
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 in a closed, oriented 3-manifold , we use representation spaces and the Lagrangian field theory framework of Wehrheim and Woodward to define a new homological knot invariant . We then use a result of Ivan Smith to show that when is a (1,1) knot in (a set of knots which includes torus knots, for example), the rank of agrees with the rank of knot Floer homology, , and we conjecture that this holds in general for any knot . In Chapter 3, we prove a somewhat strange result, giving a purely topological formula for the Jones polynomial of a 2-bridge knot . First, for any lens space , we combine the -invariants from…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
