a way that
Rg,u = R
e
g,u for all u ∈ 2n and that S eg,ϕu,v = S •g,ϕ for all edges u v . Similarly,u,v
the edge set of data for the group action can be used to construct the complete set of
data in such a way that Re n eg,u = Rg,u for all u ∈ 2 and that Sg,ϕ = Sg,ϕu,v for allu,v
edges u •v .
Proof. We begin by showing that the complete data definition of the G-action
produces the edge set of data. To do this, we need to define Reg,u and S
e
g,ϕ andu,v
show that conditions (e.i) − (e.v) hold. We require that Reg,u = Rg,u for all vertices
and that Seg,ϕ = Sg,ϕu,v for all edges, which immediately implies that conditionsu,v
(e.i) − (e.iv) follow directly from (c.i) − (c.iv). All that remains to be shown is
•v •
that condition (e.v) holds. To prove this, we consider any face u • •
w and note
v′
that in the following diagram, the two center squares commute because of condition
(c.v).
Du,v,v′,w
−1
Du,v,w Du,v′,w
D(ϕv,w)×D(v) D(ϕu,v) D(ϕu,w) D(ϕv′,w)×D(v′) D(ϕu,v′)
Se × Se e e eg,ϕv,w g,ϕ Sg,ϕ Sg,ϕ ′ × Su,v u,w v ,w g,ϕu,v′
D(ϕgv,gw)×D(gv) D(ϕgu,gv) D(ϕgu,gw) D(ϕgv′,gw)×− D(gv′) D(ϕgu,gv′)D 1gu,gv,gw Dgu,gv′,gw
Dgu,gv,gv′,gw
24
Since D −1u,v,v′,w = Du,v′,w ◦ Du,v,w the outer square of the diagram commutes and
so condition (e.v) holds, and so definition 3.3 can be used to construct the data for
definition 3.4.
To show the other direction, we need to show that the collections of maps
for the edge definition of the G-action give us a collection of maps for the complete
G-action. We must again define Rg,u = R
e
g,u. This trivially satisfies the first half of
condition (c.i) and condition (c.ii). In order to define Sg,ϕu,v we will use the idea of
maximal chains. For each pair of vertices u >k v, choose any sequence of k edges
u = zu,v • ... • zu,v • .... • zu,v0 i k = v.
to be the maximal chain mu,v from u to v. Given these choices of maximal chains
we can define D(ϕu,v) = D(ϕzu,v ,zu,v )×D(zu,v ) D(ϕzu,v ,zu,v )× ...×D(zu,v) D(ϕzu,v ,zu,v).k k−1 k−1 k−1 k−2 1 1 0
Given (xk, ..., x1) ∈ D(ϕ ) we can now apply our Seu,v g,∗ maps to each element in
the tuple (xk, ..., x1) since each element comes from an edge in the cube. This gives
us a map from D(ϕu,v) to D(ϕgzu,vk ,gz
u,v ) ×D(gzu,v ) D(ϕgzu,v ,gzu,v ) × ... ×k−1 k−1 k−1 k−2 D(gzu,v1 )
D(ϕgzu,v ,gzu,v) which sends1 0 ( )
(x , x e ek k−1, ... , x2, x1) 7−→ Sg,ϕ u,v u,v (xk), ... , Sg,ϕ u,v u,v (x1) .
z ,z
k k− z1 1 ,z0
If k = 1 then t(hen S = Seg,ϕu,v g,ϕ for the edge u •v as required. However,u,v )
if k > 1, then Se, ϕzu,v ,zu,v (xk), . . . , S
e, ϕzu,v ,zu,vg g (x1) may not be ink k−1 1 0
D(ϕgu,gv) since the action of G may not send m
u,v to mgu,gv. More explicitly,
gu = gzu,v • . . . • gzu,v • . . . • gzu,v0 i k = gv need not equal gu =
zgu,gv • ... • zgu,gv •0 i .... • z
gu,gv
k = gv. However, since both of these chains start
at gu and gv, we can apply a series of face isomorphisms D∗,∗,∗,∗ that are included
in the data of our functor to get a map from
D(ϕgzu,v ,gzu,v )×D(gzu,v D(ϕ u,v u,v )× ...×D(ϕ u,v u,v)k k−1 k−1) gzk−1,gzk−2 gz1 ,gz0
25
to
D(ϕzgu,gv ,zgu,gv)×− D(zgu,gv) D(ϕ− zgu,gv ,zgu,gv)× ...×D(ϕ gu,gv gu,gv) = D(ϕ− − z ,z gu,gv)k k 1 k 1 k 1 k 2 1 0
as required. This means, we can define Sg,ϕu,v as the composition of the product of
our Seg,∗ maps and some number of face isomorphisms. It follows immediately that
the second part of condition (c.i) holds. To check that condition (c.iii) holds, we
first note that by our description the map Sh,ϕu,v ◦Sg,ϕu,v is made up of four parts: a
product of Seg,∗ maps, a composition of face isomorphisms D
gi g1
∗,∗,∗,∗ ◦ · · · ◦ D∗,∗,∗,∗,
a product of Seh,∗ maps, and finally a second composition of face isomorphisms
Dhi h1∗,∗,∗,∗ ◦ · · · ◦D∗,∗,∗,∗. By making use of condition (e.v) we can commute the product
of Seh,∗ maps past the first set of face isomorphisms, and by using condition (e.iii),
we can compose the product of Seh,∗ maps with the product of the S
e
g,∗ maps to give
u(s )
Sh,ϕu,v ◦ Sg,ϕu,v (x)
h ′ g ( )
= D j ◦ · · · ◦Dh1 j g1 e e∗,∗,∗,∗ ∗,∗,∗,∗ ◦ (Sh,∗, . . . Sh,∗) ◦D∗,∗,∗,∗ ◦ ·(· · ◦D∗,∗,∗,∗ ◦ Sg,∗(xk), . . . Sg,∗(x1)hj′ )= D∗,∗,∗,∗◦· · ·◦ gDh1 j g1 e e e e∗,∗,∗,∗◦Dh∗,h∗,h∗,h∗◦· · ·◦Dh∗,h∗,h∗,h∗◦ (Sh,∗◦S(g,∗)(xk), . . . (Sh,∗◦Sg,∗)(x1)hj′ g )= D h j g1 e e∗,∗,∗,∗ ◦ · · · ◦ D 1∗,∗,∗,∗ ◦ Dh∗,h∗,h∗,h∗ ◦ · · · ◦ Dh∗,h∗,h∗,h∗ ◦ (Shg,∗(xk), . . . (Shg,∗(x1)
= Shg,ϕu,v(x)
as required. It is easy to check that condition (c.iv) follows directly from condition
(e.iv). Finally, for condition (c.v), we know that the map Du,v,w is defined to be
a composition of face isomorpisms between mu,v ∪ mv,w and mu,w and that these
isomorphisms commute with the group action in the sense of (e.v). Hence, the
proof of (c.v) is analogous to our proof of (c.iii). Thus, the complete set of data
for the group action can be constructed from the edge set of data.
