Difference between revisions of "The Kontsevich Integral for Knotted Trivalent Graphs"
(→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.  +  {{071352/Schematics of the Kontsevich Integral}} 
+  
+  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.  +  [[Image:071352 Renormalizing the Kontsevich Integral.png480pxcenter]] 
+  
+  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>.  +  * It has a sensible behaviour under the changing of the scale <math>\mu</math>, as follows from: 
+  
+  '''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>,  
+  
+  {{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.  +  [[Image:071352 Correctors Corrections Syzygies.png480pxcenter]] 
+  
+  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>.  +  The Drinfel'd KZ associator <math>\Phi_{KZ}</math> is reproduced already by <math>Z_2</math>: 
+  
+  [[Image:071352 Normalizing PhiKZ.png480pxcenter]] 
Latest revision as of 16:22, 6 March 2007

The information below is preliminary and cannot be trusted! (v)
Contents 
Abstract
We construct a (very) wellbehaved 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 "braidlike" 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 nonrigid 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 :