Diagonal knots and the tau invariant
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912577196392448 |
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| author | Arndt, Jackson Jansen, Malia McBurney, Payton Vance, Katherine |
| author_facet | Arndt, Jackson Jansen, Malia McBurney, Payton Vance, Katherine |
| contents | In 2003, Ozsváth, Szabó, and Rasmussen introduced the $τ$ invariant for knots, and in 2011, Sarkar published a computational shortcut for the $τ$ invariant of knots that can be represented by diagonal grid diagrams. Previously, the only knots known to have diagonal grid diagram representations were torus knots. We prove that all such knots are positive knots, and we produce an example of a knot with a diagonal grid diagram representation which is not a torus knot. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13796 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diagonal knots and the tau invariant Arndt, Jackson Jansen, Malia McBurney, Payton Vance, Katherine Geometric Topology 57K10 (Primary), 57K18 (Secondary) In 2003, Ozsváth, Szabó, and Rasmussen introduced the $τ$ invariant for knots, and in 2011, Sarkar published a computational shortcut for the $τ$ invariant of knots that can be represented by diagonal grid diagrams. Previously, the only knots known to have diagonal grid diagram representations were torus knots. We prove that all such knots are positive knots, and we produce an example of a knot with a diagonal grid diagram representation which is not a torus knot. |
| title | Diagonal knots and the tau invariant |
| topic | Geometric Topology 57K10 (Primary), 57K18 (Secondary) |
| url | https://arxiv.org/abs/2412.13796 |