06-1350/Class Notes for Tuesday November 7

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In Preparation

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 $Z=(Z_{\Gamma }):KTG\to {\mathcal {A}}$ 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 $Z=(Z_{\Gamma }):KTG\to {\mathcal {A}}^{\nu }$ from the TG-algebra of knotted trivalent graphs to the $\nu$-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 ${\mathcal {A}}^{\nu }$.

Definition. $Z$ is called R-normal if Failed to parse (unknown function "\MobiusSymbol"): {\displaystyle Z(\bigcirc)^{-1}Z(\MobiusSymbol)=\exp(\isolatedchord/4)} in ${\mathcal {A}}(\bigcirc )$, 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 ${\mathcal {A}}(\bigcirc )$.

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

[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.