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Floergåsbord

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 $ρ$-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 $σ(K)$ denote the knot signature, and let $\mathcal{O}$