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

Hyperbolic structures on link complements (equivalently, representations of the fundamental group into $\operatorname{SL}_2(\mathbb{C})$) 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 $\mathcal{U}_\xi(\mathfrak{sl}_2)$ at a root of unity $\xi$. This braiding gives coordinates on the $\operatorname{SL}_2(\mathbb{C})$ representation variety of a link and our work shows how to interpret these geometrically.