  This is a substantially revised version. 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 null-move 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 joint work of the first author and A. Kricker.
