Near extremal Khovanov homology of Turaev genus one links
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866912539381596160 |
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| author | Beldon, Theo DeStefano, Mia Lowrance, Adam M. Milgrim, Wyatt Villaseñor, Cecilia |
| author_facet | Beldon, Theo DeStefano, Mia Lowrance, Adam M. Milgrim, Wyatt Villaseñor, Cecilia |
| contents | The Turaev surface of a link diagram $D$ is a closed, oriented surface constructed from a cobordism between the all-$A$ and all-$B$ Kauffman states of $D$. The Turaev genus of a link $L$ is the minimum genus of the Turaev surface of any diagram $D$ of $L$. A link is alternating if and only if its Turaev genus is zero, and so one can view Turaev genus one links as being close to alternating links. In this paper, we study the Khovanov homology of a Turaev genus one link in the first and last two polynomial gradings where the homology is nontrivial. We show that a particular summand in the Khovanov homology of a Turaev genus one link is trivial. This trivial summand leads to a computation of the Rasmussen $s$ invariant and to bounds on the smooth four genus for certain Turaev genus one knots. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_10009 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Near extremal Khovanov homology of Turaev genus one links Beldon, Theo DeStefano, Mia Lowrance, Adam M. Milgrim, Wyatt Villaseñor, Cecilia Geometric Topology 57K10, 57K18 The Turaev surface of a link diagram $D$ is a closed, oriented surface constructed from a cobordism between the all-$A$ and all-$B$ Kauffman states of $D$. The Turaev genus of a link $L$ is the minimum genus of the Turaev surface of any diagram $D$ of $L$. A link is alternating if and only if its Turaev genus is zero, and so one can view Turaev genus one links as being close to alternating links. In this paper, we study the Khovanov homology of a Turaev genus one link in the first and last two polynomial gradings where the homology is nontrivial. We show that a particular summand in the Khovanov homology of a Turaev genus one link is trivial. This trivial summand leads to a computation of the Rasmussen $s$ invariant and to bounds on the smooth four genus for certain Turaev genus one knots. |
| title | Near extremal Khovanov homology of Turaev genus one links |
| topic | Geometric Topology 57K10, 57K18 |
| url | https://arxiv.org/abs/2211.10009 |