06-1350/Class Notes for Tuesday November 7 - Revision history

Drorbn at 12:43, 24 November 2009

2009-11-24T12:43:30Z

← Older revision		Revision as of 08:43, 24 November 2009
Line 28:		Line 28:

	{{note\|BLT}} D. Bar-Natan, T. Q. T. Le and D. P. Thurston, ''Two applications of elementary knot theory to Lie algebras and Vassiliev invariants'', [http://www.msp.warwick.ac.uk/gt/2003/07/p001.xhtml Geometry and Topology '''7-1''' (2003) 1-31], {{arXiv\|math.QA/0204311}}.		{{note\|BLT}} D. Bar-Natan, T. Q. T. Le and D. P. Thurston, ''Two applications of elementary knot theory to Lie algebras and Vassiliev invariants'', [http://www.msp.warwick.ac.uk/gt/2003/07/p001.xhtml Geometry and Topology '''7-1''' (2003) 1-31], {{arXiv\|math.QA/0204311}}.

			{{note\|Da}} Z. Dancso, ''On the Kontsevich integral for knotted trivalent graphs'', {{arXiv\|0811.4615}}.

	{{note\|MO}} J. Murakami and T. Ohtsuki, ''Topological Quantum Field Theory for the Universal Quantum Invariant'', Communications in Mathematical Physics '''188''' (1997) 501-520.		{{note\|MO}} J. Murakami and T. Ohtsuki, ''Topological Quantum Field Theory for the Universal Quantum Invariant'', Communications in Mathematical Physics '''188''' (1997) 501-520.

Drorbn at 20:35, 27 February 2007

2007-02-27T20:35:28Z

← Older revision		Revision as of 16:35, 27 February 2007
Line 3:		Line 3:
	==The Fundamental Theorem of Finite Type Invariants==		==The Fundamental Theorem of Finite Type Invariants==

			{{06-1350/The Fundamental Theorem}}
	'''Almost Theorem.''' There exists a universal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}</math> from the TG-algebra of knotted trivalent graphs to the TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.

	'''Theorem.''' (Essentially due to Murakami and Ohtsuki, {{ref\|MO}}) There exists an R-normal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}^\nu</math> from the TG-algebra of knotted trivalent graphs to the <math>\nu</math>-twisted TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.

	The above theorem is simply the accurate formulation of the almost theorem above it. The "almost theorem" is just what you would have expected, with an additional uniqueness statement. The "theorem" just adds to it a few normalizations that actually make it right. The determination of these normalizations is quite a feat; even defining them takes a page or two. I'm not entirely sure why the Gods of mathematics couldn't have just allowed the "almost theorem" to be true and make our lives a bit simpler.

	Enough whining; we just need to define "R-normal" and <math>{\mathcal A}^\nu</math>.

	'''Definition.''' <math>Z</math> is called R-normal if <math>Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)</math> in <math>{\mathcal A}(\bigcirc)</math>, where <math>(\MobiusSymbol)</math> denotes the positively-twisted Möbius band and where <math>(\isolatedchord)</math> denotes the unique degree 1 chord diagram in <math>{\mathcal A}(\bigcirc)</math>.

	'''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>), but the unzip operations on <math>{\mathcal A}^\nu</math> get "renormalized":
	;The edge-unzip operations.
	:Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following subsection. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, the two operations are related by <math>u^\nu_e=\nu^{-1/2}_{e'}\nu^{-1/2}_{e''}u_e\nu^{1/2}_e</math>. Here "<math>\nu^{1/2}_e</math>" means "inject a copy of <math>\nu^{1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{-1/2}_{e'}\nu^{-1/2}_{e''}</math>" means "inject copies of <math>\nu^{-1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".

	===The Mysterious <math>\nu</math>===		===The Mysterious <math>\nu</math>===

Drorbn: /* Some values of the invariant */

2007-01-23T16:33:00Z

Some values of the invariant

← Older revision		Revision as of 12:33, 23 January 2007
Line 33:		Line 33:
	While these are not necessary for the statement of the theorem, it is worthwhile to note that the invariants of the unknot, the unknotted <math>\theta</math>-graph and the unknotted dumbbell are as follows:		While these are not necessary for the statement of the theorem, it is worthwhile to note that the invariants of the unknot, the unknotted <math>\theta</math>-graph and the unknotted dumbbell are as follows:

	[[Image:UnknotThetaDumbbell.png\|thumb\|center\|640px\|Some noteworthy invariants ~~(the factor in the middle of the dumbbell should be removed)~~]]		[[Image:UnknotThetaDumbbell.png\|thumb\|center\|640px\|Some noteworthy invariants]]

