06-1350/Class Notes for Tuesday November 7: Difference between revisions
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'''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. |
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'''Theorem.''' (Essentially due to Murakami- |
'''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. |
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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. |
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Enough whining; we just need to define "R-normal" and <math>{\mathcal A}^\nu</math>. |
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'''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>. |
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'''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>". |
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{{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. |
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Revision as of 16:52, 6 November 2006
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The information below is preliminary and cannot be trusted! (v)
Today's handout was taken from Talks: HUJI-001116 (Knotted Trivalent Graphs, Tetrahedra and Associators).
The Fundamental Theorem of Finite Type Invariants
Almost Theorem. There exists a universal TG-morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=(Z_\Gamma):KTG\to{\mathcal A}} 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_97]) There exists an R-normal TG-morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=(Z_\Gamma):KTG\to{\mathcal A}^\nu} from the TG-algebra of knotted trivalent graphs to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} -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 .
Definition. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} is called R-normal if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}(\bigcirc)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\MobiusSymbol)} denotes the positively-twisted Möbius band and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\isolatedchord)} denotes the unique degree 1 chord diagram in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}(\bigcirc)} .
Definition. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}^\nu} is almost the same as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}} . It has the same spaces (i.e., for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , ) and the same operations except the unzip operation. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} denote the specific element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}(\uparrow)} defined in the following definition. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_e} denotes the unzip operation of an edge Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} for the TG-algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u^\nu_e} is the corresponding operation in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}^\nu} , the two operations are related by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u^\nu_e=\nu^{1/2}_{e'}\nu^{1/2}_{e''}u_e\nu^{-1/2}_e} . Here "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu^{-1/2}_e} " means "inject a copy of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu^{-1/2}} on the edge Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , and likewise, "" means "inject copies of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu^{1/2}} on the edges Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e'} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e''} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_e\Gamma} that are created by the unzip of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} ".
[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.
