A quantization of the ${\mathrm{SL}}_2(\mathbb{C})$ 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 ${\mathrm{SL}}_2(\mathbb{C})$ 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 ${\mathrm{SL}}_2(\mathbb{C})$ and we discuss how to interpret our results in this context.

v2: Results strengthened, exposition rewritten. v3: Minor corrections. 69 + 6 pages