A quantization of the $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons invariant of tangle exteriors

Abstract

We define a sequence of invariants $\mathcal{Z}_{N}^{\psi}$ of tangles with flat $\mathfrak{sl}_{2}$ connections (i.e. hyperbolic structures) on their complements. These can be interpreted as a geometric twist of the Kashaev invariant or as a quantization of the $\operatorname{SL}_{2}(\mathbb{C})$ Chern-Simons invariant. To support the second interpretation we give a new description $\mathcal{I}^{\psi}$ of the Chern-Simons invariant of a tangle exterior. $\mathcal{Z}_{N}^{\psi}$ directly recovers $\mathcal{I}^{\psi}$ when $N = 1$. We build $\mathcal{Z}_{N}^{\psi}$ using modules over unrestricted quantum $\mathfrak{sl}_{2}$ at a root of unity and the holonomy $R$-matrices previously constructed by the author and Reshetikhin (arXiv:2509.02354). Unlike most previous constructions of geometric quantum invariants $\mathcal{Z}_{N}^{\psi}$ is defined without any phase ambiguity. It is natural to conjecture that $\mathcal{Z}_{N}^{\psi}$ is related to the quantization of Chern-Simons theory with complex, noncompact gauge group $\operatorname{SL}_{2}(\mathbb{C})$ and we discuss how to interpret our results in this context.