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arXiv · 2502.17711

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

Rhea Palak Bakshi·Benjamin A. Burton·Huizheng Guo·Dionne Ibarra·Gabriel Montoya-Vega·Sujoy Mukherjee·Józef H. Przytycki
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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.

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Rhea Palak Bakshi, Benjamin A. Burton, Huizheng Guo, Dionne Ibarra, Gabriel Montoya-Vega, Sujoy Mukherjee, Józef H. Przytycki. 2025-02-24. The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings. https://arxiv.org/abs/2502.17711

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