The complexified Volume Conjecture

Let $K$ be a hyperbolic knot, meaning that its complement $S^3\setminus K$ possesses a complete hyperbolic structure. Let $J_N(K;q)$ be the $N$-dimensional colored Jones polynomial, and let $\operatorname{CS}(S^3\setminus K)$ denote the Chern--Simons invariant of the three-manifold with torus boundary. The complexified Volume Conjecture.

$$2\pi\lim_{N\to\infty}\frac{\log J_N(K;\exp(2\pi\sqrt{-1}/N))}{N}\equiv \operatorname{Vol}(S^3\setminus K)+\sqrt{-1}\operatorname{CS}(S^3\setminus K)\pmod{\pi^2\sqrt{-1}\mathbb{Z}}.$$

This conjecture incorporates the Chern--Simons invariant as the imaginary part of the complexified volume. The paper presents it as a natural strengthening of the Volume Conjecture, with the stated general claim not established in full.