Genus bounds from unrolled quantum groups at roots of unity

Abstract

For any simple complex Lie algebra $\mathfrak{g}$, we show that the degrees of the "ADO" link polynomials coming from the unrolled restricted quantum group $\overline{U}^H_q(\mathfrak{g})$ 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 $\mathfrak{u}_q(\mathfrak{g})$-invariants, where $\mathfrak{u}_q(\mathfrak{g})$ is the small quantum group. We give a second proof by showing that the invariant $P_{\mathfrak{u}_q(\mathfrak{b})}^{\theta}(K)$ of our previous work coincides with such ADO invariants, where $\mathfrak{u}_q(\mathfrak{b})$ is the Borel part of $\mathfrak{u}_q(\mathfrak{g})$. 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.