Alexander Polynomials of Periodic Knots: A Homological Proof and Twisted Extension

TL;DR

This paper provides a new homological proof of Murasugi's 1971 condition on Alexander polynomials of periodic knots and extends it to twisted Alexander polynomials, enhancing understanding of knot periodicity.

Contribution

It introduces an alternative homological proof of Murasugi's condition and generalizes it to twisted Alexander polynomials, broadening the theoretical framework.

Findings

Homological proof of Murasugi's condition

Extension to twisted Alexander polynomials

Enhanced criteria for knot periodicity

Abstract

In 1971, Kunio Murasugi proved a necessary condition on a knot's Alexander polynomial for that knot to be periodic of prime power order. In this paper I present an alternate proof of Murasugi's condition which is subsequently used to extend his result to the twisted Alexander polynomial.