  We study the relation between perturbative knot invariants and the free
energies defined by topological string theory on the character variety of the
knot. Such a correspondence between SL(2;C) Chern-Simons gauge theory and the
topological open string theory was proposed earlier on the basis of the volume
conjecture and AJ conjecture. In this paper we discuss this correspondence
beyond the subleading order in the perturbative expansion on both sides. In the
computation of the perturbative invariants for the hyperbolic 3-manifold, we
adopt the state integral model for the hyperbolic knots, and the factorized AJ
conjecture for the torus knots. On the other hand, we iteratively compute the
free energies on the character variety using the Eynard-Orantin topological
recursion relation. We check the correspondence for the figure eight knot
complement and the once punctured torus bundle over S^1 with the holonomy L^2R
up to the fourth order. For the torus knots, we find trivial the recursion
relations on both sides.
