The Kontsevich Integral for Knotted Trivalent Graphs
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The information below is preliminary and cannot be trusted! (v)
Contents |
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

We define the "naive Kontsevich integral" 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 and an infinitesimal
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 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
, as follows from:
Claim. Let be distinct complex numbers and let
be another complex number. Let
be the (
-strand) braid "rescaling braid" which is the image of the map
defined by
. Finally, in
let
be the "sum of all horizontal chords";
. Then up to normalization factors which we are getting right with probability
for some small
,

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 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. The details still need to be worked out here!
Aside - The Relationship with Drinfel'd's KZ Associator
The Drinfel'd KZ associator is reproduced already by
: