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

TL;DR

This paper expresses a quantum invariant of knots as contour integrals over hyperbolic structures, aiding in understanding its asymptotic behavior related to the Volume Conjecture.

Contribution

It provides a new integral representation of the quantum invariant, facilitating asymptotic analysis and advancing the study of quantum topology.

Findings

Expressed the invariant as a sum over contour integrals

Linked the integral form to hyperbolic structures of knot complements

Discussed obstacles to proving exponential growth of the invariant

Abstract

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 $\operatorname{Z}_{N}^{\psi}$.