Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

The non-orientable 4-genus of a knot $K$ in $S^{3}$ is defined to be the minimum first Betti number of a non-orientable surface $F$ in $B^{4}$ so that $K$ bounds $F$. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots, torus knots, and Whitehead doubles. We also will view obstructions to a knot bounding a M\"{o}bius band given by the double branched cover of $S^{3}$ branched over $K$. Additionally, we discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used techniques to obstructing sliceness. We improve bounds in general for the non-orientable 4 genus of $t$-twisted Whitehead doubles and provide genus 1 non-orientable cobordisms to cable knots. We conclude with examining knot traces, which are 4-manifolds obtained by attaching a 2-handle to $B^{4}$ along a knot $K$ with framing $r$, denoted $X_{r}(K)$. Previous work has analyzed knots $K$ that do not bound a smoothly embedded disk in $B^{4}$, yet there exists an $r$ so that a smoothly embedded $S^{2}$ generates the second homology, $H_{2}(X_{r}(K) ; \mathbb{Z}) \cong \mathbb{Z}$. We examine the non-orientable analog, a smoothly embedded $\mathbb{R}P^{2}$ generating $H_{2}(X_{r}(K); \mathbb{Z}_{2} ) \cong \mathbb{Z}_{2}$, and discuss techniques and tools for both the orientable and non-orientable setting.

Date

2-13-2025

Committee Chair

Dasbach, Oliver

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