Obstructions to free periodicity and symmetric L-space knots
Keegan Boyle, Nicholas Rouse, Ben Williams
TL;DR
This work addresses obstructions to free periodicity of knots through Hartley’s factorization of the Alexander polynomial, recasting the condition in number-theoretic terms via $n$-Hartley and the invariant $D(\Delta(t))$ to obtain a finite test for freeness using only the polynomial. The authors prove that any polynomial not a product of cyclotomic polynomials can be $n$-Hartley for only finitely many $n$, and they develop an explicit algorithm to compute the allowable $n$; they also demonstrate computational evidence up to genus bound $18$ and relate these obstructions to conjectures about symmetric L-space knots. A key contribution is linking the Hartley condition to a finite computational procedure, enabling practical ruling-out of free periodicities from Alexander polynomials. The results support a broader conjecture that periodic or freely periodic L-space knots are iterated torus knots, with implications for Heegaard Floer theory and 3-manifold topology, especially under the L-space conjecture.
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.