Difference between revisions of "The Kontsevich Integral for Knotted Trivalent Graphs"

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(Step 3 - The Corrected Kontsevich Integral)
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==Step 1 - The Naive Kontsevich Integral==
 
==Step 1 - The Naive Kontsevich Integral==
  
We define the "naive Kontsevich integral" <math>Z_1</math> of a knotted trivalent graph or a slice thereof.
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{{07-1352/Schematics of the Kontsevich Integral}}
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We define the "naive Kontsevich integral" <math>Z_1</math> 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===
 
===The Good===
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==Step 2 - The Renormalized Kontsevich Integral==
 
==Step 2 - The Renormalized Kontsevich Integral==
  
After choosing a scale <math>\mu</math> and using ideas from renormalization theory as practiced by quantum field theorists, we define the "renormalized Kontsevich integral" <math>Z_2</math> of a knotted trivalent graph or a slice thereof.
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[[Image:07-1352 Renormalizing the Kontsevich Integral.png|400px|center]]
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After choosing a scale <math>\mu</math> and using ideas from renormalization theory as practiced by quantum field theorists, we define the "renormalized Kontsevich integral" <math>Z_2</math> of a knotted trivalent graph or a slice thereof as summarized by the picture above.
  
 
===The Good===
 
===The Good===

Revision as of 11:50, 6 March 2007

In Preparation

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

07-1352 Kontsevich Integral.png
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

07-1352 Renormalizing the Kontsevich Integral.png

After choosing a scale \mu 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.

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.

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

The Drinfel'd KZ associator is reproduced already by Z_2.