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

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.