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arXiv · 2407.15561

Non-semisimple $\mathfrak{sl}_2$ quantum invariants of fibred links

Daniel López Neumann·Roland van der Veen
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Abstract

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.

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Daniel López Neumann, Roland van der Veen. 2024-07-22. Non-semisimple $\mathfrak{sl}_2$ quantum invariants of fibred links. https://arxiv.org/abs/2407.15561

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