Dynamics of Twisted Alexander Invariants

TL;DR

This paper explores the algebraic and topological properties of twisted Alexander invariants for links and knots, revealing their dynamical systems nature, entropy characteristics, and polynomial roots related to specific representations.

Contribution

It introduces the concept of total twisted representations and analyzes the zeros of twisted Alexander polynomials for certain knot classes, providing new insights into their structure and behavior.

Findings

The twisted Alexander module forms an algebraic dynamical system.

Topological entropy relates to torsion growth in cyclic covers.

Zeros of polynomials for specific representations are roots of unity.

Abstract

The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological entropy is the logarithmic growth rate of the number of torsion elements in the twisted first-homology group of r-fold cyclic covers of the knot complement, as r goes to infinity. Total twisted representations are introduced, and their properties are studied. The twisted Alexander polynomial obtained from any nonabelian parabolic SL(2,C) representation of a 2-bridge knot group is seen to be nontrivial. The zeros of any twisted Alexander polynomial of a torus knot corresponding to a parabolic SL(2,C) representation or a finite-image permutation representation are shown to be roots of unity.