  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.

Quantum Algebra;High Energy Physics - Theory;Geometric Topology