Twisted Alexander Polynomials and Representation Shifts

TL;DR

This paper explores the relationship between twisted Alexander polynomials, representation shifts, and properties of knot groups, establishing equivalences that connect topological and algebraic features of knots.

Contribution

It introduces new equivalences linking the existence of certain finite covers, representations with vanishing twisted Alexander polynomials, and subgroup properties of knot groups.

Findings

Uncountably many finite covers correspond to specific representations.

Vanishing twisted Alexander polynomials relate to representation properties.

Subgroup inclusions reflect the structure of knot groups and their covers.

Abstract

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.