Two non-semisimple quantum invariants of lens spaces

Abstract

We calculate the chromatic spherical invariant ${\mathcal{K}}_{\mathrm{Rep}(H)}(L(p,q))$ 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 $p\ge 1$. We also describe the duals ${\mathcal{S}}_{\mathcal{I}}(M{)}^{\ast }$ 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, ${\mathcal{S}}_{\mathcal{I}}(L(p,q){)}^{\ast }$ corresponds with cyclic traces on $\mathcal{I}$ satisfying a property reminiscent of the modified trace condition.

arXiv comment29 pages, 13 figures, comments welcome