Genus bounds for twisted quantum invariants

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 \operatorname{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.