Recurrence relations and asymptotics of colored Jones polynomials

We consider $q$-difference relations for the colored Jones polynomials. These sequences of polynomials are invariants for the knots and their asymptotics plays an important role in the famous volume conjecture for the complement of the knot to the 3D sphere. We give an introduction to the theory of hyperbolic volume of the knot complements and study the asymptotics of the solutions of the $q$-recurrence equations of higher order.