26
CHAPTER IV
A Z/pZ-ACTION ON THE KHOVANOV HOMOTOPY TYPE
The goal of this chapter is to prove theorem 1.1., i.e. the natural group
action of Z/pZ on a p-periodic link L induces an action on XKh(L) that makes
XKh(L) a naive G-spectrum.
We begin by explaining how the action of Z/pZ on L induces an action on
2n. Recall from section 2.5 that if we fix an ordering of the n crossings of a knot
diagram for L, then given a vertex u = {u1, . . . , un} ∈ Ob(2n) performing the ui-
resolution on the ith crossing of L forms a collection of embedded circles Lu. Since
g ∈ Z/pZ acts on L by rotating the knot diagram 2gπ radians about a central axis,
p
we can define gLu to be the image of Lu under this rotation. It is clear that gLu is
identical to some other resolution Lu′ in Khovanov’s cube of resolution where u
′ =
{u′1, . . . u′n} is another vertex in Ob(2n), and so it makes sense to define gu = u′.
Clearly |u| = |gu| since the rotation action does not change any of the resolutions
in Lu. Further, for any vertices u >k v, we know that gu >k gv, and since the Hom
sets in 2n contain either 0 or 1 elements the only way to satisfy the condition in
definition 3.1 is for g to map ϕu,v to ϕgu,gv.
A more succinct way to state the above is that the action of Z/pZ of
L induces a permutation of the n crossings of L, and so Z/pZ acts on 2n by
permuting the n-coordinates of {0, 1}n in the same manner.
We can extend the induced action of Z/pZ on 2n to an action on 2̂n, by
defining g to act on objects by sending (u→ v → w)→7 (gu→ gv → gw) and to act
on morphisms by sending (ϕu,u′ , ϕv′,v, ϕw,w′) 7→ (ϕgu,gu′ , ϕgv′,gv, ϕgw,gw′).
27
Since we now have a description for the induced group action on 2n and
2̂n, we can now describe how Z/pZ acts on the Khovanov functor FKh and the
thickened Khovanov functor F̂Kh.
Proposition 4.1. Let L be a p-periodic link and let FKh(L) : 2
n → B be the
associated Khovanov functor. The natural action of Z/pZ on L induces an action
2n which induces an action on FKh(L).
Proof. We will prove this proposition by describing a collection of maps that
satisfy the edge definition for a group action on a strictly-unitary-lax-2-functor
(definition 3.4). That is, we will define maps Reg,u : FKh(u) → FKh(gu) and
Seg,ϕ : FKh(ϕu,v)→ FKh(ϕgu,gv) that meet conditions (e.i)− (e.v).u,v
For any vertex u, FKh(u) = {x : π0(Lu) → {x+, x−}} is the set of
labellings of the circles in Lu by the preferred Khovanov generators. Since Lgu is
just a rotation of Lu, there is an induced map g∗ : π0(Lu) → π0(Lgu). This means
that for any x ∈ FKh(u), we can define Rg,u(x) = x ◦ (g −1∗) . It is clear that this
definition of Rg,u satisfies the first half of condition (e.i) and condition (e.ii).
For the edge morphism u •v , FKh(ϕu,v) is defined by FKh(ϕu,v) = {(y, x) ∈
FKh(v)× FKh(u)|x,y = 1}, so we define Sg,ϕu,v : FKh(ϕu,v)→ FKh(ϕgu,gv) by setting
Seg,ϕ ((y, x)) = (R
e e
g,v(y), Rg,u(x)). It is clear that this satisfied the second half ofu,v
condition (e.i). We can check directly that condition (e.iii) is satisfied since
(Sh,ϕgu,gv ◦ Sg,ϕu,v)((y, x)) = ((Sh,ϕgu,gv)(Rg,v(y), Rg,u(x)) )
= (Rh,gv ◦Rg,v)(y), (Rh,gu ◦Rg,u)(x)
= (Rhg,v(y), Rhg,u(x))
= (Shg,ϕu,v(y, x)).
Again, checking directly, we get that
28
(s ◦ Sg,ϕu,v)((y, x)) = s(Rg,v(y), Rg,u(x)) = Rg,u(x) = Rg,u(s(y, x)) = (Rg,u ◦ s)((y, x))
and that
(t ◦ Sg,ϕu,v)((y, x)) = t(Rg,v(y), Rg,u(x)) = Rg,v(y) = Rg,v(t(y, x)) = (Rg,u ◦ t)((y, x))
and so condition (e.iv) is satisfied.
All that remains for us to check is that our definitions satisfy condition
•v •
(e.v). That is, we need to check that for any face u •w• the following diagram
v′
commutes
Fu,v,v′,w
FKh(ϕv,w)×F F (ϕ ) F (ϕ ′ )× ′Kh(v) Kh u,v Kh v ,w FKh(v ) FKh(ϕu,v′)
Seg,ϕ × Se Se ee × e Sv,w Rg,v g,ϕu,v g,ϕv′ R,w g,v′ g,ϕu,v′
FKh(ϕgv,gw)×F (gv) FKh(ϕgu,gv) FKh(ϕgv′,gw)×F ′ F (ϕKh Kh(gv ) Kh gu,gv′)
Fgu,gv,gv′,gw .
In section 2.6, we noted that for x ∈ FKh(u) and z ∈ FKh(w) we can define the sets
Ax,z := s
−1(x) ∩ t−1(z) ⊆ FKh(ϕv,w)×FKh(v) FKh(ϕu,v)
A′x,z := s
−1(x) ∩ t−1(z) ⊆ FKh(ϕv′,w)×F (v′) FKh(ϕKh u,v′)
Applying the vertical maps in the diagram above to these two sets sends the sets
A ′x,z and Ax,z to
A := s−1(R (x)) ∩ t−1Rg,u(x),Rg,w(z) g,u (Rg,w(z)) ⊆ FKh(ϕgv,gw)×F (gv) FKh(ϕgu,gv)Kh
and
A′ −1 −1 ′ ′Rg,u(x),R (z) := s (Rg,u(x)) ∩ t (Rg,w(z)) ⊆ FKh(φgv ,gw)×F (gv′) FKh(φg,w Kh gu,gv )
29
respectively. We know the cardinality of these four sets will always be the same
and will always be 0,1, or 2. When the cardinality of the sets is either 0 or 1 the
above square obviously commutes, so we only need to consider the case when the
cardinality is 2. In this case, we used the ladybug configuration which involved
drawing arcs av and av′ , which allowed us to define a right pair of arcs in the circle
Cw. Labeling one of the right pair arcs as 1 and the other as 2 allowed us to label
the two circles in Lv and Lv′ and define the elements of Ax,z as
α = ((Cw → x+), (C1v , C2v )→ (x−, x+), (Cu → x−))
β = ((C → x ), (C1, C2w + v v )→ (x+, x−), (Cu → x−))
and the two elements of A′x,z as
α′ = ((Cw → x+), (C1 , C2v′ v′)→ (x−, x+), (Cu → x−))
β′ = ((C → x ), (C1w + v′ , C2v′)→ (x+, x−), (Cu → x−)).
