Patterns of the $V_2$-polynomial of knots
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908907453022208 |
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| author | Garoufalidis, Stavros Li, Shana Yunsheng |
| author_facet | Garoufalidis, Stavros Li, Shana Yunsheng |
| contents | Recently, Kashaev and the first author constructed an $R$-matrix from a Nichols algebra with an automorphism, that leads, via the Reshetikhin--Turaev functor, to a multivariable polynomial invariant of knots. Applying this to a rank 2 Nichols algebra, results in a sequence $V_n$ of 2-variable knot polynomials with integer coefficients, the first polynomial been identified with the Links--Gould polynomial. In this note we present the results of the computation of the $V_n$-polynomials for $n=1,2,3,4$. This leads to the discovery of emerging patterns, including the genus bound for $V_2$ being an equality for all 352.2 million knots with at most $19$ crossings, as well as unexpected Conway mutations that seem undetected by the $V_n$-polynomials as well as by Heegaard Floer Homology and Khovanov Homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03557 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Patterns of the $V_2$-polynomial of knots Garoufalidis, Stavros Li, Shana Yunsheng Geometric Topology High Energy Physics - Theory Recently, Kashaev and the first author constructed an $R$-matrix from a Nichols algebra with an automorphism, that leads, via the Reshetikhin--Turaev functor, to a multivariable polynomial invariant of knots. Applying this to a rank 2 Nichols algebra, results in a sequence $V_n$ of 2-variable knot polynomials with integer coefficients, the first polynomial been identified with the Links--Gould polynomial. In this note we present the results of the computation of the $V_n$-polynomials for $n=1,2,3,4$. This leads to the discovery of emerging patterns, including the genus bound for $V_2$ being an equality for all 352.2 million knots with at most $19$ crossings, as well as unexpected Conway mutations that seem undetected by the $V_n$-polynomials as well as by Heegaard Floer Homology and Khovanov Homology. |
| title | Patterns of the $V_2$-polynomial of knots |
| topic | Geometric Topology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2409.03557 |