Signs of knot polynomial evaluations from a topological perspective
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912843271503872 |
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| author | Jost, Luana Lewark, Lukas |
| author_facet | Jost, Luana Lewark, Lukas |
| contents | We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of the double branched covering. We provide explicit formulae for these evaluations in terms of the linking pairing. The proof proceeds via so-called singular determinants, from which we also extract new lower bounds for the unknotting numbers of knots and links. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16588 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Signs of knot polynomial evaluations from a topological perspective Jost, Luana Lewark, Lukas Geometric Topology 57K10, 57K14 We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of the double branched covering. We provide explicit formulae for these evaluations in terms of the linking pairing. The proof proceeds via so-called singular determinants, from which we also extract new lower bounds for the unknotting numbers of knots and links. |
| title | Signs of knot polynomial evaluations from a topological perspective |
| topic | Geometric Topology 57K10, 57K14 |
| url | https://arxiv.org/abs/2601.16588 |