Burnside groups and $n$-moves for links

TL;DR

This paper studies the $n$th Burnside groups of links, providing a necessary condition to distinguish them from trivial links, and demonstrates the existence of links that cannot be simplified to trivial links via $p$-moves for any odd prime $p$.

Contribution

It introduces a necessary condition for $p$th Burnside groups to match those of trivial links, aiding in distinguishing complex links from trivial ones.

Findings

Established a criterion to identify non-trivial links via Burnside groups.

Proved the existence of links not reducible to trivial links by $p$-moves for any odd prime $p$.

Abstract

Let $n$ be a positive integer. M. K. Dabkowski and J. H. Przytycki introduced the $n$th Burnside group of links which is preserved by $n$-moves, and proved that for any odd prime $p$ there exist links which are not equivalent to trivial links up to $p$-moves by using their $p$th Burnside groups. This gives counterexamples for the Montesinos-Nakanishi $3$-move conjecture. In general, it is hard to distinguish $p$th Burnside groups of a given link and a trivial link. We give a necessary condition for which $p$th Burnside groups are isomorphic to those of trivial links. The necessary condition gives us an efficient way to distinguish $p$th Burnside groups of a given link and a trivial link. As an application, we show that there exist links, each of which is not equivalent to a trivial link up to $p$-moves for any odd prime $p$.