Difference Equation of the Colored Jones Polynomial for Torus Knot

The N-colored Jones polynomial JK (N) is one of the quantum invariants for knot K . It is associated with the N-dimensional irreducible representation of sl(2), and is powerful to classify knots. Motivated by “volume conjecture” [12, 18] saying that a hyperbolic volume of the knot complement dominates an asymptotic behavior of the colored Jones polynomial JK (N) at q = e2πi/N , it receives much interests toward a geometrical and topological interpretation of the quantum invariants. Recently another intriguing structure of the colored Jones polynomial was put forward; it was shown that the N-colored Jones polynomial JK (N) can be written in a q-hypergeometric form, and that it satisfies a recursion relation with respect to N [6]. It was further demonstrated for trefoil and figure-eight knot [5] that a recursion relation is related to the A-polynomial AK (L, M) (see also Ref. 4), which denotes an algebraic curve of eigenvalues of the S L(2,C) representation of the boundary torus of knot K [3]. As the A-polynomial contains many geometrical informations such as the boundary slopes of the knot, this “AJ conjecture” may help our geometrical understanding of the colored Jones polynomial. In this article, we study torus knot Ts,t where s and t are relatively prime integers. We prove that the N-colored Jones polynomial JK (N) for the torus knots K = Ts,t satisfies the second order recursion relation (7) [Theorem 4], which reduces to the first order (9) only in a case of K = T2,2m+1 [Theorem 6]. Furthermore we shall show that this difference operator gives the A-polynomial of the torus knot as was demonstrated in Ref. 5 [Theorem 7]. Throughout this article, we normalize the colored Jones polynomial to be