Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations
Geometric Topology
2009-03-11 v1
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.
Keywords
Cite
@article{arxiv.0903.1689,
title = {Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations},
author = {Mikami Hirasawa and Kunio Murasugi},
journal= {arXiv preprint arXiv:0903.1689},
year = {2009}
}