The colored Jones polynomial $V_N(K,q)$ of a knot $K$ is a knot invariant. Several years ago, Kashaev and followers proposed the “volume conjecture” for the large $N$ asymptotics of Jones polynomial, namely that: $1/N Log(V_N(e^(2ipi/N))) to$ Volume of the knot complement (i.e. the volume of $S^3/K$, measured with the complete hyperbolic metric). This conjecture has been proved only for very few knots. Recently, Dijkgraaf, Fuji, Manabe, Gukov, Sulkowski, proposed an all order asymptotic formula for the log of Jones polynomial at large $N$, such that the leading order is indeed the volume. Their conjecture claims that the higher order terms in the expansion are the symplectic invariants (E.O. topological recursion) of the character variety of the knot. We refined their conjecture, and checked it to the first few orders for several knots, and interpret it in terms of Baker-Akhiezer functions (wave function) of some integrable system. At least for torus knots we have a heuristic matrix model proof of that conjectured formula.

IPhT