Khovanov homology and rational unknotting
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866918218029858816 |
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| author | Iltgen, Damian Lewark, Lukas Marino, Laura |
| author_facet | Iltgen, Damian Lewark, Lukas Marino, Laura |
| contents | Building on work by Alishahi-Dowlin, we extract a new knot invariant $λ\ge 0$ from universal Khovanov homology. While $λ$ is a lower bound for the unknotting number, in fact more is true: $λ$ is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that for all $n \ge 0$, there exists a knot K with $λ(K) = n$. Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_15107 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Khovanov homology and rational unknotting Iltgen, Damian Lewark, Lukas Marino, Laura Geometric Topology 57K10, 57K18 Building on work by Alishahi-Dowlin, we extract a new knot invariant $λ\ge 0$ from universal Khovanov homology. While $λ$ is a lower bound for the unknotting number, in fact more is true: $λ$ is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that for all $n \ge 0$, there exists a knot K with $λ(K) = n$. Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles. |
| title | Khovanov homology and rational unknotting |
| topic | Geometric Topology 57K10, 57K18 |
| url | https://arxiv.org/abs/2110.15107 |