Mathematics Theses and Dissertations

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https://hdl.handle.net/1794/2044

This collection contains some of the theses and dissertations produced by students in the University of Oregon Mathematics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.

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Browsing Mathematics Theses and Dissertations by Author "Bodish, Holt"

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    Reducible Dehn Surgeries, Ribbon Concordance, and Satellite Knots
    (University of Oregon, 2024-08-07) Bodish, Holt; Lipshitz, Robert
    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 \emph{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 \ref{reduciblesurgeries} and develop a lower bound on the slice genus of knots that admit reducible surgeries in terms of the surgery parameters and study when a slope on an almost L-space knot is a reducing slope. In particular, we show that when $g(K)$ is odd and $>3$, the only possible reducing slope on an almost L-space knot is $g(K)$ and in that case the complement of an almost L-space knot does not contain any punctured projective planes. In Chapter \ref{chaptersatellite} we investigate the effect of satellite operations on knot Floer homology using techniques from bordered Floer homology \cite{LOT} and the immersed curve reformulation \cites{HRW,Chen, chenhanselman}. In particular we study the functions $n \mapsto g(P_n(K)), \epsilon(P_n(K))$ and $\tau(P_n(K))$ for some families of $(1,1)$ patterns $P$ from the immersed curve perspective. We also consider the function $n \mapsto \dim(\HFKhat(S^3,P_n(K),g(P_n(K)))$, and use this together with the fibered detection property of knot Floer homology \cite{Nifibered} to determine, for a given pattern $P$, for which $n \in \Z$ the twisted pattern $P_n$ is fibered in the solid torus. In Chapter \ref{Chapterribbon} we answer positively a question posed by Lipshitz and Sarkar about the existence of Steenrod operations on the Khovanov homology of prime knots \cite[Question 3]{MR3966803}. The proof relies on a construction of a particular type of surface, called a ribbon concordance in $S^3 \times I$, interpolating 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 \cite{MR3122052,MR4041014}.