	===References===		===References===

Drorbn at 16:15, 23 January 2007

2007-01-23T16:15:31Z

← Older revision		Revision as of 12:15, 23 January 2007
Line 13:		Line 13:
	'''Definition.''' <math>Z</math> is called R-normal if <math>Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)</math> in <math>{\mathcal A}(\bigcirc)</math>, where <math>(\MobiusSymbol)</math> denotes the positively-twisted Möbius band and where <math>(\isolatedchord)</math> denotes the unique degree 1 chord diagram in <math>{\mathcal A}(\bigcirc)</math>.		'''Definition.''' <math>Z</math> is called R-normal if <math>Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)</math> in <math>{\mathcal A}(\bigcirc)</math>, where <math>(\MobiusSymbol)</math> denotes the positively-twisted Möbius band and where <math>(\isolatedchord)</math> denotes the unique degree 1 chord diagram in <math>{\mathcal A}(\bigcirc)</math>.

	'''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>), but ~~two~~ ~~of the~~ operations on <math>{\mathcal A}^\nu</math> get "renormalized":		'''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>), but the unzip operations on <math>{\mathcal A}^\nu</math> get "renormalized":
	;The edge-unzip operations.		;The edge-unzip operations.
	:Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following subsection. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, the two operations are related by <math>u^\nu_e=\nu^{-1/2}_{e'}\nu^{-1/2}_{e''}u_e\nu^{1/2}_e</math>. Here "<math>\nu^{1/2}_e</math>" means "inject a copy of <math>\nu^{1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{-1/2}_{e'}\nu^{-1/2}_{e''}</math>" means "inject copies of <math>\nu^{-1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".		:Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following subsection. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, the two operations are related by <math>u^\nu_e=\nu^{-1/2}_{e'}\nu^{-1/2}_{e''}u_e\nu^{1/2}_e</math>. Here "<math>\nu^{1/2}_e</math>" means "inject a copy of <math>\nu^{1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{-1/2}_{e'}\nu^{-1/2}_{e''}</math>" means "inject copies of <math>\nu^{-1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".
	;The graph-connect operations.
	:Following any "blank" graph-connect operation as in <math>{\mathcal A}</math>, inject a copy of <math>\nu^{-1/2}</math> on the "new" edge created for the connection.

	===The Mysterious <math>\nu</math>===		===The Mysterious <math>\nu</math>===
Line 35:		Line 33:
	While these are not necessary for the statement of the theorem, it is worthwhile to note that the invariants of the unknot, the unknotted <math>\theta</math>-graph and the unknotted dumbbell are as follows:		While these are not necessary for the statement of the theorem, it is worthwhile to note that the invariants of the unknot, the unknotted <math>\theta</math>-graph and the unknotted dumbbell are as follows:

	[[Image:UnknotThetaDumbbell.png\|thumb\|center\|640px\|Some noteworthy invariants]]		[[Image:UnknotThetaDumbbell.png\|thumb\|center\|640px\|Some noteworthy invariants (the factor in the middle of the dumbbell should be removed)]]

	===References===		===References===

Drorbn: /* Some values of the invariant */

2006-11-08T21:50:36Z

Some values of the invariant

← Older revision		Revision as of 17:50, 8 November 2006
Line 33:		Line 33:
	===Some values of the invariant===		===Some values of the invariant===

	While these are not necessary for the statement of the theorem, it is worthwhile to note that the invariants of the unknot, the unknotted \theta-graph and the unknotted dumbbell are as follows:		While these are not necessary for the statement of the theorem, it is worthwhile to note that the invariants of the unknot, the unknotted <math>\theta</math>-graph and the unknotted dumbbell are as follows:

	[[Image:UnknotThetaDumbbell.png\|thumb\|center\|640px\|Some noteworthy invariants]]		[[Image:UnknotThetaDumbbell.png\|thumb\|center\|640px\|Some noteworthy invariants]]

Drorbn at 21:07, 8 November 2006

2006-11-08T21:07:03Z

← Older revision		Revision as of 17:07, 8 November 2006
Line 1:		Line 1:
	{{06-1350/Navigation}}		{{06-1350/Navigation}}

	{{In Preparation}}

	==The Fundamental Theorem of Finite Type Invariants==		==The Fundamental Theorem of Finite Type Invariants==
Line 15:		Line 13:
	'''Definition.''' <math>Z</math> is called R-normal if <math>Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)</math> in <math>{\mathcal A}(\bigcirc)</math>, where <math>(\MobiusSymbol)</math> denotes the positively-twisted Möbius band and where <math>(\isolatedchord)</math> denotes the unique degree 1 chord diagram in <math>{\mathcal A}(\bigcirc)</math>.		'''Definition.''' <math>Z</math> is called R-normal if <math>Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)</math> in <math>{\mathcal A}(\bigcirc)</math>, where <math>(\MobiusSymbol)</math> denotes the positively-twisted Möbius band and where <math>(\isolatedchord)</math> denotes the unique degree 1 chord diagram in <math>{\mathcal A}(\bigcirc)</math>.

