Asymptotic behavior of unknotting numbers of links in a twist family
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916674996797440 |
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| author | Baker, Kenneth L. Miyazawa, Yasuyuki Motegi, Kimihiko |
| author_facet | Baker, Kenneth L. Miyazawa, Yasuyuki Motegi, Kimihiko |
| contents | By twisting a given link $L$ along an unknotted circle $c$, we obtain an infinite family of links $\{ L_n \}$. We introduce the ``stable unknotting number'' which describes the asymptotic behavior of unknotting numbers of links in the twist family. We show the stable unknotting number for any twist family of links depends only on the winding number of $L$ about $c$ (the minimum geometric intersection number of $L$ with a Seifert surface of $c$) and is independent of the wrapping number of $L$ about $c$ (the minimum geometric intersection number of $L$ with a disk bounded by $c$). Thus there are twist families for which the discrepancy between the wrapping number and the stable unknotting number is arbitrarily large. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_03867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of unknotting numbers of links in a twist family Baker, Kenneth L. Miyazawa, Yasuyuki Motegi, Kimihiko Geometric Topology 57K10, 57K14, 57K16 By twisting a given link $L$ along an unknotted circle $c$, we obtain an infinite family of links $\{ L_n \}$. We introduce the ``stable unknotting number'' which describes the asymptotic behavior of unknotting numbers of links in the twist family. We show the stable unknotting number for any twist family of links depends only on the winding number of $L$ about $c$ (the minimum geometric intersection number of $L$ with a Seifert surface of $c$) and is independent of the wrapping number of $L$ about $c$ (the minimum geometric intersection number of $L$ with a disk bounded by $c$). Thus there are twist families for which the discrepancy between the wrapping number and the stable unknotting number is arbitrarily large. |
| title | Asymptotic behavior of unknotting numbers of links in a twist family |
| topic | Geometric Topology 57K10, 57K14, 57K16 |
| url | https://arxiv.org/abs/2504.03867 |