Twisting Alexander Invariants with Periodic Representations

TL;DR

This paper introduces a generalization of twisted Alexander invariants for knots using periodic representations, exploring their properties, reciprocality, zero bounds, and topological interpretations of their Mahler measures.

Contribution

It extends twisted Alexander invariants to periodic representations of the knot group, providing new properties and topological insights.

Findings

Properties of the new invariants are established.

Reciprocality and zero bounds are derived.

A topological interpretation of Mahler measure is provided.

Abstract

Twisted Alexander invariants have been defined for any knot and linear representation of its group. The invariants are generalized for any periodic representation of the commutator subgroup of the knot group. Properties of the new twisted invariants are given. Under suitable hypotheses, reciprocality and bounds on the moduli of zeros are obtained. A topological interpretation of the Mahler measure of the invariants is presented. Keywords: Knot, twisted Alexander polynomial, representation shift, Mahler measure.