Census L-space knots are braid positive, except for one that is not
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866917255401439232 |
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| author | Baker, Kenneth L. Kegel, Marc |
| author_facet | Baker, Kenneth L. Kegel, Marc |
| contents | We exhibit braid positive presentations for all L-space knots in the SnapPy census except one, which is not braid positive. The normalized HOMFLY polynomial of o9_30634, when suitably normalized is not positive, failing a condition of Ito for braid positive knots.
We generalize this knot to a 1-parameter family of hyperbolic L-space knots that might not be braid positive. Nevertheless, as pointed out by Teragaito, this family yields the first examples of hyperbolic L-space knots whose formal semigroups are actual semigroups, answering a question of Wang. Furthermore, the roots of the Alexander polynomials of these knots are all roots of unity, disproving a conjecture of Li-Ni. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_12013 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Census L-space knots are braid positive, except for one that is not Baker, Kenneth L. Kegel, Marc Geometric Topology 2020 57K10 (primary) 57K18 (secondary) We exhibit braid positive presentations for all L-space knots in the SnapPy census except one, which is not braid positive. The normalized HOMFLY polynomial of o9_30634, when suitably normalized is not positive, failing a condition of Ito for braid positive knots. We generalize this knot to a 1-parameter family of hyperbolic L-space knots that might not be braid positive. Nevertheless, as pointed out by Teragaito, this family yields the first examples of hyperbolic L-space knots whose formal semigroups are actual semigroups, answering a question of Wang. Furthermore, the roots of the Alexander polynomials of these knots are all roots of unity, disproving a conjecture of Li-Ni. |
| title | Census L-space knots are braid positive, except for one that is not |
| topic | Geometric Topology 2020 57K10 (primary) 57K18 (secondary) |
| url | https://arxiv.org/abs/2203.12013 |