06-1350/Class Notes for Tuesday November 7: Difference between revisions
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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)]. |
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==The Fundamental Theorem of Finite Type Invariants== |
==The Fundamental Theorem of Finite Type Invariants== |
Revision as of 20:57, 6 November 2006
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The information below is preliminary and cannot be trusted! (v)
The Fundamental Theorem of Finite Type Invariants
Almost Theorem. There exists a universal TG-morphism 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 from the TG-algebra of knotted trivalent graphs to the -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. is called R-normal if Failed to parse (unknown function "\MobiusSymbol"): {\displaystyle Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)} in , where Failed to parse (unknown function "\MobiusSymbol"): {\displaystyle (\MobiusSymbol)} denotes the positively-twisted Möbius band and where Failed to parse (unknown function "\isolatedchord"): {\displaystyle (\isolatedchord)} denotes the unique degree 1 chord diagram in .
Definition. is almost the same as . It has the same spaces (i.e., for any , ), but the operations on get "renormalized":
- The edge-unzip operations.
- 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 "" means "inject a copy of on the edge of , and likewise, "" means "inject copies of on the edges and of that are created by the unzip of ".
- The edge-delete operations.
- To be written.
- The graph-connect operations.
- To be written.
It remains to define . 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, means "exponentiation in the disjoint union sense", is the "wheel with 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 (unknown function "\fourwheel"): {\displaystyle \omega_4=\fourwheel,} etc.) and the 's are the "modified Bernoulli numbers" defined by the power series expansion
(so , , , 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.