On cycles and coverings associated to a knot

TL;DR

This paper studies a finite dynamical system arising from knot groups and finite abelian representations, linking cycle lengths to Alexander polynomial roots, and classifies certain knot coverings.

Contribution

It generalizes previous results to arbitrary finite abelian groups and provides a complete classification of depth 2 solvable knot coverings.

Findings

Cycle lengths are described by roots of the Alexander polynomial.

Generalization from prime to arbitrary finite abelian groups.

Complete classification of depth 2 solvable coverings.

Abstract

We consider the space of all representations of the commutator subgroup of a knot group into a finite abelian group {\Sigma}, together with a shift map {\sigma}_x. This is a finite dynamical system, introduced by D.Silver and S. Williams. We describe the lengths of its cycles in terms of the roots of the Alexander polynomial of the knot. This generalizes our previous result for {\Sigma}= Z/p, p is prime, and gives a complete classification of depth 2 solvable coverings of the knot complement.