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

TL;DR

This paper investigates the properties of non-semisimple quantum invariants, specifically ADO invariants, for fibered links in three-dimensional space, revealing their dependence on topological features like genus and Hopf invariants.

Contribution

It establishes a direct link between the degree and top coefficient of ADO invariants and topological invariants of fibered links, extending understanding of quantum invariants in non-semisimple categories.

Findings

Degree of ADO invariant depends on the genus of the fiber surface.

Top coefficient of ADO invariant is a root of unity.

Top coefficient is determined by the Hopf invariant of the associated plane field.

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.