Stavros Garoufalidis and Lev Rozansky
Abstract: 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 by joint work of Kricker and the first author.
Key words: Kontsevich integral, claspers, null-move, S-equivalence, Blanchfield pairing, Euler degree, beads, trivalent graphs, hair map, hairy struts, hairy vertices, finite type invariants, n-equivalence.
Notes: 20 pages.