The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912244388855808 |
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| author | Bakshi, Rhea Palak Burton, Benjamin A. Guo, Huizheng Ibarra, Dionne Montoya-Vega, Gabriel Mukherjee, Sujoy Przytycki, Józef H. |
| author_facet | Bakshi, Rhea Palak Burton, Benjamin A. Guo, Huizheng Ibarra, Dionne Montoya-Vega, Gabriel Mukherjee, Sujoy Przytycki, Józef H. |
| contents | 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 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_17711 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings Bakshi, Rhea Palak Burton, Benjamin A. Guo, Huizheng Ibarra, Dionne Montoya-Vega, Gabriel Mukherjee, Sujoy Przytycki, Józef H. Geometric Topology 2020 Primary: 57K10, Secondary: 57-08, 57-04 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 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. |
| title | The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings |
| topic | Geometric Topology 2020 Primary: 57K10, Secondary: 57-08, 57-04 |
| url | https://arxiv.org/abs/2502.17711 |