On the Colored Jones Polynomial and the Kashaev Invariant

We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the q-Weyl algebra of q-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it would lead to a proof of the volume conjecture. 0. Introduction For a knot K in R, the colored Jones polynomial J ′ K(N) is a Laurent polynomial, J ′ K(N) ∈ R := Z[q ], see [J, MMo]. Here N is a positive integer standing for the N -dimensional prime sl2-module. We use the unframed version and the normalization in which J ′ K(N) = 1 when K is the unknot. The colored Jones polynomial J ′ K(N) is defined using the R-matrix of the quantized enveloping algebra of sl2(C). Here we present the colored Jones polynomial as the inverse of the quantum determinant of an almost quantum matrix whose entries are in the q-Weyl algebra of q-operators acting on the polynomial rings, evaluated at the constant function 1. The proof is based on the quantum MacMahon Master theorem proved in [GLZ]. Actually, it was an attempt to get a determinant formula for the colored Jones polynomial that led the second author to the conjecture that eventually became the quantum MacMahon’s Master theorem in [GLZ]. We will then give an application to the case of the Kashaev invariant 〈K〉N := J ′ K(N)|q=exp 2πi/N . We show that a special evaluation of the determinant will give the Kashaev invariant. Our interpretation of the Kashaev invariant suggests that the natural generalization of the Kashaev invariant for other simple Lie algebra should be the quantum invariant of knots colored by the Verma module of highest weight −δ, where δ is the half-sum of positive roots. Finally we point out how the hyperbolic volume of the knot complement, through the theory of L-torsion, has a determinant formula that looks strikingly similar to the one of Kashaev invariants: In both we have non-commutative deformations of the Burau matrices, but in one case quantum determinant is use, in the other the Fuglede-Kadison determinant is used. This suggests an approach to the volume conjecture using quantum determinant as an approximation of the infinite-dimensional Fuglede-Kadison determinant. 0.1. A determinant formula for the colored Jones polynomial. 0.1.1. Right-quantum matrices and quantum determinants. A 2× 2 matrix ( a b c d ) is right-quantum if ac = qca (q-commutation of the entries in a column) bd = qdb (q-commutation of the entries in a column) ad = da+ qcb− qbc (cross commutation relation). An m×m matrix is right-quantum if any 2× 2 submatrix of it is right-quantum. The meaning is a rightquantum matrix preserves the structure of quantum m-spaces (see [Ma]). The product of 2 right-quantum Date: February 8, 2008 First edition: January 31, 2005. The second author was supported in part by National Science Foundation. 1991 Mathematics Classification. Primary 57M25.