On the genera of symmetric unions of knots
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908680280080384 |
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| author | Boileau, Michel Kitano, Teruaki Nozaki, Yuta |
| author_facet | Boileau, Michel Kitano, Teruaki Nozaki, Yuta |
| contents | In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09881 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the genera of symmetric unions of knots Boileau, Michel Kitano, Teruaki Nozaki, Yuta Geometric Topology 57K10, 57K14 (Primary) 57M05, 57R18 (Secondary) In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union. |
| title | On the genera of symmetric unions of knots |
| topic | Geometric Topology 57K10, 57K14 (Primary) 57M05, 57R18 (Secondary) |
| url | https://arxiv.org/abs/2407.09881 |