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

2025/02/24 by Bakshi, Rhea Palak, Burton, Benjamin A., Guo, Huizheng +4
#2020 Primary: 57K10 #57-04 #FOS: Mathematics #Geometric Topology (math.GT) #Secondary: 57-08

paper · doi:10.48550/arxiv.2502.17711

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, Mieczysław D\c abkowski 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.

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