The Kontsevich Integral for Knotted Trivalent Graphs
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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
We define the "naive Kontsevich integral" of a knotted trivalent graph or a slice thereof.
- 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 and using ideas from renormalization theory as practiced by quantum field theorists, we define the "renormalized Kontsevich integral" of a knotted trivalent graph or a slice thereof.
- 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 .
- 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 and , and the two types of trivalent vertices with factors and , we define the "corrected Kontsevich integral" of a knotted trivalent graph or a slice thereof. It has all the good properties we can wish for.
Aside - The Relationship with Drinfel'd's KZ Associator
The Drinfel'd KZ associator is reproduced already by .