The Kontsevich Integral for Knotted Trivalent Graphs - Revision history

Drorbn: /* The Good */

2007-03-06T20:22:55Z

The Good

← Older revision		Revision as of 16:22, 6 March 2007
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===

Drorbn: /* Aside - The Relationship with Drinfel'd's KZ Associator */

2007-03-06T20:07:39Z

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

← Older revision		Revision as of 16:07, 6 March 2007
Line 47:		Line 47:
	==Aside - The Relationship with Drinfel'd's KZ Associator==		==Aside - The Relationship with Drinfel'd's KZ Associator==

	The Drinfel'd KZ associator <math>\Phi_{KZ}</math> 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]]

Drorbn at 19:58, 6 March 2007

2007-03-06T19:58:08Z

← Older revision		Revision as of 15:58, 6 March 2007
Line 41:		Line 41:
	==Step 3 - The Corrected Kontsevich Integral==		==Step 3 - The Corrected Kontsevich Integral==

			[[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.

		⚫	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>.

Drorbn at 15:52, 6 March 2007

2007-03-06T15:52:44Z

← Older revision		Revision as of 11:52, 6 March 2007
Line 23:		Line 23:
	==Step 2 - The Renormalized Kontsevich Integral==		==Step 2 - The Renormalized Kontsevich Integral==

	[[Image:07-1352 Renormalizing the Kontsevich Integral.png\|~~400px~~\|center]]		[[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===

Drorbn at 15:50, 6 March 2007

2007-03-06T15:50:04Z

← Older revision		Revision as of 11:50, 6 March 2007
Line 7:		Line 7:
	==Step 1 - The Naive Kontsevich Integral==		==Step 1 - The Naive Kontsevich Integral==

			{{07-1352/Schematics of the Kontsevich Integral}}
⚫	We define the "naive Kontsevich integral" <math>Z_1</math> of a knotted trivalent graph or a slice thereof.

		⚫	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===
Line 21:		Line 23:
	==Step 2 - The Renormalized Kontsevich Integral==		==Step 2 - The Renormalized Kontsevich Integral==

			[[Image:07-1352 Renormalizing the Kontsevich Integral.png\|400px\|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.

		⚫	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===

Drorbn: /* Step 3 - The Corrected Kontsevich Integral */

2007-02-28T01:38:55Z

Step 3 - The Corrected Kontsevich Integral

← Older revision		Revision as of 21:38, 27 February 2007
Line 37:		Line 37:
	==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.		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.

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

Drorbn at 01:36, 28 February 2007

2007-02-28T01:36:18Z

New page

{{Paperlets Navigation}}
{{In Preparation}}

==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" <math>Z_1</math> of a knotted trivalent graph or a slice thereof.

===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 <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.

===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 <math>\mu</math>.

===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" <math>Z=Z_3</math> 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 <math>Z_2</math>.