This allowed us to define Fu,v,v′,w as the map sending α 7→ α′ and β →7 β′. Using
the ladybug configuration for Lgw, we can also define the elements of ARg,u(x),Rg,w(z)
as
αg = ((C → x ), (C1 , C2gw + gv gv)→ (x−, x+), (Cgu → x−))
βg = ((Cgw → x+), (C1 2gv, Cgv)→ (x+, x−), (Cgu → x−))
and the two elements of A′R asg,u(x),Rg,w(z)
α′g = ((Cgw → x+), (C1 2gv′ , Cgv′)→ (x−, x+), (Cgu → x−))
β′g = ((Cgw → x+), (C1 2gv′ , Cgv′)→ (x+, x−), (Cgu → x−)).
FKh gu,gv,gv′,gw is defined to be the map that sends αg 7→ α′g and βg 7→ β′g. If we let
gav and gav′ be the image of the arcs av and av′ under the rotation action of g, it is
30
clear that gav = agv, gav′ = agv′ and that the image of the right pair of arcs in Lw
is the same as the right pair of arcs in Lgw. However, the image of the right pair arc
labeled 1 in Lw may not coincide with the right pair arc labeled 1 in Lgw since we
independently chose these labellings when defining FKh u,v,v′,w and FKh gu,gv,gv′,gw.
If the image of the right pair arc labeled 1 in Lw coincides with the right pair arc
labeled 1 in Lgw, then the vertical maps in the diagram send α 7→ αg, β 7→ βg,
α′ 7→ α′g, and β′ 7→ β′g. If the image of the right pair arc labeled 1 in Lw coincides
with the right pair arc labeled 2 in Lgw, then the vertical maps in the diagram send
α 7→ βg, β 7→ αg, α′ 7→ β′g, and β′ 7→ α′g. In either case, the diagram commutes.
The proof of the previous proposition used the edge definition of a group
acting on a strictly-unitary-lax-2-functor to show that there is an induced action
of Z/pZ on FKh(L) : 2n → B. By proposition 3.5, we know that the we can
use the maps Reg,u : FKh(u) → FKh(gu) and Seg,ϕ : Fu,v Kh(ϕu,v) → FKh(ϕgu,gv)
defined in the previous proof to construct maps Rg,u : FKh(u) → FKh(gu) and
Sg,ϕu,v : FKh(ϕu,v) → FKh(ϕgu,gv) that satisfy conditions (c.i) through (c.v) from
the complete definition of g acting on FKh(L) : 2
n → B. We will now use these
complete definition maps to prove the following proposition.
Proposition 4.2. Let L be an p-periodic link and let FKh(L) : 2
n → B be the
Khovanov functor. The natural action of Z/pZ on L induces an action on the
thickened Khovanov functor F̂ k : 2̂nKh → Top•.
Proof. As in definition 3.2, we need to construct a map
R kg,u→v→w : F̂Kh(u→ v → w)→ F̂ kKh(gu→ gv → gw)
that satisfies the three conditions in definition 3.2. This means we need a map
31
∨ ∏ ∨ ∏
R : Sk −→ Skg,u→v→w
a∈FKh(ϕu,v) b∈FKh(ϕv,w) Sg,ϕu,v (a)∈FKh(ϕgu,gv) Sg,ϕv,w (b)∈FKh(ϕgv,gw)
s(b)=t(a) s(Sg,ϕv,w (b))=t(Sg,ϕu,v (a))
We will define this map on each component of the wedge sum, and then extend this
across the wedge sum by mapping the component corresponding to a ∈ FKh(ϕu,v)
to the wedge component corresponding to Sg,ϕu,v(a) ∈ FKh(ϕgu,gv). So fix an
a ∈ FKh(ϕu,v), label the elements of {b ∈ FKh(ϕv,w); s(b) = t(a)} arbitrarily by
b1, . . . , b`, and let pb be a point in the S
k component of the product corresponding
i
to bi. Then we can defin(e )
Rg,u→v→w (pb1 , ... , pb ) = (p` Sg,ϕ , ... , p ).v,w (b1) Sg,ϕv,w (b`)
The facts that Re,u→v→w is the identity morphism and that Rh,gu→gv→gw ◦
Rg,u→v→w = Rhg,u→v→w follow directly from conditions (c.i) and (c.ii) of the group
action on F nKh : 2 → B.
All that remains is to verify the diagram in condition (3) of definition 3.2
commutes. Recall that in section 2.7, we defined the map F̂ kKh(ϕu,u′ , ϕv′,v, ϕw,w′)
from F̂ kKh(u → v → w) to F̂ k (u′Kh → v′ → w′) on each component of the wedge sum
and then extended it to the whole wedge sum. Since Rg,u→v→w is similarly defined,
it suffices to check the commutative diagram in definition 3.2 component-wise as
well. So fix an a ∈ F̂ kKh(ϕu,v). Also recall that in 2̂n the morphism
ϕu,v ϕv,w
u v w
ϕu,u′ ϕv′,v ϕw,w′
u′ v′ϕ w
′
u′,v′ ϕv′,w′
gives us two isomorphisms
FKh(ϕu,v) ∼= FKh(ϕv′,v)×F (v′) FKh(ϕu′,v′)×Kh FKh(u′) FKh(ϕu,u′)
and
32
F (ϕ ′ ′) ∼Kh v ,w = FKh(ϕw,w′)×FKh(w) FKh(ϕv,w)×F FKh(v) Kh(ϕv′,v).
So for each a ∈ FKh(ϕu,v) there is a triple (y, a′, x) in the composition, and similarly
for each b′ ∈ FKh(ϕv′,w′) there is a corresponding triple (z, b̄, ȳ). The induced group
action on 2̂n also gives us the following isomorphisms
FKh(ϕgu,gv) ∼= FKh(ϕgv′,gv)×F ′ F (ϕKh(gv ) Kh gu′,gv′)×FKh(gu′) FKh(ϕgu,gu′)
and
FKh(ϕgv′ ∼,gw′) = FKh(ϕgw,gw′)×F (gw) FKh(ϕKh gv,gw)×FKh(gv) FKh(ϕgv′,gv).
