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

$Z_{0}(K)=\ \ \ \ \ \ \ \ \ \ \int \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum _{m;\ t_{1}<\ldots <t_{m};\ P=\{(z_{i},z'_{i})\}}{\frac {(-1)^{\#P_{\downarrow }}}{(2\pi i)^{m}}}D_{P}\bigwedge _{i=1}^{m}{\frac {dz_{i}-dz'_{i}}{z_{i}-z'_{i}}}$

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"; $c=\sum _{1\leq i<j\leq n}t^{ij}$. 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 $Z_{2}$: