Hyperbolic knots with arbitrarily large torsion order in knot Floer homology
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915084660375552 |
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| author | Himeno, Keisuke Teragaito, Masakazu |
| author_facet | Himeno, Keisuke Teragaito, Masakazu |
| contents | In knot Floer homology, there are two types of torsion order. One is the minimal power of the action of the variable $U$ to annihilate the $\mathbb{F}_2[U]$-torsion submodule of the minus version of knot Floer homology $\mathrm{HFK}^-(K)$. This is introduced by Juhász, Miller and Zemke, and denoted by $\mathrm{Ord}(K)$. The other, $\mathrm{Ord}'(K)$, introduced by Gong and Marengon, is similarly defined for the $\mathbb{F}_2[U]$-torsion submodule of the unoriented knot Floer homology $\mathrm{HFK}'(K)$.
For both torsion orders, it is known that arbitrarily large values are realized by torus knots. In this paper, we prove that they can be realized by hyperbolic knots, most of which are twisted torus knots. Two torsion orders are argued in a unified way by using the Upsilon torsion function introduced by Allen and Livingston. We also give the first infinite family of hyperbolic knots which shares a common Upsilon torsion function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperbolic knots with arbitrarily large torsion order in knot Floer homology Himeno, Keisuke Teragaito, Masakazu Geometric Topology 57K10, 57K18 In knot Floer homology, there are two types of torsion order. One is the minimal power of the action of the variable $U$ to annihilate the $\mathbb{F}_2[U]$-torsion submodule of the minus version of knot Floer homology $\mathrm{HFK}^-(K)$. This is introduced by Juhász, Miller and Zemke, and denoted by $\mathrm{Ord}(K)$. The other, $\mathrm{Ord}'(K)$, introduced by Gong and Marengon, is similarly defined for the $\mathbb{F}_2[U]$-torsion submodule of the unoriented knot Floer homology $\mathrm{HFK}'(K)$. For both torsion orders, it is known that arbitrarily large values are realized by torus knots. In this paper, we prove that they can be realized by hyperbolic knots, most of which are twisted torus knots. Two torsion orders are argued in a unified way by using the Upsilon torsion function introduced by Allen and Livingston. We also give the first infinite family of hyperbolic knots which shares a common Upsilon torsion function. |
| title | Hyperbolic knots with arbitrarily large torsion order in knot Floer homology |
| topic | Geometric Topology 57K10, 57K18 |
| url | https://arxiv.org/abs/2412.20652 |