061350/Class Notes for Tuesday November 7

The Fundamental Theorem of Finite Type Invariants
Almost Theorem. There exists a universal TGmorphism from the TGalgebra of knotted trivalent graphs to the TGalgebra of Jacobi diagrams. Furthermore, any two such TGmorphisms are twist equivalent.
Theorem. (Essentially due to Murakami and Ohtsuki, [MO]; see also Dancso [Da]) There exists an Rnormal TGmorphism from the TGalgebra of knotted trivalent graphs to the twisted TGalgebra of Jacobi diagrams. Furthermore, any two such TGmorphisms 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 "Rnormal" and .
Definition. is called Rnormal 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 , where Failed to parse (unknown function "\MobiusSymbol"): {\displaystyle (\MobiusSymbol)} denotes the positivelytwisted 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 unzip operations on get "renormalized":
 The edgeunzip operations.
 Let denote the specific element of defined in the following subsection. If denotes the unzip operation of an edge for the TGalgebra 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 Mysterious
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 (unknown function "\twowheel"): {\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 's are the "modified Bernoulli numbers" defined by the power series expansion
(so , , , 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 graph and the unknotted dumbbell are as follows:
References
[BGRT] ^ D. BarNatan, S. Garoufalidis, L. Rozansky and D. P. Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot, Israel Journal of Mathematics 119 (2000) 217237, arXiv:qalg/9703025.
[BLT] ^ D. BarNatan, T. Q. T. Le and D. P. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geometry and Topology 71 (2003) 131, arXiv:math.QA/0204311.
[Da] ^ Z. Dancso, On the Kontsevich integral for knotted trivalent graphs, arXiv:0811.4615.
[MO] ^ J. Murakami and T. Ohtsuki, Topological Quantum Field Theory for the Universal Quantum Invariant, Communications in Mathematical Physics 188 (1997) 501520.