Volume Conjecture and WKB Asymptotics

We consider q-difference equations for colored Jones polynomials. These polynomials are invariants for the knots and their asymptotics plays an important role in the famous Volume Conjecture (VC) for the complement of the knot to the 3-d sphere. We study WKB asymptotic behavior of the $n$-th colored Jones polynomial at the point exp {$ {2\pi i/N}$} when $n$ and $N$ tends to infinity and limit of $n/N$ belongs to [0, 1]. We state a Theorem on asymptotic expansion of \underline{general solutions} of the q-difference equations. For the \underline{partial solutions}, corresponding to the colored Jones polynomials, using some heuristic and numeric consideration, we suggest a conjecture on their WKB asymptotics. For the special knots under consideration, this conjecture is in accordance with the VC.
This is a joint work with Dmitrii Toulyakov and Tatyana Dudnikova. The work was done in Moscow Center of Fundamental and Applied Mathematics (agreement with Ministry of Science and Higher Education RF № 075-15-2022-283).