Difference between revisions of "The Kontsevich Integral for Knotted Trivalent Graphs"
(→The Good) |
|||
(3 intermediate revisions by one user not shown) | |||
Line 23: | Line 23: | ||
==Step 2 - The Renormalized Kontsevich Integral== | ==Step 2 - The Renormalized Kontsevich Integral== | ||
− | [[Image:07-1352 Renormalizing the Kontsevich Integral.png| | + | [[Image:07-1352 Renormalizing the Kontsevich Integral.png|480px|center]] |
− | 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. | + | 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=== | ||
Line 33: | Line 33: | ||
* 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=== | ||
Line 41: | Line 45: | ||
==Step 3 - The Corrected Kontsevich Integral== | ==Step 3 - The Corrected Kontsevich Integral== | ||
− | 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. | + | [[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>. | + | The Drinfel'd KZ associator <math>\Phi_{KZ}</math> is reproduced already by <math>Z_2</math>: |
+ | |||
+ | [[Image:07-1352 Normalizing PhiKZ.png|480px|center]] |
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) 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 :