On the Kontsevich integral for knotted trivalent graphs

TL;DR

This paper extends the Kontsevich integral from knots to knotted trivalent graphs, ensuring compatibility with various graph operations, using elementary methods to define a family of well-behaved invariants.

Contribution

It provides a step-by-step elementary construction of a one-parameter family of invariants for knotted trivalent graphs, extending previous work by Murakami and Ohtsuki.

Findings

Constructed an extension of the Kontsevich integral for knotted trivalent graphs.

Ensured invariance under orientation switches, edge deletions, unzips, and connected sums.

Developed a family of corrections yielding well-behaved invariants.

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 [MO] 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.