  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.
