The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings

TL;DR

This paper proves the Montesinos-Nakanishi 3-move conjecture for all links up to 19 crossings and most 20-crossing links, classifying them modulo 3-moves using computational methods.

Contribution

It extends the verification of the 3-move conjecture to links with up to 20 crossings, including new computational tools in Regina.

Findings

Proved the conjecture for links up to 19 crossings.

Classified most 20-crossing links modulo 3-moves.

Developed new computational methods for link classification.

Abstract

Yasutaka Nakanishi formulated the following conjecture in 1981: every link is 3-move equivalent to a trivial link. While the conjecture was proved for several specific cases, it remained an open question for over twenty years. In 2002, Mieczys{\l}aw D{\c a}bkowski and the last author showed that it does not hold, in general. In this article, we prove the Montesinos-Nakanishi $3$-move conjecture for links with up to 19 crossings and, with the exception of six pairwise non-isotopic links including the Chen link and its mirror image, for links with 20 crossings. Our work completely classifies links up 20 crossings modulo $3$-moves. This work includes computational methods, including new code in Regina that generalises pre-existing knot functions to work with links.