Floergåsbord

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 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 S(K) otimes C agrees with the rank of knot Floer homology, hat{HFK}(K) otimes 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 susbet 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

Ip,q : Z/pZ → 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 O denote the set of relative orientations of K, which has cardinality 2^{(# of components of K)-1}. Then we prove the following formula for the Jones polynomial J(K):

i^-sigma(K) q^(3sigma(K)) J(K) = \sum_{o \in O} (iq)^{2sigma(K^o)} + q^(-1) - q^1 \sum_{s \in Z/pZ} (iq)^(I_{p,q}(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 hat{CF}(M) is particularly simple, i.e., M is what we call a "strong L-space," then \pi_1(M) is not left-orderable.