# The Kontsevich Integral for Knotted Trivalent Graphs

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In Preparation

The information below is preliminary and cannot be trusted! (v)

## Abstract

We construct a (very) well-behaved invariant of knotted trivalent graphs using only the Kontsevich integral, in three steps.

## Step 1 - The Naive Kontsevich Integral $Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots

We define the "naive Kontsevich integral" $Z_1$ of a knotted trivalent graph or a slice thereof as in the "standard" picture above, except generalized to graphs in the obvious manner.

### The Good

• It has a factorization property.
• For the "braid-like" factors, it has invariance under horizontal deformations.
• It is morally a universal finite type invariant.

### The Bad

• It is divergent.

## Step 2 - The Renormalized Kontsevich Integral

After choosing a scale $\mu$ and an infinitesimal $\epsilon$ and using ideas from renormalization theory as practiced by quantum field theorists, we define the "renormalized Kontsevich integral" $Z_2$ of a knotted trivalent graph or a slice thereof as summarized by the picture above.

### The Good

• It retains all the good properties of the naive Kontsevich integral.
• It is convergent.
• It is invariant under rigid motions of critical points and trivalent vertices.
• It has sensible behaviour under vertical connect sum, delete and unzip operations.
• It has a sensible behaviour under the changing of the scale $\mu$, as follows from:

Claim. Let $w_1,\ldots,w_n$ be distinct complex numbers and let $\beta$ be another complex number. Let $B$ be the ( $n$-strand) braid "rescaling braid" which is the image of the map $[0,1]\to[0,1]\times{\mathbb C}^n$ defined by $t\mapsto(t, e^{\beta t}w_1,\ldots,e^{\beta t}w_n)$. Finally, in ${\mathcal A}(\uparrow_n)$ let $c$ be the "sum of all horizontal chords"; $c=\sum_{1\leq i. Then up to normalization factors which we are getting right with probability $\frac{1}{2^k}$ for some small $k\in{\mathbb N}$, $Z(B)=\exp\frac{\beta c}{2\pi i}\in{\mathcal A}(\uparrow_n)$.

### The Bad

• It is not invariant under non-rigid motions of vertices and critical points.

## Step 3 - The Corrected Kontsevich Integral

By correcting caps and cups with factors $n$ and $u$, and the two types of trivalent vertices with factors $y$ and $\lambda$, we define the "corrected Kontsevich integral" $Z=Z_3$ of a knotted trivalent graph or a slice thereof. It has all the good properties we can wish for. The details still need to be worked out here!

## Aside - The Relationship with Drinfel'd's KZ Associator

The Drinfel'd KZ associator $\Phi_{KZ}$ is reproduced already by $Z_2$: