Obstructions to free periodicity and symmetric L-space 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.
Keywords
Cite
@article{arxiv.2310.01705,
title = {Obstructions to free periodicity and symmetric L-space knots},
author = {Keegan Boyle and Nicholas Rouse and Ben Williams},
journal= {arXiv preprint arXiv:2310.01705},
year = {2026}
}
Comments
Added Ben Williams as an author, and corrected some arguments involved in the main results. The newest version fixes a typo in the statement of Proposition 1.5