Genus bounds from unrolled quantum groups at roots of unity

Abstract For any simple complex Lie algebra , we show that the degrees of the “ADO” link polynomials coming from the unrolled restricted quantum group at a root of unity give lower bounds to the Seifert genus of the link. We give a direct simple proof of this fact relying on a Seifert surface formula involving universal ‐invariants, where is the small quantum group. As a special case, we get a genus bound for the Harper polynomial which allows to detect the genera of the Kinoshita–Terasaka and Conway knots. We give a second proof of our main theorem by showing that the invariant of our previous work [22] coincides with such ADO invariants, where is the Borel part of . To prove this, we show that equivariantizations of relative Drinfeld centers of crossed products essentially contain unrolled restricted quantum groups, a fact that could be of independent interest.