Date of Degree

6-2022

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Ilya Kofman

Committee Members

Abhijit Champanerkar

Dennis Sullivan

Joshua Sussan

Subject Categories

Geometry and Topology

Keywords

topology, geometric topology, knot theory, quantum topology, jones polynomial, knots

Abstract

This dissertation has two parts, each motivated by an open problem related to the Jones polynomial. The first part addresses the Volume Conjecture of Kashaev, Murakami, and Murakami. We define a polynomial invariant, JTn, of links in the thickened torus, which we call the nth toroidal colored Jones polynomial, and we show JTn satisfies many properties of the original colored Jones polynomial. Most significantly, JTn exhibits volume conjecture behavior. We prove a volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions of JTn, one as a generalized operator invariant we call a pseudo-operator invariant, and another using the Kauffman bracket skein module of the torus. Finally, we show JTn produces invariants of biperiodic and virtual links. To our knowledge, JTn gives the first example of volume conjecture behavior in a virtual (non-classical) link.

The second part of this dissertation addresses the challenge of defining the Jones polynomial in a diagram-free way. We give an explicit algorithm for calculating the Kauffman bracket of a link diagram from a Goeritz matrix for that link, and we show how the Jones polynomial can be recovered when the corresponding checkerboard surface is orientable, or when more information is known about its Gordon-Litherland form. In the process we develop a theory of Goeritz matrices for cographic matroids, which extends the bracket polynomial to any symmetric integer matrix. We place this work in the context of links in thickened surfaces.

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