Genus bounds from unrolled quantum groups at roots of unity

For any simple complex Lie algebra (Formula presented.), we show that the degrees of the “ADO” link polynomials coming from the unrolled restricted quantum group (Formula presented.) 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 (Formula presented.) -invariants, where (Formula presented.) 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 (Formula presented.) of our previous work [22] coincides with such ADO invariants, where (Formula presented.) is the Borel part of (Formula presented.). 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.