Non-semisimple $\mathfrak{sl}_2$ quantum invariants of fibred links
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911964004876288 |
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| author | Neumann, Daniel López van der Veen, Roland |
| author_facet | Neumann, Daniel López van der Veen, Roland |
| contents | The Akutsu-Deguchi-Ohtsuki (ADO) invariants are the most studied quantum link invariants coming from a non-semisimple tensor category. We show that, for fibered links in $S^3$, the degree of the ADO invariant is determined by the genus and the top coefficient is a root of unity. More precisely, we prove that the top coefficient is determined by the Hopf invariant of the plane field of $S^3$ associated to the fiber surface. Our proof is based on the genus bounds established in our previous work, together with a theorem of Giroux-Goodman stating that fiber surfaces in the three-sphere can be obtained from a disk by plumbing/deplumbing Hopf bands. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15561 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-semisimple $\mathfrak{sl}_2$ quantum invariants of fibred links Neumann, Daniel López van der Veen, Roland Quantum Algebra Geometric Topology 57K16, 57K10, 20G42 The Akutsu-Deguchi-Ohtsuki (ADO) invariants are the most studied quantum link invariants coming from a non-semisimple tensor category. We show that, for fibered links in $S^3$, the degree of the ADO invariant is determined by the genus and the top coefficient is a root of unity. More precisely, we prove that the top coefficient is determined by the Hopf invariant of the plane field of $S^3$ associated to the fiber surface. Our proof is based on the genus bounds established in our previous work, together with a theorem of Giroux-Goodman stating that fiber surfaces in the three-sphere can be obtained from a disk by plumbing/deplumbing Hopf bands. |
| title | Non-semisimple $\mathfrak{sl}_2$ quantum invariants of fibred links |
| topic | Quantum Algebra Geometric Topology 57K16, 57K10, 20G42 |
| url | https://arxiv.org/abs/2407.15561 |