Twisted Alexander polynomials, character varieties and Reidemeister torsion of double branched covers

TL;DR

This paper extends Fox's Alexander polynomial formula to double branched covers, linking Reidemeister torsion, twisted Alexander polynomials, and character varieties, and uses these to distinguish two-bridge knots up to mirror images.

Contribution

It introduces a new formula connecting Reidemeister torsion with twisted Alexander polynomials and character varieties for double branched covers.

Findings

The formula computes Reidemeister torsion using twisted Alexander polynomials.

The product of two factors distinguishes two-bridge knots up to mirror images.

Application to knot classification demonstrates the formula's effectiveness.

Abstract

We give an extension of Fox's formula of the Alexander polynomial for double branched covers over the three-sphere. Our formula provides the Reidemeister torsion of a double branched cover along a knot for a non-trivial one dimensional representation by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety. As an application, we show that these products distinguish isotopy classes of two-bridge knots up to mirror images.