5-move equivalence classes of links and their algebraic invariants

TL;DR

This paper systematically studies links under 5-move equivalence using algebraic invariants like Jones and Kauffman polynomials, aiming to develop tools for skein modules related to 5-moves, and partially classifies certain classes of links.

Contribution

It introduces a framework for analyzing 5-move equivalence of links using algebraic invariants and group-theoretic tools, advancing the classification of links under these moves.

Findings

Links related by one (2,2)-move are not 5-move equivalent.

Partial classification of 3-braids, pretzel, and Montesinos links up to 9 crossings.

Development of algebraic tools for skein modules based on 5-move relations.

Abstract

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S^3 along a link. We use also the fact that a 5-move is a composition of two rational \pm (2,2)-moves (i.e. \pm 5/2-moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2,2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.