  Hyperbolic structures (equivalently, principal $\operatorname{PSL}_2(\mathbb
C)$-bundles with connection) on link complements can be described algebraically
by using the octahedral decomposition, which assigns an ideal triangulation to
any diagram of the link. 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 are closely related to a certain
presentation of the Kac-de Concini quantum group
$\mathcal{U}_q(\mathfrak{sl}_2)$ in terms of cluster algebras at $q = \xi$ a
root of unity. Specifically, we identify ratios of the shape parameters of the
octahedral decomposition with central characters of
$\mathcal{U}_\xi(\mathfrak{sl}_2)$. The quantum braiding on these characters is
known to be closely related to $\operatorname{SL}_2(\mathbb C)$-bundles on link
complements, and our work provides a geometric perspective on this
construction.
