The complexified Kashaev conjecture for hyperbolic links

Let $L$ be a hyperbolic link, and let $J_N(L)$ be its colored Jones polynomial evaluated at $\exp\left(2\pi\sqrt{-1}/N\right)$. Write $\operatorname{vol}(L)$ for the hyperbolic volume of the complement of $L$ and $\operatorname{CS}(L)$ for its Chern–Simons invariant. Complexified Kashaev conjecture. As $N\to\infty$,

$$J_N(L)\sim\exp\left(\frac{N}{2\pi}\left(\operatorname{vol}(L)+\sqrt{-1}\,\operatorname{CS}(L)\right)\right).$$

This conjecture refines the volume asymptotics by incorporating the Chern–Simons invariant. In the paper it is proposed on the basis of numerical observations for the knots $6_3$, $8_9$, and $8_{20}$ and the Whitehead link; the complement is a hyperbolic manifold with cusps.