Unknotting via null-homologous twists and multi-twists
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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_ | 1866917730509127680 |
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| author | Allen, Samantha Ince, Kenan Kim, Seungwon Ruppik, Benjamin Matthias Turner, Hannah |
| author_facet | Allen, Samantha Ince, Kenan Kim, Seungwon Ruppik, Benjamin Matthias Turner, Hannah |
| contents | The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by a null-homologous twist on 2 strands. While the unknotting number gives an upper bound on the smooth 4-genus, the untwisting number gives an upper bound on the topological 4-genus. The surgery description number, which allows multiple null-homologous twists in a single twisting region to count as one operation, lies between the topological 4-genus and the untwisting number. We show that the untwisting and surgery description numbers are different for infinitely many knots, though we also find that the untwisting number is at most twice the surgery description number plus 1. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_04621 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unknotting via null-homologous twists and multi-twists Allen, Samantha Ince, Kenan Kim, Seungwon Ruppik, Benjamin Matthias Turner, Hannah Geometric Topology 57K40, 57K10 The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by a null-homologous twist on 2 strands. While the unknotting number gives an upper bound on the smooth 4-genus, the untwisting number gives an upper bound on the topological 4-genus. The surgery description number, which allows multiple null-homologous twists in a single twisting region to count as one operation, lies between the topological 4-genus and the untwisting number. We show that the untwisting and surgery description numbers are different for infinitely many knots, though we also find that the untwisting number is at most twice the surgery description number plus 1. |
| title | Unknotting via null-homologous twists and multi-twists |
| topic | Geometric Topology 57K40, 57K10 |
| url | https://arxiv.org/abs/2211.04621 |