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 10 years ago. Our results complement recent work of Huynh and Le, who also give a non-commutative formulae for the colored Jones function of a knot, starting from a non-commutative formula for the R matrix of the quantum group ; see Huynh and Le (in math.GT/0503296).