Thistlethwaite Theorems for Knotoids and Linkoids
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913614710964224 |
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| author | Chmutov, Sergei Deng, Qingying Ellis-Monaghan, Joanna A. Lando, Sergei Moltmaker, Wout |
| author_facet | Chmutov, Sergei Deng, Qingying Ellis-Monaghan, Joanna A. Lando, Sergei Moltmaker, Wout |
| contents | The classical Thistlethwaite theorem for links can be phrased as asserting that the Kauffman bracket of a link can be obtained from an evaluation of the Bollobás-Riordan polynomial of a ribbon graph associated to one of the link's Kauffman states. In this paper, we extend this result to knotoids, which are a generalization of knots that naturally arises in the study of protein topology. Specifically we extend the Thistlethwaite theorem to the twisted arrow polynomial of knotoids, which is an invariant of knotoids on compact, not necessarily orientable, surfaces. To this end, we define twisted knotoids, marked ribbon graphs, and their arrow- and Bollobás-Riordan polynomials. We also extend the Thistlethwaite theorem to the loop arrow polynomial of knotoids in the plane, and to spherical linkoids. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12357 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Thistlethwaite Theorems for Knotoids and Linkoids Chmutov, Sergei Deng, Qingying Ellis-Monaghan, Joanna A. Lando, Sergei Moltmaker, Wout Geometric Topology 57K12, 57K14 The classical Thistlethwaite theorem for links can be phrased as asserting that the Kauffman bracket of a link can be obtained from an evaluation of the Bollobás-Riordan polynomial of a ribbon graph associated to one of the link's Kauffman states. In this paper, we extend this result to knotoids, which are a generalization of knots that naturally arises in the study of protein topology. Specifically we extend the Thistlethwaite theorem to the twisted arrow polynomial of knotoids, which is an invariant of knotoids on compact, not necessarily orientable, surfaces. To this end, we define twisted knotoids, marked ribbon graphs, and their arrow- and Bollobás-Riordan polynomials. We also extend the Thistlethwaite theorem to the loop arrow polynomial of knotoids in the plane, and to spherical linkoids. |
| title | Thistlethwaite Theorems for Knotoids and Linkoids |
| topic | Geometric Topology 57K12, 57K14 |
| url | https://arxiv.org/abs/2412.12357 |