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

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(The Good)
 
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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|480px|center]]
 +
 
 +
After choosing a scale <math>\mu</math> and an infinitesimal <math>\epsilon</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===
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* It is invariant under rigid motions of critical points and trivalent vertices.
 
* 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 sensible behaviour under vertical connect sum, delete and unzip operations.
* It has a sensible behaviour under the changing of the scale <math>\mu</math>.
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* It has a sensible behaviour under the changing of the scale <math>\mu</math>, as follows from:
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'''Claim.''' Let <math>w_1,\ldots,w_n</math> be distinct complex numbers and let <math>\beta</math> be another complex number. Let <math>B</math> be the (<math>n</math>-strand) braid "rescaling braid" which is the image of the map <math>[0,1]\to[0,1]\times{\mathbb C}^n</math> defined by <math>t\mapsto(t, e^{\beta t}w_1,\ldots,e^{\beta t}w_n)</math>. Finally, in <math>{\mathcal A}(\uparrow_n)</math> let <math>c</math> be the "sum of all horizontal chords"; <math>c=\sum_{1\leq i<j\leq n}t^{ij}</math>. Then up to normalization factors which we are getting right with probability <math>\frac{1}{2^k}</math> for some small <math>k\in{\mathbb N}</math>,
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{{Equation*|<math>Z(B)=\exp\frac{\beta c}{2\pi i}\in{\mathcal A}(\uparrow_n)</math>.}}
  
 
===The Bad===
 
===The Bad===
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==Step 3 - The Corrected Kontsevich Integral==
 
==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" <math>Z=Z_3</math> of a knotted trivalent graph or a slice thereof. It has all the good properties we can wish for.
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[[Image:07-1352 Correctors Corrections Syzygies.png|480px|center]]
 +
 
 +
By correcting caps and cups with factors <math>n</math> and <math>u</math>, and the two types of trivalent vertices with factors <math>y</math> and <math>\lambda</math>, we define the "corrected Kontsevich integral" <math>Z=Z_3</math> 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==
 
==Aside - The Relationship with Drinfel'd's KZ Associator==
  
The Drinfel'd KZ associator is reproduced already by <math>Z_2</math>.
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The Drinfel'd KZ associator <math>\Phi_{KZ}</math> is reproduced already by <math>Z_2</math>:
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[[Image:07-1352 Normalizing PhiKZ.png|480px|center]]

Latest revision as of 16:22, 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 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

07-1352 Correctors Corrections Syzygies.png

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:

07-1352 Normalizing PhiKZ.png