Twisted Alexander polynomials with the adjoint action for some classes of knots

TL;DR

This paper computes twisted Alexander polynomials with the adjoint action for certain knots, providing explicit formulas for nonabelian Reidemeister torsion of torus and twist knots, improving previous results.

Contribution

It introduces new formulas for nonabelian Reidemeister torsion of torus and twist knots using twisted Alexander polynomials with the adjoint action.

Findings

Explicit formulas for nonabelian Reidemeister torsion of torus knots

Improved formulas for nonabelian Reidemeister torsion of twist knots

Calculations of twisted Alexander polynomials with the adjoint action

Abstract

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in \cite{DHY}.