Recurrence Relations and Asymptotics of Colored Jones Polynomials

Abstract We consider $$q$$-difference equations for 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 $$3$$d sphere. We give an introduction to the theory of hyperbolic volume of the knots complements and study the asymptotics of the solutions of $$q$$-recurrence relations of high order.