06-1350/Class Notes for Tuesday November 7: Difference between revisions
No edit summary |
|||
| 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 {{ref| |
'''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. |
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>". |
'''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>". |
||
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}}: |
|||
| ⚫ | |||
<center><math>\nu=\chi\left(\exp_\cup\left(\sum_{n=1}^\infty b_{2n}\omega_{2n}\right)\right).</math></center> |
|||
In the above formula \chi denotes the PBW "symmetrization" map, <math>\exp_\cup</math> means "exponentiation in the disjoint union sense", <math>\omega_{2n}</math> is the "wheel with <math>2n</math> legs" (so <math>\omega_2=\twowheel,</math> <math>\omega_4=\fourwheel,</math> etc.) and the <math>b_{2n}</math>'s are the "modified Bernoulli numbers" defined by the power series expansion |
|||
<center><math>\sum_{n=0}^\infty b_{2n}x^{2n} = \frac12\log\frac{\sinh x/2}{x/2}</math></center> |
|||
(so <math>b_2=1/48</math>, <math>b_4=-1/5760</math>, <math>b_6=1/362880</math>, etc.). |
|||
===References=== |
|||
{{note|BGRT}} D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston, ''Wheels, wheeling, and the Kontsevich integral of the unknot'', Israel Journal of Mathematics '''119''' (2000) 217-237, {{arXiv|q-alg/9703025}}. |
|||
{{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}}. |
|||
| ⚫ | |||
Revision as of 17:15, 6 November 2006
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 and Ohtsuki, [MO]) 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 . It has the same spaces (i.e., for any , ) and the same operations except the unzip operation. Let denote the specific element of defined in the following definition. If denotes the unzip operation of an edge for the TG-algebra and is the corresponding operation in , the two operations are related by . 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, "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'}\nu^{1/2}_{e''}} " 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 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} ".
It remains to define 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\in{\mathcal A}(\uparrow)} . 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 [BGRT] and [BLT]:
In the above formula \chi denotes the PBW "symmetrization" map, 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 \exp_\cup} means "exponentiation in the disjoint union sense", 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 \omega_{2n}} is the "wheel with 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 2n} legs" (so 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 \omega_2=\twowheel,} 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 \omega_4=\fourwheel,} etc.) and 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 b_{2n}} 's are the "modified Bernoulli numbers" defined by the power series expansion
(so 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 b_2=1/48} , 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 b_4=-1/5760} , 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 b_6=1/362880} , etc.).
References
[BGRT] ^ D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot, Israel Journal of Mathematics 119 (2000) 217-237, arXiv:q-alg/9703025.
[BLT] ^ D. Bar-Natan, T. Q. T. Le and D. P. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geometry and Topology 7-1 (2003) 1-31, arXiv:math.QA/0204311.
[MO] ^ J. Murakami and T. Ohtsuki, Topological Quantum Field Theory for the Universal Quantum Invariant, Communications in Mathematical Physics 188 (1997) 501-520.
