Two non-semisimple quantum invariants of lens spaces

Abstract

We calculate the chromatic spherical invariant $\mathscr{K}_{\mathrm{Rep}(H)}(L(p,q))$ of lens spaces $L(p, q)$ for arbitrary finite-dimensional unibalanced, unimodular Hopf algebras $H$. In the case of the index-2 Nichols Hopf algebra $\mathcal{K}_2$ and $q=1$, we find that the invariant of $L(p, 1)$ depends only on the parity of $p$ for $p\geq 1$. We also describe the duals $\mathcal{S_I}(M)^*$ of the admissible skein modules of handlebodies $M=H_g$ and of $M=L(p, q)$ for arbitrary tensor ideals $\mathcal{I}$ of a ribbon tensor category. In particular, $\mathcal{S_I}(L(p, q))^*$ corresponds with cyclic traces on $\mathcal{I}$ satisfying a property reminiscent of the modified trace condition.