A Non-commutative Formula for the Colored Jones Function

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation value of Wilson loops of a 3-dimensional gauge theory, the Chern-Simons theory. We present the colored Jones function as an evaluation of the inverse of a non-commutative fermionic partition function. This result is in the form familiar in quantum field theory, namely the inverse of a generalized determinant. Our formula also reveals a direct relation between the Alexander polynomial and the colored Jones function of a knot and immediately implies the extensively studied Melvin-Morton-Rozansky conjecture, first proved by Bar-Natan and the first author about ten years ago. Our results complement recent work of T.T.Q. Le and H. Vu, who also give a noncommutative formule for the colored Jones function of a knot, starting from a noncommutative formula for the R matrix of the quantum group Uq(sl2); see [LV].