Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations

TL;DR

This paper investigates the structure of twisted Alexander polynomials for 2-bridge knots with specific metabelian representations, proposing a conjecture and proving it for a class of knots with certain group properties.

Contribution

It formulates a conjecture on the form of twisted Alexander polynomials for knots with metabelian representations and proves it for 2-bridge knots with particular group mappings.

Findings

Confirmed the conjecture for 2-bridge knots with G(K) mapping onto Z/2*Z/3

Proposed a general form of twisted Alexander polynomials for metabelian representations

Discussed extensions to more general metabelian representations

Abstract

Suppose the knot group G(K) of a knot K has a non-abelian representation \rho on A_4 \subset GL(4,Z). We conjecture that the twisted Alexander polynomial of K associated to \rho is of the form: \Delta_K(t)/(1-t) \phi(t^3), where \Delta_K (t) is the Alexander polynomial of K and \phi(t^3) is an integer polynomial in t^3. We prove the conjecture for 2-bridge knots K whose group G(K) can be mapped onto a free product Z/2*Z/3. Later, we discuss more general metabelian representations of the knot groups and propose a similar conjecture on the form of the twisted Alexander polynomials.