By applying condition (c.v) from(definition 3.3, we see that for Sg),ϕu,v(a) ∈
FKh(ϕgu,gv) we have Sg,ϕu,v(a) = Sg,ϕ ′ (y), S
′
g,ϕ( ′ (a ), Sg,ϕ ′ (x) , and that forv ,v u,v u,u′ )Sg,ϕ ′ (b ) ∈ FKh(ϕgv′,gw) we have S ′g,ϕ ′ (b ) = Sv ,w v ,w g,ϕw,w′ (z), Sg,ϕu,v′ (b̄), Sg,ϕ ′ (ȳ) .u,u
With these isomorphisms in mind, we let mi = |{b′ = (z, b̄, ȳ) ∈ FKh(ϕv′,w′) | b̄ =
bi ȳ = y}|. Since a is fixed, we know that y in the above isomorphisms is also fixed.
This means each triple (z, b̄, ȳ) is uniquely determined by bi and a z ∈ FKh(ϕw,w′).
So we will label the these triples as (bi, zi,1), · · · , (bi, zi,m ), so under the diagonali
map Πb∆b, we will have that pb 7→ (p , p , · · ·, p ).i bi,zi,1 bi,zi,2 bi,zi,mi
We need to show that the following diagram commutes, but first for
clarity we will describe the maps involved in the diagram. The two horizontal
maps are given by the group action. More specifically, the top horizontal map
is given by Rg,u→v→w and the bottom horizontal map Rg,u′→v′→w′ . The first pair
of vertical maps is just a diagonal map applied to each element in FKh(ϕu,v)
and FKh(ϕgu,gv). The second pair of vertical maps is just a relabeling of the
e(lements in the product under th)e bijections b′ = (z, bi, ȳ) and S ′g,ϕv′ (b ) =,w
Sg,ϕ ′ (z), Sg,ϕ ′ (b̄), Sg,ϕ ′ (ȳ) described above. For the the last pair ofw,w u,v u,u
vertical maps, recall from section 2.7 that {b′ ∈ F (ϕ ′ ′)|b′Kh u ,v = (z, b, y)}
33
is a subset of {b′ ∈ FKh(ϕu′,v′)|t(a′) = s(b′)} and that {S ′g,ϕv,w(b ) ∈
FKh(ϕgu′,gv′)|S ′g,ϕv,w(b ) = (Sg,ϕ ′ (z), Sw,w g,ϕv,w(b), Sg,ϕ ′ (y))} is a subset ofv ,v
{S ′g,ϕv,w(b ) ∈ FKh(ϕgu′,gv′)|t(S ′g,ϕu,v(a )) = s(S ′g,ϕv,w(b ))}. So these final verical
maps are just extensions of the previous vertical maps that map to the base point
in the remaining∏factors. ∏
k
Sk Rg,u→v→w S
b ∈ FKh(ϕv,w) Sg,ϕv,w (b) ∈ FKh(ϕgv,gw)
s(b)=t(a) s(Sg,ϕv,w (b)) = t(Sg,ϕu,v (a))
∏ ∏
Sk
b ∈ FKh(ϕ ′v,w) b =(z,b̄,ȳ) ∈ FKh(ϕv′,w′ )
s(b)=t(a) b̄=b
ȳ=y ∏ ∏
Sk
∼ S ′∗ ∗= g,ϕv,w (b) ∈ FKh(ϕgv,gw) b =(z ,b̄∗,ȳ∗) ∈ FKh(ϕgv′,gw′ )
s(Sg,ϕv,w (b))=t(Sg,ϕu,v (a)) b̄∗=Sg,ϕv,w (b)
ȳ∗=Sg,ϕ (y)
∏ v′,v
Sk
b′ ∈ FKh(ϕv′ ∼,w′ ) =
b′=(z,b,y)
s(b)=t(a) ∏
Sk
S ′g,ϕ ′ ′ (b ) ∈ FKh(ϕgv′,gw′ )v ,w
S ′g,ϕ ′ ′ (b ) = (Sg,ϕ ′ (z), Sg,ϕv,w (b), Sg,ϕ ′ (y))v ,w w,w v ,v
s(Sg,ϕv,w (b)) = t(Sg,ϕu,v (a))
∏ ∏
k SkS
b′ ∈ F (ϕ ′ ′ ) Sg,ϕ ′ ′ (b
′) ∈ FKh(ϕgv′,gw′ )Kh v ,w
′ R ′ ′ ′
v ,w
s(b )=t(a′) g,u →v →w s(S ′g,ϕ ′ ′ (b ))=t(S ′gv ,gw g,ϕgu′,gv′ (a ))
34
Goi(ng down and)then right we get that
pb1(, . . . , pbl )
→7 (pb1,z1,1 , . . . , pb1,z1,m , p1 b2,z2,1 , . . . , p)bl,zl,ml
→7 (pb′ , . . . , p1,1 b1,m , p1 b2,1 , . . . , pb′`,m` )
→7 (pb′ , . . . , pb1,m , pb2,1 , . . . , pb′ , ∗, . . . , ∗1,1 1 `,m` )
7→ pS (b′g,ϕ ′ ′ 1.1), . . . , pS ′ ′ ′v ,w g,ϕ , pv′,w′ (b1,m ) Sg,ϕ ′ ′ (b2,1), . . . , pSv ,w g,ϕ (b , ∗, . . . , ∗` v′,w′ `,m )`
where ∗ denotes the base point.
Goi(ng right and )then down we get
pb1(, . . . , pbl )
→7 (pSg,ϕ (b1), . . . , pv,w Sg,ϕv,w (bl)
7→ pSg,ϕ ′ ′ (b1),Sv ,w g,ϕw,w′ (z1,1), . . . , pSg,ϕ (b ),S (z ),v′,w′ 1 g,ϕw,w′ 1,m1
( pSg,ϕv,w (b2),Sg,ϕ (z2,1), . . . , p′ Sg,ϕv,w (b`),Sw,w g,ϕw,w′ (z`,m )` )
7→ (pSg,ϕ ′ ′ (b′ ), . . . , p ′1.1 Sg,ϕ ′ ′ (b1,m ), pSg,ϕ ′ ′ (b′ , . . . , p ′v ,w v ,w v ,w 2,1) Sg,ϕ (b )` v′,w′ `,m` )
7→ pSg,ϕ ′ ′ (b′ ), . . . , p ′1.1 Sg,ϕ ′ ′ (b1,m ), pS ′g,ϕ , . . . , p′ ′ (b2,1) S (b′g,ϕ , ∗, . . . , ∗ .v ,w v ,w ` v ,w v′,w′ `,m )`
Thus, the diagram commutes and so condition (3) is satisfied.
