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 $\exp ((u+2p\pi \mathrm{ı})\mathrm{/}N)$, 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 $\exp (4N{\pi }^2\mathrm{/}(u+2p\pi \mathrm{ı}))$ 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.