Genus bounds for twisted quantum invariants

TL;DR

This paper establishes bounds on the degree of twisted quantum invariants of knots, linking algebraic properties of Hopf algebras to the knot's Seifert genus, and recovers known bounds for special cases like twisted Alexander polynomials.

Contribution

It introduces a general bound on the degree of twisted quantum invariants based on the Hopf algebra's top degree and the knot's genus, unifying and extending previous results.

Findings

Degree of invariants bounded by 2g(K)·d(H)

Recovers Friedl and Kim's bounds for twisted Alexander polynomials

Provides new genus bounds for ADO invariants at roots of unity

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.