Periodicity and Free Periodicity of Alternating Knots

TL;DR

This paper investigates the symmetries of alternating knots, showing how free and periodic actions can be visualized in projections and providing criteria to detect such symmetries using decomposition techniques.

Contribution

It introduces a method to represent free $q$-actions on alternating knots via flypes on a single twisted band diagram, linking symmetry detection to knot decomposition.

Findings

Free $q$-actions are virtually visible in alternating projections.

Criteria are provided to detect $q$-actions from virtually visible projections.

Decomposition into atoms helps determine the visibility type of freely periodic knots.

Abstract

A knot $K$ in $S^3$ is $q$-periodic if it admits a symmetry that is conjugate to a rotation of order $q$ of $S^3$. If $K$ admits a symmetry which is a homeomorphism without fixed point of period $q$ of $S^3$, then $K$ is called freely $q$-periodic. In a previous paper, we obtained, as a consequence of Flyping Theorem due to Menasco and Thislethwaite, that the $q$-periodicity with $q>2$ can be visualized in an alternating projection as a rotation of the projection sphere. In this paper, we show that the free $q$-action of an alternating knot can be represented on some alternating projection as a composition of a rotation of order $q$ with some flypes all occurring on the same twisted band diagram of its essential Conway decomposition. Therefore, for an alternating knot to be freely periodic, its essential decomposition must satisfy certain conditions. We show that any free or non-free $q$-action is somehow visible (virtually visible) and give some sufficient criteria to detect the existence of $q$-actions from virtually visible projections. Finally, we show how the Murasugi decomposition into atoms enables us to determine the visibility type $(q,r)$ of the freely $q$-periodic alternating knots ($(q,r)$-lens knots); in fact, we only need to focus on a certain atom of their Murasugi decomposition to deduce their visibility type.