Since we know that Z/pZ acts on all of the F̂ kKh’s, we will now check that
the action commutes with the natural transformations Sn ∧ F̂ kKh(L) → F̂
k+n
Kh (L).
That is, we will check that the following diagram commutes:
35

 ∨ ∏ 
Sn ∧ Sk
a∈FKh(ϕu,v) b∈FKh(ϕv,w)
s(b)=t(a)
∨ ∏ id ∧Rg,u→v→w
Sn+k
a∈FKh(ϕu,v) b∈FKh(ϕv,w)
s(b)=t(a)
  ∨ ∏ Sn ∧ Sk
Sg,ϕu,v (a)∈FKh(ϕgu,gv) Sg,ϕv,w (b)∈FKh(ϕgv,gw)
s(Sg,ϕv,w (b))=t(Sg,ϕu,v (a))
Rg,u→v→w
∨ ∏
Sn+k.
Sg,ϕu,v (a)∈FKh(ϕgu,gv) Sg,ϕu,v (b)∈ϕgv,gw
s(Sg,ϕu,v (b))=t(Sg,ϕu,v (a))
Recall that the suspension maps involve commuting the smash product past the
wedge sum and then applying the following map to each summand
∏ ∏
σn : Sn ∧ Xi → Sn ∧Xi
i i
where we view Sn ∧ X as [0, 1]n × X/(∂[0, 1]n × X ∪ [0, 1]n × {∗}) and where
σn(y, x1, . . . , xn) = ((y, x1) . . . (y, xn)).
The action of Z/pZ on F̂ kKh and F̂
k+n
Kh permutes the parts of the wedge
sum and then permutes the Sk’s in each of the products. Since we commute the
smash product past the wedge sum the first permutation of the parts of the wedge
sum, will not affect the natural transformation. Similarly, permuting the Sk’s just
36
∏
corresponds to permuting the xi’s in the description of S
n ∧ iXi which clearly
commutes with applying the map σn.
We now know G acts on F̂Kh(L) : 2̂n → B, which can be extended by to an
+
action on F̂Kh (L) by having G fix the added point ∗. We can now prove theorem
1.1.
Theorem 1.1 For a p-periodic link L, the natural action of Z/pZ on L induces a
Z/pZ action on XKh(L), which makes XKh(L) a naive Z/pZ-spectrum.
Proof. Let Z = u → v → w, Z ′ = u′ → v′ → w′, gZ = gu → gv → gw, and
gZ ′ = gu′
+
→ gv′ → gw′ be objects in 2̂n. Since XKh(L) = hocolim(F̂Kh (L)), we
+
know that the kth space in the spectrum is hocolim(F̂ kKh(L)), which is defined as ⊔ ⊔∞ ′ +Cn(Z,Z )× In × F̂ k (Z)Kh ∪ {∗}/ ∼.
Z,Z′∈ 2̂n n=0
For g ∈ Z/pZ, g acts on hocolim(F̂ kKh(L)) by sending the above collection of cells to ⊔ ⊔∞ ′ ( + )Cn(gZ, gZ )× In ×R kg,Z F̂Kh (Z)  ∪ {∗}/ ∼.
gZ,gZ′∈ 2̂n n=0
In more detail, G acts on the homotopy colimit by sending a chain of composible
morphisms in 2̂n to its image under the action of G on 2̂n (that is it sends an
element of C ′n(Z,Z ) to an element of Cn(gZ, gZ ′)), by sending In to In by the
+ +
identity map, and by sending elements of F̂ kKh (Z) to the elements of F̂
k
Kh (gZ) by
+
using the map Rg,Z given by the action of G on F̂ kKh (L) : 2̂
n → B. The relations
∼ only involve the composition of morphisms in 2̂n which we know commutes
with the action of G. Similarly, the conditions (c.i) - (c.v) ensure that this action
+
satisfies the conditions for G acting on hocolim(F̂ kKh (L)) as a topological space.
+
Furthermore, the suspension map between hocolim(F̂ kKh (L)) and
+
hocolim(F̂ k+1Kh (L)) is given by natural transformation S
1 ∧ F̂ k (L) → F̂ k+1Kh Kh (L),
37
and we know that this commutes with the action of g. Thus, g acts on each space
in XKh(L) and the action of g commutes with the suspension in XKh(L), and so
XKh(L) is a naive Z/pZ-spectrum as desired.
38
CHAPTER V
PROOF OF INVARIANCE
The goal of this chapter is to prove theorem 1.2. In order to do that, we
need to show that if L and L′ are two equivariantly isotopic p-periodic links, then
+
XKh(L) = hocolim(F̂Kh (L)) is Borel stable homotopy equivalent to XKh(L′) =
+
hocolim(F̂ ′Kh (L )). This is equivalent to showing that the Khovanov homotopy
type is invariant under the three equivariant Reidemeister moves (def 2.6). To do
this, we need the following definition of an insular subfunctor, which is a special
case of [LLS17, Def 3.25].
Definition 5.1. Given a strictly-unitary-lax-2-functor D : 2n → B an insular
subfunctor E of D is a collection of subsets E(u) ⊂ D(u) for each u ∈ Ob(2n) such
that for all u > v
s−1(E(u)) ∩ t−1(D(v) \ E(v)) = ∅ ⊂ D(ϕu,v).
We can extend E to a strictly-unitary-lax-2-functor by defining E(ϕu,v) ⊂
D(ϕu,v) = s
−1(E(u))∩t−1(E(v)) and letting Eu,v,w : E(ϕv,w)×E(v)E(ϕu,v)→ E(ϕu,w)
be the map induced by Du,v,w : D(ϕv,w)×D(v) D(ϕu,v)→ D(ϕu,w).
Given an insular subfunctor E of D, we can define the corresponding
quotient functor (D/E) : 2n → B be setting (D/E)(u) = D(u) \ E(u),
(D/E)(ϕu,v) = s
−1(D/E(u)) ∩ t−1(D/E)(v) and letting (D/E)u,v,w be the
map induced by Du,v,w. This is again a special case of the corresponding quotient
functor described in [LLS17].
Our reason for introducing insular subfunctors and their quotient functors is
the following important lemma.
Lemma 5.2. Given an insular subfunctor E of a strictly-unitary-lax-2-functor D
there exists a cofiber sequence
39
+
hocolim(Ê+) ↪−→ hocolim(D̂+) −→ hocolim(D̂/E ).
In particular, if the inclusion map (resp. quotient map) in the sequence above
corresponds to an acyclic subcomplex of Tot(A(D)), then the quotient map (resp.
inclusion map) is a stable homotopy equivalence.
