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.
"On the Kontsevich integral for knotted trivalent graphs." Algebr. Geom. Topol. 10 (3) 1317 - 1365, 2010. https://doi.org/10.2140/agt.2010.10.1317