	'''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>), but the operations on <math>{\mathcal A}^\nu</math> get "renormalized":		'''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>), but two of the operations on <math>{\mathcal A}^\nu</math> get "renormalized":
	;The edge-unzip operations.		;The edge-unzip operations.
	:Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following ~~definition~~. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, the two operations are related by <math>u^\nu_e=\nu^{1/2}_{e'}\nu^{1/2}_{e''}u_e\nu^{-1/2}_e</math>. Here "<math>\nu^{-1/2}_e</math>" means "inject a copy of <math>\nu^{-1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{1/2}_{e'}\nu^{1/2}_{e''}</math>" means "inject copies of <math>\nu^{1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".		:Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following subsection. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, the two operations are related by <math>u^\nu_e=\nu^{-1/2}_{e'}\nu^{-1/2}_{e''}u_e\nu^{1/2}_e</math>. Here "<math>\nu^{1/2}_e</math>" means "inject a copy of <math>\nu^{1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{-1/2}_{e'}\nu^{-1/2}_{e''}</math>" means "inject copies of <math>\nu^{-1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".
	;The edge-delete operations.
	:To be written.
	;The graph-connect operations.		;The graph-connect operations.
			:Following any "blank" graph-connect operation as in <math>{\mathcal A}</math>, inject a copy of <math>\nu^{-1/2}</math> on the "new" edge created for the connection.
	:To be written.

			===The Mysterious <math>\nu</math>===

	It remains to define <math>\nu\in{\mathcal A}(\uparrow)</math>. Well, it is the element often called "the invariant of the unknot", for indeed, by a long chain of reasoning, it is the invariant of the unknot. It is also given by the following explicit formula of {{ref\|BGRT}} and {{ref\|BLT}}:		It remains to define <math>\nu\in{\mathcal A}(\uparrow)</math>. Well, it is the element often called "the invariant of the unknot", for indeed, by a long chain of reasoning, it is the invariant of the unknot. It is also given by the following explicit formula of {{ref\|BGRT}} and {{ref\|BLT}}:
Line 32:		Line 30:

	(so <math>b_2=1/48</math>, <math>b_4=-1/5760</math>, <math>b_6=1/362880</math>, etc.).		(so <math>b_2=1/48</math>, <math>b_4=-1/5760</math>, <math>b_6=1/362880</math>, etc.).

			===Some values of the invariant===

			While these are not necessary for the statement of the theorem, it is worthwhile to note that the invariants of the unknot, the unknotted \theta-graph and the unknotted dumbbell are as follows:

			[[Image:UnknotThetaDumbbell.png\|thumb\|center\|640px\|Some noteworthy invariants]]

	===References===		===References===

Drorbn at 00:57, 7 November 2006

2006-11-07T00:57:29Z

← Older revision		Revision as of 20:57, 6 November 2006
Line 2:		Line 2:

	{{In Preparation}}		{{In Preparation}}

	Today's handout was taken from [http://www.math.toronto.edu/~drorbn/Talks/HUJI-001116/index.html Talks: HUJI-001116 (Knotted Trivalent Graphs, Tetrahedra and Associators)].

	==The Fundamental Theorem of Finite Type Invariants==		==The Fundamental Theorem of Finite Type Invariants==

Drorbn at 00:53, 7 November 2006

2006-11-07T00:53:38Z

← Older revision		Revision as of 20:53, 6 November 2006
Line 17:		Line 17:
	'''Definition.''' <math>Z</math> is called R-normal if <math>Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)</math> in <math>{\mathcal A}(\bigcirc)</math>, where <math>(\MobiusSymbol)</math> denotes the positively-twisted Möbius band and where <math>(\isolatedchord)</math> denotes the unique degree 1 chord diagram in <math>{\mathcal A}(\bigcirc)</math>.		'''Definition.''' <math>Z</math> is called R-normal if <math>Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)</math> in <math>{\mathcal A}(\bigcirc)</math>, where <math>(\MobiusSymbol)</math> denotes the positively-twisted Möbius band and where <math>(\isolatedchord)</math> denotes the unique degree 1 chord diagram in <math>{\mathcal A}(\bigcirc)</math>.