Proof. This follows directly from [LLS17]
In order to prove that XKh(L) is Borel homotopy equivalent to X ′Kh(L ),
it suffices to find insular subfunctors of FKh(L) that are closed under the induced
Z/pZ-action, that have corresponding chain complexes which are acyclic, and whose
quotient functors are isomorphic to the Khovanov functor of L. Then we can apply
the previous lemma to get the desired Borel homotopy equivalence.
Proposition 5.3. If L and L′ are two equivariantly isotopic p-periodic links that
differ by a equivariant Reidemeister move of type I, then XKh(L) is Borel homotopy
equivalent to X ′Kh(L ).
Proof. Let KC(L) and KC(L′) be the respective Khovanov chain complexes for
L and L′. We know that L′ differs from L by an equivariant Reidemeister move of
type I, which is the same as performing p copies of a regular Reidemeister I move.
We can depict these crossings as
.
40
(a) (b) (c) (d)
Figure 5. (a) The 1-resolution of a Reidemeister I move, (b) the 0-resolution of
a Reidemeister I move, (c) a 0-resolution indicating both x+ and x− generators,
(d) a resolution indicating all the generators for both the 1-resolution and the
0-resolution.
We will label these crossings 1 to p. Now in KC(L′) these crossing are
resolved as either the 1-resolution or the 0-resolution (respectively (a) and (b) in
figure 5). Additionally, the 0-resolution contains a unique circle that can be labeled
as + or − to specify a part of the complex that only contains one of the Khovanov
generators x+ or x− for that circle. Sometimes we want to allow for both generators
in which case we will label the circle +/− like in figure 5 (c). Similarly, when we
want to refer to part of the complex containing the generators for any of the above
resolutions, we will use the notation in figure 5 (d).
We will let C1 be the subcomplex of KC(L
′) containing all the generators
for vertices where at least one of the crossings involved in the p Reidemeister moves
of type I is the 1-resolution or at least one of the circles in the p 0-resolutions is
labeled with an x+. It is clear that C1 is closed under the natural group action.
The quotient KC(L′)/C1, which is depicted below, is the quotient complex that
contains all the generators for the vertices where all the p Reidemeister I moves are
the 0-resolution with the circle labeled x−.
41
It is clear that KC(L′)/C1 is isomorphic to KC(L) and that KC(L
′)/C1
corresponds to an insular subfunctor FKC(L′)/C of FKh(L
′). If we can show that
1
C1 is an acyclic subcomplex, then we will be able to apply lemma 5.2 to get that
+ +
hocolim(F̂ ′KC(L′)/C ) → hocolim(F̂Kh (L )) is a homotopy equivalence and that1
XKh(L) is Borel homotopy equivalent to X ′Kh(L ) as required. To show this fact, we
will describe how C1 is built out of a series of acyclic subcomplexes (similar to the
ones described by Bar-Natan [BN02]).
Let C1.1 be the subcomplex of KC(L
′) depicted below. That is, the
subcomplex of all the generators for vertices where the first Reidemeister I move
is the 0-resolution with the an x+ labeled circle or the first Reidemeister I move is
the 1-resolution.
m
C1.1
42
Note that the two types of vertices in C1.1 are connected by an edge where the x+
circle in one of 0-resolutions merges to form a 1-resolution. Each of these merge
maps is an isomorphism, which means C1.1 is an acyclic subcomplex. In a similar
fashion, let C1.2 be the subcomplex of KC(L
′) containing all generators for the
vertices where
• the first Reidemeister I move is the 0-resolution with an x− marked circle,
and the second Reidemeister I move is the 0-resolution with an x+ marked
circle, or
• the first Reidemeister I move is the 0-resolution with an x− marked circle,
and the second Reidemeister I move is the 1-resolution.
m
C1.2
Again, the merge maps between the 0-resolutions and 1-resolutions in the second
Reidemeister I move are all isomorphisms, so C1.2 is an acyclic subcomplex.
Continuing in this manner, let C1.i be the subcomplex containing all the generators
for vertices where
• the first i − 1 Reidemeister I moves are the 0-resolution with an x− marked
circle and the ith Reidemeister I move is the 0-resolution with an x+ marked
circle, or
43
• the first i − 1 Reidemeister I moves are the 0-resolution with an x− marked
circle and the ith Reidemeister I move is the 1-resolution.
m
C1.i
Again C1.i is acyclic. By construction, C1 = C1.1 ⊕ · · · ⊕ C1.n, and since each of the
C1.i’s is acyclic, it follows that C1 is an acyclic subcomplex.
In the previous proposition, we described in detail a subcomplex
corresponding to an insular subfunctor that was closed under the group action
and was made up of acyclic subcomplexes corresponding to each of the p copies
of the Reidemeister I move. We will use a similar proof technique for equivariant
Reidemeister move of type II.
Proposition 5.4. If L and L′ are two equivariantly isotopic p-periodic links
that differ by an equivariant Reidemeister move of type II, then XKh(L) is Borel
homotopy equivalent to XKh(L′).
Proof. We again let KC(L) and KC(L′) be the respective Khovanov chain
complexes for L and L′, and we note that L′ differs from L by p copies of
the normal Reidemeister move of type II. Each of these p type II moves in L′
introduces two crossings. A depiction of two of the p type II moves can be seen in
figure 6 (a) below. We will refer to the left move as the first of the p Reidemeister
44
II moves, and the right one as the ith Reidemeister II move. Each of the two
crossings can be resolved in two ways, and so each Reidemeister II move represents
four vertices in the cube of resolutions. We can label these four vertices as the 00-
resolution, 10-resolution, 11-resolution, and the 01-resolution. (See figure 6 (b)).
Note that the 01-resolution of each of the p-copies of the Reidemeister II move
contains a central circle.
Let C1 be the subcomplex of KC(L
′) consisting of all the generators
corresponding to vertices where
• one of the p copies of the Reidemeister II move is the 01-resolution with the
central circle labeled x+, or
• one of the p copies of the Reidemeister II move is the 11-resolution.
It is clear this is an equivariant subcomplex. To see that C1 is acyclic, we will let
C1.1 be the subcomplex containing all the generators for vertices where
• the first Reidemeister II move is the 01-resolution with the central circle
labeled x+, or
• the first Reidemeister II move is the 11-resolution.
(See figure 6 (c)). The merge maps connecting the vertices in C1.1 are
isomorphisms, which means C1.1 is acyclic. Similar to our method in the previous
proof, we let C1.i be the subcomplex containing all generators corresponding to
vertices where
• the first i − 1 Reidemeister II moves are the 01-resolution with the central
circle labeled x− and the ith Reidemeister II move is the 01-resolution with
the central circle labeled x+, or
45
• the first i − 1 Reidemeister II moves are the 01-resolution with the central
circle labeled x− and the ith Reidemeister II move is the 11-resolution.
