On the Kontsevich integral for knotted trivalent graphs

Abstract

We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips and connected sums. In 1997 Murakami and Ohtsuki [Comm. Math. Phys. 188 (1997) 501–520] first constructed such an extension, building on Drinfel’d’s theory of associators. We construct a step-by-step definition, using elementary Kontsevich integral methods, to get a one-parameter family of corrections that all yield invariants well behaved under the graph operations above.