A rationality conjecture about Kontsevich integral of knots and its implications to the structure of the colored Jones polynomial

We formulate a conjecture about the structure of the Kontsevich integral of a knot. We describe its value in terms of the generating functions for the numbers of external edges attached to closed 3-valent diagrams. We conjecture that these functions are rational functions of the exponentials of their arguments, their denominators being the powers of the Alexander–Conway polynomial. This conjecture implies the existence of an expansion of a colored Jones (HOMFLY) polynomial in powers of q−1 whose coefficients are rational functions of qα (α being the color assigned to the knot). We show how to derive the first Kontsevich integral polynomial associated to the θ-graph from the rational expansion of the colored SU(3) Jones polynomial.