We study the asymptotic behavior of the N-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp \bigl ((u+2p\pi \sqrt {-1})/N\bigr )$, 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 $\exp \bigl (4N\pi ^2/(u+2p\pi \sqrt {-1})\bigr )$ 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.

The optimistic limit is a mathematical formulation of the classical limit, which is a physical method to estimate the actual limit by using the saddle-point method of a certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. This modified version is easier to handle and more combinatorial than the original one. On the other hand, it is known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at a certain root of unity. This optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it is very complicated and needs many unnatural assumptions. In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo $4{\it\pi}^{2}$. This new version is easier to handle and more combinatorial than the old version, and has many advantages over the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation.

The colored Jones polynomial is a function $J_K:{\mathbb{N}}\longrightarrow{\mathbb{Z}}[t,t^{-1}]$ associated with a knot $K$ in 3-space. We will show that for an alternating knot $K$ the absolute values of the first and the last three leading coefficients of $J_K(n)$ are independent of $n$ when $n$ is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading coefficient of the colored Jones polynomial for alternating knots. As a corollary we get a volume-ish theorem for the colored Jones polynomial.