Morse matchings and Khovanov homology of 4-strand torus links
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916852821655552 |
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| author | Kelomäki, Tuomas |
| author_facet | Kelomäki, Tuomas |
| contents | Given a link or a tangle diagram, we define algorithmic Morse theoretic simplifications on their Khovanov homology. In contrast to Bar-Natan's scanning algorithm, the cancellations are postponed until the end and performed in one go. Although our novel approach is computationally inferior to Bar-Natan's algorithm, it side-steps the need for a large amount of iterations, making it more fitting for theoretical analysis. Our main application is towards integral Khovanov homology of 4-strand torus links, for which we compute non-trivial Khovanov homology groups in all homological degrees and find an abundance of $4$-torsion. At the limit $T(4,\infty)$, our computations agree with a conjecture of Gorsky, Oblomkov and Rasmussen. For finite $n$, we use the $λ$-invariant of Lewark, Marino and Zibrowius to derive lower bounds on proper rational Gordian distances from $T(4,n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Morse matchings and Khovanov homology of 4-strand torus links Kelomäki, Tuomas Geometric Topology Combinatorics 57K18 Given a link or a tangle diagram, we define algorithmic Morse theoretic simplifications on their Khovanov homology. In contrast to Bar-Natan's scanning algorithm, the cancellations are postponed until the end and performed in one go. Although our novel approach is computationally inferior to Bar-Natan's algorithm, it side-steps the need for a large amount of iterations, making it more fitting for theoretical analysis. Our main application is towards integral Khovanov homology of 4-strand torus links, for which we compute non-trivial Khovanov homology groups in all homological degrees and find an abundance of $4$-torsion. At the limit $T(4,\infty)$, our computations agree with a conjecture of Gorsky, Oblomkov and Rasmussen. For finite $n$, we use the $λ$-invariant of Lewark, Marino and Zibrowius to derive lower bounds on proper rational Gordian distances from $T(4,n)$. |
| title | Morse matchings and Khovanov homology of 4-strand torus links |
| topic | Geometric Topology Combinatorics 57K18 |
| url | https://arxiv.org/abs/2507.15060 |