Remark on the Alexander polynomials of periodic knots

TL;DR

This paper demonstrates that for certain prime-period knots with specific Alexander polynomial properties, the polynomial is uniquely determined solely by the period p.

Contribution

It establishes a uniqueness result linking the Alexander polynomial to the prime period p for a class of knots.

Findings

Alexander polynomial is uniquely determined by p for prime-period knots with monic degree p-1

The result applies specifically to knots with prime period p>2 and monic Alexander polynomial of degree p-1

Provides a new insight into the relationship between knot periodicity and Alexander polynomial structure

Abstract

We will show that if $K$ is a knot of prime period $p>2$ and whose Alexander polynomial $\Delta_K(t)$ is monic and of degree $p-1$, then $\Delta_K(t)$ is uniquely determined only by $p$.