A family of slice-torus invariants from the divisibility of Lee classes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918155221204992 |
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| author | Sano, Taketo Sato, Kouki |
| author_facet | Sano, Taketo Sato, Kouki |
| contents | We give a family of slice-torus invariants $\tilde{ss}_c$, each defined from the $c$-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements $c$ in any principal ideal domain $R$. For the special case $(R, c) = (F[H], H)$ where $F$ is any field, we prove that $\tilde{ss}_c$ coincides with the Rasmussen invariant $s^F$ over $F$. Compared with the unreduced invariants $ss_c$ defined by the first author in a previous paper, we prove that $ss_c = \tilde{ss}_c$ for $(R, c) = (F[H], H)$ and $(\mathbb{Z}, 2)$. However for $(R, c) = (\mathbb{Z}, 3)$, computational results show that $ss_3$ is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_02494 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A family of slice-torus invariants from the divisibility of Lee classes Sano, Taketo Sato, Kouki Geometric Topology 57K18 We give a family of slice-torus invariants $\tilde{ss}_c$, each defined from the $c$-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements $c$ in any principal ideal domain $R$. For the special case $(R, c) = (F[H], H)$ where $F$ is any field, we prove that $\tilde{ss}_c$ coincides with the Rasmussen invariant $s^F$ over $F$. Compared with the unreduced invariants $ss_c$ defined by the first author in a previous paper, we prove that $ss_c = \tilde{ss}_c$ for $(R, c) = (F[H], H)$ and $(\mathbb{Z}, 2)$. However for $(R, c) = (\mathbb{Z}, 3)$, computational results show that $ss_3$ is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants. |
| title | A family of slice-torus invariants from the divisibility of Lee classes |
| topic | Geometric Topology 57K18 |
| url | https://arxiv.org/abs/2211.02494 |