	'''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>) ~~and~~ the ~~same~~ operations except the unzip operation. Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following definition. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, ~~the two operations are related by <math>u^\nu_e=\nu^{1/2}_{e'}\nu^{1/2}_{e''}u_e\nu^{-1/2}_e</math>. Here~~ "<math>\nu^{-1/2}_e</math>" means "inject a copy of <math>\nu^{-1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{1/2}_{e'}\nu^{1/2}_{e''}</math>" means "inject copies of <math>\nu^{1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".		'''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>), but the operations on <math>{\mathcal A}^\nu</math> get "renormalized":
			;The edge-unzip operations.
			:Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following definition. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, the two operations are related by <math>u^\nu_e=\nu^{1/2}_{e'}\nu^{1/2}_{e''}u_e\nu^{-1/2}_e</math>. Here "<math>\nu^{-1/2}_e</math>" means "inject a copy of <math>\nu^{-1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{1/2}_{e'}\nu^{1/2}_{e''}</math>" means "inject copies of <math>\nu^{1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".
			;The edge-delete operations.
			:To be written.
			;The graph-connect operations.
			:To be written.

	It remains to define <math>\nu\in{\mathcal A}(\uparrow)</math>. Well, it is the element often called "the invariant of the unknot", for indeed, by a long chain of reasoning, it is the invariant of the unknot. It is also given by the following explicit formula of {{ref\|BGRT}} and {{ref\|BLT}}:		It remains to define <math>\nu\in{\mathcal A}(\uparrow)</math>. Well, it is the element often called "the invariant of the unknot", for indeed, by a long chain of reasoning, it is the invariant of the unknot. It is also given by the following explicit formula of {{ref\|BGRT}} and {{ref\|BLT}}:

Drorbn at 22:15, 6 November 2006

2006-11-06T22:15:26Z

@@ Line 9: / Line 9: @@
 '''Almost Theorem.''' There exists a universal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}</math> from the TG-algebra of knotted trivalent graphs to the TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.
-'''Theorem.''' (Essentially due to {{ref|Murakami-Ohtsuki_97}}) There exists an R-normal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}^\nu</math> from the TG-algebra of knotted trivalent graphs to the <math>\nu</math>-twisted TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.
+'''Theorem.''' (Essentially due to Murakami and Ohtsuki, {{ref|MO}}) There exists an R-normal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}^\nu</math> from the TG-algebra of knotted trivalent graphs to the <math>\nu</math>-twisted TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.
 The above theorem is simply the accurate formulation of the almost theorem above it. The "almost theorem" is just what you would have expected, with an additional uniqueness statement. The "theorem" just adds to it a few normalizations that actually make it right. The determination of these normalizations is quite a feat; even defining them takes a page or two. I'm not entirely sure why the Gods of mathematics couldn't have just allowed the "almost theorem" to be true and make our lives a bit simpler.
@@ Line 19: / Line 19: @@
 '''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>) and the same operations except the unzip operation. Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following definition. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, the two operations are related by <math>u^\nu_e=\nu^{1/2}_{e'}\nu^{1/2}_{e''}u_e\nu^{-1/2}_e</math>. Here "<math>\nu^{-1/2}_e</math>" means "inject a copy of <math>\nu^{-1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{1/2}_{e'}\nu^{1/2}_{e''}</math>" means "inject copies of <math>\nu^{1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".

Drorbn: /* The Fundamental Theorem of Finite Type Invariants */

2006-11-06T21:52:13Z

The Fundamental Theorem of Finite Type Invariants

← Older revision		Revision as of 17:52, 6 November 2006
Line 9:		Line 9:
	'''Almost Theorem.''' There exists a universal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}</math> from the TG-algebra of knotted trivalent graphs to the TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.		'''Almost Theorem.''' There exists a universal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}</math> from the TG-algebra of knotted trivalent graphs to the TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.

	'''Theorem.''' (Essentially due to Murakami-~~Ohtsuki<ref>test</ref>~~) There exists an R-normal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}^\nu</math> from the TG-algebra of knotted trivalent graphs to the <math>\nu</math>-twisted TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.		'''Theorem.''' (Essentially due to {{ref\|Murakami-Ohtsuki_97}}) There exists an R-normal TG-morphism <math>Z=(Z_\Gamma):KTG\to{\mathcal A}^\nu</math> from the TG-algebra of knotted trivalent graphs to the <math>\nu</math>-twisted TG-algebra of Jacobi diagrams. Furthermore, any two such TG-morphisms are twist equivalent.