See figure 6 (d). Again, the merge maps between these vertices are isomorphisms,
and so C1.i is acyclic. Since C1 = C1.1 ⊕ · · · ⊕ C1.n, we see that C1 is an acyclic
subcomplex that is closed under the group action.
Letting C2 = KC(L
′)/C1 we see that C2 is the complex pictured in figure
7 (a). We now let C3 be the subcomplex of C2 consisting of all the generators for
vertices where
• at least one of the p copies of the Reidemeister 11 moves is the 00-resolution,
or
• at least one of the p copies of the Reidemeister II moves is the 01-resolution
with the central circle labeled x−.
It is clear from the description that C3 is closed under the group action.
To show that C3 is acyclic, we will let C3.1 be the subcomplex of C2 that
contains all the generators for vertices where
• the first copy of the p type II Reidemeister moves is the 00-resolution, or
• the first copy of the p type II Reidemeister moves is the 01-resolution with
the central circle labeled x−
See figure 7 (b). Since C2 only contains generators for vertices where the 01-
resolutions have an x− labeled central circle, the splitting maps between the
vertices in C3.1 are isomorphisms, which means C3.1 is acyclic. Defining C3.i in
the same manner described above, it is clear that C3.i is acyclic and that C3 =
C3.1 ⊕ · · · ⊕ C3.n (See figure 7 (c)).
46
10 11 01 00
00 01 11 10
(a) (b)
(c) (d)
Figure 6. (a) a depiction of two of the p Reidemeister II moves that make up the
equivariant Reidemeister II move, (b) the resolutions of the crossings in the two
Reidemeister II moves, (c) the subcomplex C1.1, (d) the subcomplex C1.i
47
(a) (b)
(c) (d)
Figure 7. (a) the quotient complex C2, (b) the subcomplex C3.1, (c) the
subcomplex C3.i, (d) the quotient complex C2/C3.
48
Now, we note that C2 corresponds to an insular subfuctor FC2 of FKh(L
′),
+
so applying lemma 5.2 we get that the inclusion map hocolim(F̂C2 ) ↪→
+
hocolim(F̂Kh (L
′)) is a homotopy equivalence. Additionally, we note that C3
corresponds to an insular subfunctor FC3 functor of FC2 , which means we can
+
apply lemma 5.2 a second time to get that the quotient map hocolim(F̂C2 ) →
+
hocolim(F̂C2/C ) is a homotopy equivalence. Since C2/C3 is isomorphic to KC(L)3
+
(see figure 7 (d)), it follows that hocolim(F̂C2/C ) is homotopy equivalent to3
+
hocolim(F̂Kh (L)), and so it follows that XKh(L′) is Borel homotopy equivalent
to XKh(L).
Lipshitz and Sarkar note that the above two proofs depend upon the
following facts.
Remark 5.5. [LS14] Let u and v be vertices in a (partial) cube of resolution such
that there is an arrow from v to u, and one of the following holds.
(1) The arrow from v to u merges a circle U of the (partial) resolution at v. Let
S be the set of all generators that correspond to u; and let T be the set of all
generators corresponding to v with the circle U labeled by x+.
(2) The arrow from v to u splits off a circle U in the (partial) resolution at U .
Let S be the set of all generators that correspond to u with the circle U
labeled by x−; and let T be the set of all generators that correspond to v.
Let C be the chain complex generated by S and T ; it is an acyclic complex, and
therefore we can delete it without changing the homology. If, in addition, C is a
subcomplex or a quotient complex of the original chain complex, then in deleting
C, we do not introduce any new boundary maps.
49
In addition, we also note that if we want to delete an equivariant
subcomplex, we can take the subcomplex where at least one of p copies of the non-
equivariant Reidemeister moves contains one of the deletions described above. We
can then show that this equivariant subcomplex (or the corresponding equivariant
quotient complex) is acyclic by finding further acyclic subcomplexes. With this in
mind we can now proceed to the equivariant Reidemeister move of type III, and
note that since we have already shown that X (L) is invariant under equivariant
Reidemeister moves of type I and type II it suffices to check the following braid-like
version of the Reidemeister move of type III.
For more infomation about this braid-like Reidemeister move, see [Bal10, Section
7.3].
Proposition 5.6. If L and L′ are two equivariantly isotopic p-periodic links that
differ by the braid-like version of the equivariant Reidemeister move of type III,
then XKh(L) is Borel homotopy equivalent to XKh(L′).
Proof. We again let KC(L) and KC(L′) correspond to the Khovanov complex of
L and L′ respectively. This means that L′ differs from L by a series of p-copies of
the braid version of a Reidemeister move of type III. We know from the previous
remarks that it suffices to find acyclic equivariant subcomplexes corresponding to
insular subfunctors whose inclusion/quotient result in a complex isomorphic to
KC(L). We also know that these equivariant subcomplexes can be built up from
50
smaller complexes, each focusing on a single Reidemeister III type move. It suffices
to use the complexes described by Lipshitz and Sarkar in [LS14, Proposition 6.4].
Let C1 be the subcomplex of KC(L
′) where at least one of the p
Reidemeister type III braid-like moves contains a generator for one of the
vertices depicted in figure 8 with all of the + marked circles corresponding to
the x+ generator. Note that C1 is equivariant by construction, and it can be
checked directly that the quotient complex C2 = KC(L
′)/C1 corresponds to an
insular subfunctor FC2 . This means that if C1 is acyclic, then the inclusion map
+ +
hocolim(F̂C2 )→ hocolim(F̂ (L′Kh )) will induce a homotopy equivalence by lemma
5.2.
Let C3 be the subcomplex of C2 where at least one of the p Reidemeister
type III braid-like moves contains a generator corresponding to one of the vertices
depicted in figure 9. Again, the − marked circle corresponds only to the x−
generator. By construction C3 is equivariant and it can be checked that C3
corresponds to an insular subfunctor FC3 . If we can show that C3 is acyclic,
+
then it will again follow by lemma 5.2 that the quotient map hocolim(F̂C2 ) →
+
hocolim(F̂C2/C ) is a homotopy equivalence. The subcomplex C2/C3 results in p3
copies of the following 111000 resolution
which is clearly isotopic to KC(L). So if we can show that C1 and C3 are
+
acyclic then it will follow that hocolim(F̂C2/C ) will be homotopy equivalent to3
+
hocomlimF̂Kh (L) and that X ′Kh(L) is Borel homotopy equivalent XKh(L ).
