On the $\mathrm{SL}(2,\mathbb{C})$ quantum $6j$-symbols and their relation to the hyperbolic volume

Abstract

We generalize the colored Alexander invariant of knots to an invariant of graphs and we construct a face model for this invariant by using the corresponding $6j$-symbols, which come from the non-integral representations of the quantum group ${\mathcal{U}}_q({\mathrm{sl}}_2)$. We call it the $\mathrm{SL}(2,\mathbb{C})$-quantum $6j$-symbols, and show their relation to the hyperbolic volume of a truncated tetrahedron.