Floerg{\aa}sbord

TL;DR

This thesis introduces a new homological knot invariant using $SU(2)$ representation spaces, establishes a topological formula for the Jones polynomial of 2-bridge knots, and links Heegaard Floer homology to the orderability of 3-manifold fundamental groups.

Contribution

It defines a novel homological invariant for knots, provides a topological formula for the Jones polynomial of 2-bridge knots, and connects Heegaard Floer homology with the orderability of fundamental groups.

Findings

Rank of the new invariant matches knot Floer homology for certain knots.

Derived a topological formula for the Jones polynomial of 2-bridge knots.

Proved that strong L-spaces have non-left-orderable fundamental groups.

Abstract

In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot $K$ in a closed, oriented 3-manifold $M$, we use $SU(2)$ representation spaces and the Lagrangian field theory framework of Wehrheim and Woodward to define a new homological knot invariant $\mathcal{S}(K)$. We then use a result of Ivan Smith to show that when $K$ is a (1,1) knot in $S^3$ (a set of knots which includes torus knots, for example), the rank of $\mathcal{S}(K)\otimes \mathbb{C}$ agrees with the rank of knot Floer homology, $\widehat{HFK}(K)\otimes \mathbb{C}$, and we conjecture that this holds in general for any knot $K$. In Chapter 3, we prove a somewhat strange result, giving a purely topological formula for the Jones polynomial of a 2-bridge knot $K\subset S^3$. First, for any lens space $L(p,q)$, we combine the $d$-invariants from Heegaard Floer homology with certain Atiyah-Patodi-Singer/Casson-Gordon $\rho$-invariants to define a function $$I_{p,q}: \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}$$ Let $K = K(p,q)$ denote the 2-bridge knot in $S^3$ whose double-branched cover is $L(p,q)$, let $\sigma(K)$ denote the knot signature, and let $\mathcal{O}$ denote the set of relative orientations of $K$, which has cardinality $2^{(\# \text{ of components of } K) - 1}$. Then we prove the following formula for the Jones polynomial $J(K)$: $$i^{-\sigma(K)}q^{3\sigma(K)}J(K)= \sum_{o\in\mathcal{O}}(iq)^{2\sigma(K^{o})} +\left(q^{-1}-q^{1}\right)\sum_{\mathfrak{s}\in\mathbb{Z}/p\mathbb{Z}}(iq)^{I_{p,q}(\mathfrak{s})}$$ (here, $i = \sqrt{-1}$). In Chapter 4, we present joint work with Adam Levine, concerning Heegaard Floer homology and the orderability of fundamental groups. Namely, we prove that if $\widehat{CF}(M)$ is particularly simple, i.e., $M$ is what we call a "strong $L$-space," then $\pi_1(M)$ is not left-orderable.