51
Figure 8. A subcomplex of C1 for one of the p copies of the Reidemeister III
move. The symbol + indicates the corresponding circle is labeled x+.
52
Figure 9. A subcomplex of C3 for one of the p copies of the Reidemeister III
move. The symbol − indicates the corresponding circle is labeled x−.
53
To see why these complexes are acyclic, we note the following list of deletions
described by Lipshitz and Sarkar.
Top half 1∗1111, 1∗1110, 1∗1101, 1∗1011, 110∗11, 0111∗1, ∗01111, 1∗1100,
1∗1010, 1∗1001, 0111∗01, 110∗01, ∗01110, 01101∗, ∗01011, 110∗10,
010∗11, 1∗0011, 01∗101, ∗01101, 1001∗1, 10∗101.
Bottom half 0000∗0, 1000∗0, 0100∗0, 0010∗0, 00∗100, 0∗0001, 00001∗, 0110∗0,
1100∗0, 1∗1000, 10101∗0, 01∗100, 1∗0001, ∗01001, 00∗101, 01∗001,
01001∗, 010∗10, 0∗0110, ∗01100, ∗01010, 100∗10, 1∗0100, 10∗100.
Here 1∗1111 means to cancel along the edge from the 101111 resolution to the
111111 resolution. It can be checked directly that each of these cancellations
corresponds to taking either a subcomplex or a quotient complex. This means
during all of the cancellations no additional maps are introduced.
We can build C1 and C3 out of subcomplexes C1,i and C3,i as we did in
propositions 5.3 and 5.4 above. That is, we can apply the deletions described by
Lipshitz and Sarkar to each of the p copies of the Reidemeister III moves in turn.
Each of the C1,i’s and the C3,i’s will be acyclic by construction and so C1 and C3
are acyclic as required.
We can now prove our second main theorem.
Theorem 1.2 For a p-periodic link L, the natural action of Z/pZ on L induces
a Z/pZ action on XKh(L) which makes XKh(L) a naive Z/pZ-spectrum.
Furthermore, if a link L′ is equivariantly isotopic to L, then XKh(L′) is Borel
homotopy equivalent to XKh(L).
Proof. If L is equivariantly isotopic to L′, we know that L can be transformed into
L′ by a series of equivariant Reidemeister moves. Applying propositions 5.3, 5.4,
54
and 5.6 in the same order as the series of equivariant Reidemeister moves gives us a
roof of morphisms from XKh(L) to XKh(L′).
55
REFERENCES CITED
[Bal10] John A. Baldwin. On the spectral sequence from Khovanov homology to
Heegaard Floer homology. International Mathematics Research Notices,
2010.
[Béa67] J. Béabou. Introduction to bicategory, volume 47 of Reports of the Midwest
Category Seminar. Springer, Berlin, 1967.
[BN02] Dror Bar-Natan. On Khovanov’s categorification of the Jones polynomial.
Algebr. Geom. Topol., 2:337–370, 2002.
[BP17] Maciej Borodzik and Wojciech Politarczyk. Khovanov homology and
periodic links. arXiv:1704.07316, 2017.
[BPS18] Maciej Borodzik, Wojciech Politarczyk, and Marithania Silvero. Khovanov
homotopy type and periodic links. arXiv:1807.08795, 2018.
[CF94] Louis Crane and Igor B. Frenkel. Four-dimensional topological quantum field
theory, Hopf categories, and the canonical bases. Journal of Mathematical
Physics, 35(10):5136–5154, Oct 1994.
[Chb07] Nafaa Chbili. Equivariant Khovanov homology associated with symmetric
links. arXiv Mathematics e-prints, page math/0702359, Feb 2007.
[CJS95] R. L. Cohen, J.D.S. Jones, and G. B. Segal. Floers inifinite-dimensional
Morse theory and homotopy theory . The Floer memorial volume, Progr.
Math, 133:297–325, 1995.
[Fox61] R. H. Fox. A quick trip through Knot Theory. Topology of 3-Manifolds,
pages 120–167, 1961.
[HKK12] Po Hu, Daniel Kriz, and Igor Kriz. Field theories, stable homotopy theory
and Khovanov homology. arXiv e-prints, page arXiv:1203.4773, Mar 2012.
[Jon85] Vaughan F. R. Jones. A polynomial invariant for knots via von Neumann
algebras. Bull. Amer. Math. Soc. (N.S.), 12(1):103–111, 1985.
[Kau87] Louis H. Kauffman. State models and the Jones polynomial. Topology,
36:395–407, 1987.
[Kho00] Mikhail Khovanov. A categorification of the Jones polynomial. Duke Math.
J., 101(3):359–426, 2000.
56
[LLS15a] Tyler Lawson, Robert Lipshitz, and Sucharit Sarkar. Khovanov homotopy
type, Burnside category, and products. arXiv:1505.00213, 2015.
[LLS15b] Tyler Lawson, Robert Lipshitz, and Sucharit Sarkar. The cube and the
Burnside category. arXiv e-prints, page arXiv:1505.00512, May 2015.
[LLS17] Tyler Lawson, Robert Lipshitz, and Sucharit Sarkar. Khovanov spectra for
tangles. arXiv e-prints, page arXiv:1706.02346, Jun 2017.
[LS14] Robert Lipshitz and Sucharit Sarkar. A Khovanov stable homotopy type. J.
Amer. Math. Soc., 27(4):983–1042, 2014.
[MT93] William Menasco and Morwen Thistlethwaite. The classification of
alternating links. Annal of Mathematics, 138(1):113–171, 1993.
[Mur87] Kunio Murasugi. Jones polynomials and classical conjectures in knot
theory. Topology, 26(2):187–194, 1987.
[Mur88] Kunio Murasugi. Jones polynomials of periodic links. Pacific J. Math.,
131(2):319–329, 1988.
[Pol15] Wojciech Politarczyk. Equivariant Khovanov homology of periodic links.
arXiv e-prints, page arXiv:1504.00376, Apr 2015.
[Rei74] K. Reidemeister. Knotentheorie. Springer-Verlag, Berlin-New York, 1974.
Reprint.
[SZ18] Matthew Stoffregen and Melissa Zhang. Localization in Khovanov homology.
arXiv:1810.04769, 2018.
[Tai98] P. G. Tait. On knots I, II, III. Scientific Papers, 1:273–347, 1898.
[Thi87] Morwen Thistlethwaite. A spanning tree expansion of the Jones polynomial.
Topology, 26(3):297–309, 1987.
[Thi88] Morwen Thistlethwaite. Kauffman’s polynomial and alternating links.
Topology, 27(3):311–318, 1988.
[Vog73] Rainer M. Vogt. Homotopy limits and colimits. Mathematische Zeitung,
134:11–52, 1973.
57