Twisted Alexander polynomials of genus one two-bridge knots

TL;DR

This paper computes the twisted Alexander polynomial and Reidemeister torsion for genus one two-bridge knots, extending previous work on twist knots, and provides a formula for torsion after Dehn surgery.

Contribution

It introduces explicit formulas for twisted Alexander polynomials and Reidemeister torsion for a broader class of knots, including genus one two-bridge knots.

Findings

Computed twisted Alexander polynomials for genus one two-bridge knots.

Derived a formula for Reidemeister torsion after Dehn surgery.

Extended previous results from twist knots to a larger class.

Abstract

Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.