Obstructions to free periodicity and symmetric L-space knots

TL;DR

This paper explores the algebraic conditions on Alexander polynomials of freely periodic knots, providing a number-theoretic framework, and demonstrates that certain polynomials can only arise from finitely many freely p-periodic knots, especially for L-space knots with genus up to 16.

Contribution

It introduces a number-theoretic approach to analyze Alexander polynomials of freely periodic knots and establishes finiteness results for polynomials not composed of cyclotomic factors.

Findings

Polynomials not factored into cyclotomic polynomials correspond to finitely many freely p-periodic knots.

Alexander polynomials of freely-periodic L-space knots with genus ≤ 16 are products of cyclotomic polynomials.

Conjecture: all periodic or freely periodic L-space knots are iterated torus knots.

Abstract

We investigate a polynomial factorization problem that naturally arises from Hartley's factorization condition on the Alexander polynomial of freely periodic knots. We give a number-theoretic interpretation of this factorization condition, which allows for efficient computation. As an application, we prove that any polynomial which is not a product of cyclotomic polynomials can be the Alexander polynomial of a freely p-periodic knot for only finitely many p. As a demonstration of the computational efficiency of these methods, we also show that the Alexander polynomial of any freely-periodic L-space knot with genus at most 16 must be a product of cyclotomic polynomials. We conjecture that any periodic or freely periodic L-space knot must be an iterated torus knot.