State integrals for the quantized $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons invariant

Previous work of the author and N. Reshetikhin defines an invariant $\operatorname{Z}_{N}^{\psi}(K, \rho, \mu)$ of a knot $K$, a representation $\rho : \pi_{1}(S^{3} \setminus K) \to \operatorname{SL}_2(\mathbb{C})$, and a logarithm $\mu$ of a meridian eigenvalue of $\rho$. It can be interpreted as a geometric twist of the Kasahev invariant or as a quantization of the $\operatorname{SL}(\mathbb{C})$ Chern-Simons invariant and is defined using a discrete state-sum involving quantum dilogarithms. In this paper we show how to express $\operatorname{Z}_{N}^{\psi}(K, \rho, \mu)$ as a sum over contour integrals in a space parametrizing hyperbolic structures on the knot complement. Such integral presentations are an important step in determining the asymptotics of quantum invariants as predicted by the Volume Conjecture. We discuss this perspective and the remaining obstacles to establishing exponential growth of $\operatorname{Z}_{N}^{\psi}$.