The loop expansion of the Kontsevich integral, the null move and S-equivalence

Details The loop expansion of the Kontsevich integral, the null move and S-equivalence The loop expansion of the Kontsevich integral, the null move and S-equivalence

2000-03-28

arxiv.org/abs/math/0003187v3

Mathematics

Geometric Topology

AMS-LaTeX, 20 pages with 31 figures

Scientific paper

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.

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