The Loop Expansion of the Kontsevich Integral, the Null-move and S-equivalence

The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric nullmove on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null move leads in a natural way to the introduction of trivalent graphs with beads, and to a conjecture on a rational version of the Kontsevich integral, formulated by the second author and proven in [GK1].