Hyperbolic structures on link complements, octahedral decompositions, and quantum ${\mathfrak{sl}}_2$

Abstract

Hyperbolic structures on link complements (equivalently, representations of the fundamental group into ${\mathrm{SL}}_2(\mathbb{C})$) can be described algebraically by using the octahedral decomposition determined by a link diagram. The decomposition (like any ideal triangulation) gives a set of gluing equations in shape parameters whose solutions are hyperbolic structures. We show that these equations can be obtained from Kashaev-Reshetikhin's braiding on the Kac-de Concini quantum group ${\mathcal{U}}_{\hat{I}}¾({\mathfrak{sl}}_2)$ at a root of unity $\hat{I}¾$. This braiding gives coordinates on the ${\mathrm{SL}}_2(\mathbb{C})$ representation variety of a link and our work shows how to interpret these geometrically.

39 pages + refs. v3: substantial revision