	The above theorem is simply the accurate formulation of the almost theorem above it. The "almost theorem" is just what you would have expected, with an additional uniqueness statement. The "theorem" just adds to it a few normalizations that actually make it right. The determination of these normalizations is quite a feat; even defining them takes a page or two. I'm not entirely sure why the Gods of mathematics couldn't have just allowed the "almost theorem" to be true and make our lives a bit simpler.		The above theorem is simply the accurate formulation of the almost theorem above it. The "almost theorem" is just what you would have expected, with an additional uniqueness statement. The "theorem" just adds to it a few normalizations that actually make it right. The determination of these normalizations is quite a feat; even defining them takes a page or two. I'm not entirely sure why the Gods of mathematics couldn't have just allowed the "almost theorem" to be true and make our lives a bit simpler.

			Enough whining; we just need to define "R-normal" and <math>{\mathcal A}^\nu</math>.
	<div class="references-small">

	<references/>
			'''Definition.''' <math>Z</math> is called R-normal if <math>Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)</math> in <math>{\mathcal A}(\bigcirc)</math>, where <math>(\MobiusSymbol)</math> denotes the positively-twisted Möbius band and where <math>(\isolatedchord)</math> denotes the unique degree 1 chord diagram in <math>{\mathcal A}(\bigcirc)</math>.
	</div>

			'''Definition.''' <math>{\mathcal A}^\nu</math> is almost the same as <math>{\mathcal A}</math>. It has the same spaces (i.e., for any <math>\Gamma</math>, <math>{\mathcal A}^\nu(\Gamma)={\mathcal A}(\Gamma)</math>) and the same operations except the unzip operation. Let <math>\nu</math> denote the specific element of <math>{\mathcal A}(\uparrow)</math> defined in the following definition. If <math>u_e</math> denotes the unzip operation of an edge <math>e</math> for the TG-algebra <math>{\mathcal A}</math> and <math>u^\nu_e</math> is the corresponding operation in <math>{\mathcal A}^\nu</math>, the two operations are related by <math>u^\nu_e=\nu^{1/2}_{e'}\nu^{1/2}_{e''}u_e\nu^{-1/2}_e</math>. Here "<math>\nu^{-1/2}_e</math>" means "inject a copy of <math>\nu^{-1/2}</math> on the edge <math>e</math> of <math>\Gamma</math>, and likewise, "<math>\nu^{1/2}_{e'}\nu^{1/2}_{e''}</math>" means "inject copies of <math>\nu^{1/2}</math> on the edges <math>e'</math> and <math>e''</math> of <math>u_e\Gamma</math> that are created by the unzip of <math>e</math>".

			{{note\|Murakami-Ohtsuki_97}} J. Murakami and T. Ohtsuki, ''Topological Quantum Field Theory for the Universal Quantum Invariant'', Communications in Mathematical Physics '''188''' (1997) 501-520.

← Older revision		Revision as of 08:43, 24 November 2009
Line 28:		Line 28:

	{{note\|BLT}} D. Bar-Natan, T. Q. T. Le and D. P. Thurston, ''Two applications of elementary knot theory to Lie algebras and Vassiliev invariants'', [http://www.msp.warwick.ac.uk/gt/2003/07/p001.xhtml Geometry and Topology '''7-1''' (2003) 1-31], {{arXiv\|math.QA/0204311}}.		{{note\|BLT}} D. Bar-Natan, T. Q. T. Le and D. P. Thurston, ''Two applications of elementary knot theory to Lie algebras and Vassiliev invariants'', [http://www.msp.warwick.ac.uk/gt/2003/07/p001.xhtml Geometry and Topology '''7-1''' (2003) 1-31], {{arXiv\|math.QA/0204311}}.

			{{note\|Da}} Z. Dancso, ''On the Kontsevich integral for knotted trivalent graphs'', {{arXiv\|0811.4615}}.

	{{note\|MO}} J. Murakami and T. Ohtsuki, ''Topological Quantum Field Theory for the Universal Quantum Invariant'', Communications in Mathematical Physics '''188''' (1997) 501-520.		{{note\|MO}} J. Murakami and T. Ohtsuki, ''Topological Quantum Field Theory for the Universal Quantum Invariant'', Communications in Mathematical Physics '''188''' (1997) 501-520.