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

Abstract

We formulate a conjecture (already proven by A. Kricker) about the structure of 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 q^color. We show how to derive the first Kontsevich integral polynomial associated to the theta-graph from the rational expansion of the colored SU(3) Jones polynomial.