Dissertations, Theses, and Capstone Projects

Date of Degree

9-2024

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Abhijit Champanerkar

Committee Members

Ilya Kofman

Joshua Sussan

Subject Categories

Geometry and Topology

Keywords

Twisted Alexander Polynomial, Ptolemy Variety, Twist Knot, Punctured Torus Bundle, Triangulation, Hyperbolic Volume

Abstract

The first focus of this dissertation is to compute Ptolemy varieties for triangulations of two infinite families of manifolds. Given an ideal triangulation of a cusped manifold, one can compute the Ptolemy variety and using it, obtain parabolic representations of the fundamental group. We compute certain obstruction classes for these manifolds, which are necessary to obtain the discrete faithful representation. This leads to our second focus of the dissertation, the twisted Alexander polynomial. The twisted Alexander polynomial (TAP) is a variation of the classical Alexander polynomial twisted by a representation of the fundamental group into a linear group. It was discovered by Lin in 1990 for knots, and extended to group presentations by Wada in 1994. Since then, the invariant has been shown to detect many properties of knots which the classical polynomial does not. In 2012, Dunfield-Friedl-Jackson conjectured that the TAP twisted by the discrete faithful representation detects genus and fibering for all hyperbolic knots. With DFJ's conjecture in mind, we compute the TAP using the representations we obtained from the Ptolemy variety. In doing so, we also introduce a new way to parameterize TAPs when given an ideal triangulation of the underlying manifold.

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