The generalized volume conjecture for the figure-eight knot parametrized by a complex number with small imaginary part

We study the asymptotic behavior, as $N$ tends to infinity, of the $N$-dimensional colored Jones polynomial of the figure-eight knot, evaluated at $\exp (\xi \mathrm{/}N)$ for a complex parameter $\xi$ with $0<\mathrm{I}\mathrm{m}\xi <\pi \mathrm{/}2$. We prove that if $\mathrm{R}\mathrm{e}\xi$ is large the colored Jones polynomial grows exponentially with growth rate expressed by the Chern–Simons invariant, and that if $\mathrm{R}\mathrm{e}\xi$ is small it converges to the reciprocal of the Alexander polynomial evaluated at $\exp \xi$.