Genus bounds for twisted quantum invariants

Abstract

By twisted quantum invariants we mean polynomial invariants of knots in the three-sphere endowed with a representation of the fundamental group into the automorphism group of a Hopf algebra $H$. These are obtained by the Reshetikhin–Turaev construction extended to the $\mathrm{Aut}(H)$-twisted Drinfeld double of $H$, provided $H$ is finite-dimensional and ${\mathbb{N}}^m$-graded. We show that the degree of these polynomials is bounded above by $2g(K)\cdot d(H)$, where $g(K)$ is the Seifert genus of a knot $K$ and $d(H)$ is the top degree of the Hopf algebra. When $H$ is an exterior algebra, our theorem recovers Friedl and Kim’s genus bounds for twisted Alexander polynomials. When $H$ is the Borel part of restricted quantum ${\mathfrak{sl}}_2$ at an even root of unity, we show that our invariant is the ADO invariant, therefore giving new genus bounds for these invariants.