On the colored Links--Gould polynomial
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908718614970368 |
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| author | Garoufalidis, Stavros Harper, Matthew Kashaev, Rinat Kohli, Ben-Michael Wagner, Emmanuel |
| author_facet | Garoufalidis, Stavros Harper, Matthew Kashaev, Rinat Kohli, Ben-Michael Wagner, Emmanuel |
| contents | We give a cabling formula for the Links--Gould polynomial of knots colored with a $4n$-dimensional irreducible representation of $\mathrm{U}^H_q\mathfrak{sl}(2|1)$ and identify them with the $V_n$-polynomial of knots for $n=2$. Using the cabling formula, we obtain genus bounds and a specialization to the Alexander polynomial for the colored Links--Gould polynomial that is independent of $n$, which implies corresponding properties of the $V_n$-polynomial for $n=2$ conjectured in previous work of two of the authors, and extends the work done for $n=1$. Combined with work of one of the authors arXiv:2409.03557, our genus bound for $\mathrm{LG}^{(2)}=V_2$ is sharp for all knots with up to $16$ crossings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10911 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the colored Links--Gould polynomial Garoufalidis, Stavros Harper, Matthew Kashaev, Rinat Kohli, Ben-Michael Wagner, Emmanuel Quantum Algebra Geometric Topology 57K16, 57K14, 17B37 We give a cabling formula for the Links--Gould polynomial of knots colored with a $4n$-dimensional irreducible representation of $\mathrm{U}^H_q\mathfrak{sl}(2|1)$ and identify them with the $V_n$-polynomial of knots for $n=2$. Using the cabling formula, we obtain genus bounds and a specialization to the Alexander polynomial for the colored Links--Gould polynomial that is independent of $n$, which implies corresponding properties of the $V_n$-polynomial for $n=2$ conjectured in previous work of two of the authors, and extends the work done for $n=1$. Combined with work of one of the authors arXiv:2409.03557, our genus bound for $\mathrm{LG}^{(2)}=V_2$ is sharp for all knots with up to $16$ crossings. |
| title | On the colored Links--Gould polynomial |
| topic | Quantum Algebra Geometric Topology 57K16, 57K14, 17B37 |
| url | https://arxiv.org/abs/2509.10911 |