On the Kashaev signature conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913433425805312 |
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| author | Cimasoni, David Ferretti, Livio |
| author_facet | Cimasoni, David Ferretti, Livio |
| contents | In 2018, Kashaev introduced a square matrix indexed by the regions of a link diagram, and conjectured that it provides a novel way of computing the Levine-Tristram signature and Alexander polynomial of the corresponding oriented link. In this article, we show that for the classical signature (i.e. the Levine-Tristram signature at -1), this conjecture follows from the seminal work of Gordon-Litherland. We also relate Kashaev's matrix to Kauffman's "Formal Knot Theory" model of the Alexander polynomial. As a consequence, we establish the Alexander polynomial and classical signature parts of the conjecture for arbitrary links, as well as the full conjecture for definite knots. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_16729 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Kashaev signature conjecture Cimasoni, David Ferretti, Livio Geometric Topology 57K10 In 2018, Kashaev introduced a square matrix indexed by the regions of a link diagram, and conjectured that it provides a novel way of computing the Levine-Tristram signature and Alexander polynomial of the corresponding oriented link. In this article, we show that for the classical signature (i.e. the Levine-Tristram signature at -1), this conjecture follows from the seminal work of Gordon-Litherland. We also relate Kashaev's matrix to Kauffman's "Formal Knot Theory" model of the Alexander polynomial. As a consequence, we establish the Alexander polynomial and classical signature parts of the conjecture for arbitrary links, as well as the full conjecture for definite knots. |
| title | On the Kashaev signature conjecture |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2310.16729 |