The complexification of the volume conjecture

Let $\mathscr{H}\subset S^3$ be a hyperbolic knot. Define its complex volume by

$$\operatorname{cv}(\mathscr{H})=\sqrt{-1}\operatorname{Vol}(\mathscr{H})-2\pi^2\operatorname{CS}^{\rm{SO(3)}}(\mathscr{H}),$$

where $\operatorname{CS}^{\rm{SO(3)}}(\mathscr{H})$ is the Chern–Simons invariant modulo $\pi^2$. Complexification of the volume conjecture. For any hyperbolic knot $\mathscr{H}$ in $S^3$,

$$\lim_{N\to\infty}\frac{\log J_N(\mathscr{H};e^{2\pi\sqrt{-1}/N})}{N}=\frac{\operatorname{cv}(\mathscr{H})}{2\pi\sqrt{-1}}.$$

This extends the volume conjecture from exponential growth rates involving real volume to a complex asymptotic incorporating the Chern–Simons invariant. Its status is not resolved in the source.