The colored Jones polynomial of the figure-eight knot and a quantum modularity

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp\bigl((u+2p\pi\i)/N\bigr)$, where $u$ is a small real number and $p$ is a positive integer. We show that it is asymptotically equivalent to the product of the $p$-dimensional colored Jones polynomial evaluated at $\exp\bigl(4N\pi^2/(u+2p\pi\i)\bigr)$ and a term that grows exponentially with growth rate determined by the Chern--Simons invariant. This indicates a quantum modularity of the colored Jones polynomial.