On knots that divide ribbon knotted surfaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909928955838464 |
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| author | Boden, Hans U. Elmacioglu, Ceyhun Guha, Anshul Karimi, Homayun Rushworth, William Tang, Yun-chi Jun, Bryan Wang Peng |
| author_facet | Boden, Hans U. Elmacioglu, Ceyhun Guha, Anshul Karimi, Homayun Rushworth, William Tang, Yun-chi Jun, Bryan Wang Peng |
| contents | We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of which K is a cross-section. We compute this genus for all prime knots up to 12 crossings, and many 13-crossing knots. The same approach yields new computations of the doubly slice genus. We also introduce the half fusion number of a knot K, that measures the complexity of ribbon 2-knots of which K is a cross-section. We show that it is bounded from below by the Levine-Tristram signatures, and differs from the standard fusion number by an arbitrarily large amount. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_15577 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On knots that divide ribbon knotted surfaces Boden, Hans U. Elmacioglu, Ceyhun Guha, Anshul Karimi, Homayun Rushworth, William Tang, Yun-chi Jun, Bryan Wang Peng Geometric Topology 57K10, 57K45 We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of which K is a cross-section. We compute this genus for all prime knots up to 12 crossings, and many 13-crossing knots. The same approach yields new computations of the doubly slice genus. We also introduce the half fusion number of a knot K, that measures the complexity of ribbon 2-knots of which K is a cross-section. We show that it is bounded from below by the Levine-Tristram signatures, and differs from the standard fusion number by an arbitrarily large amount. |
| title | On knots that divide ribbon knotted surfaces |
| topic | Geometric Topology 57K10, 57K45 |
| url | https://arxiv.org/abs/2209.15577 |