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

${\displaystyle Z_{0}(K)=\ \ \ \ \ \ \ \ \ \ \int \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum _{m;\ t_{1}<\ldots

We define the "naive Kontsevich integral" ${\displaystyle Z_{1}}$ 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.

• It is divergent.

Step 2 - The Renormalized Kontsevich Integral

After choosing a scale ${\displaystyle \mu }$ and an infinitesimal ${\displaystyle \epsilon }$ and using ideas from renormalization theory as practiced by quantum field theorists, we define the "renormalized Kontsevich integral" ${\displaystyle Z_{2}}$ 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 ${\displaystyle \mu }$, as follows from:

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

${\displaystyle Z(B)=\exp {\frac {\beta c}{2\pi i}}\in {\mathcal {A}}(\uparrow _{n})}$.

By correcting caps and cups with factors ${\displaystyle n}$ and ${\displaystyle u}$, and the two types of trivalent vertices with factors ${\displaystyle y}$ and ${\displaystyle \lambda }$, we define the "corrected Kontsevich integral" ${\displaystyle Z=Z_{3}}$ 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!
The Drinfel'd KZ associator ${\displaystyle \Phi _{KZ}}$ is reproduced already by ${\displaystyle Z_{2}}$: