On the twisted Alexander polynomial for representations into SL_2(C)

TL;DR

This paper investigates the properties of twisted Alexander polynomials for non-abelian representations of certain 2-bridge knots, including twist knots, focusing on monic polynomials and degree deviations from the known fibered case.

Contribution

It computes the number of non-abelian representations with monic twisted Alexander polynomials and those with degree less than the maximal for specific non-fibered 2-bridge knots.

Findings

Calculated the count of non-abelian representations with monic $\

,

the degree of $\

Abstract

We study the twisted Alexander polynomial $\Delta_{K,\rho}$ of a knot $K$ associated to a non-abelian representation $\rho$ of the knot group into $SL_2(\BC)$. It is known for every knot $K$ that if $K$ is fibered, then for every non-abelian representation, $\Delta_{K,\rho}$ is monic and has degree $4g(K)-2$ where $g(K)$ is the genus of $K$. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot $K$ is non-fibered, then all but finitely many non-abelian representations on some component have $\Delta_{K,\rho}$ non-monic and degree $4g(K)-2$. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where $\Delta_{K,\rho}$ is monic and calculate the number of non-abelian representations where the degree of $\Delta_{K,\rho}$ is less than $4g(K)-2$.