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

TL;DR

This paper links hyperbolic structures on link complements with quantum group representations, showing how octahedral decompositions and quantum braiding encode the geometric data of the link complement.

Contribution

It establishes a connection between hyperbolic structures, octahedral decompositions, and quantum group braiding at roots of unity, providing a new geometric interpretation.

Findings

Octahedral decompositions encode hyperbolic structures.

Quantum braiding relates to shape parameters of the decomposition.

Coordinates on the SL_2(C) representation variety are interpreted geometrically.

Abstract

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.