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 + 2ppv-1)/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(4Np(2)/(u + 2ppv-1)) 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.