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<ref>test</ref>) 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.

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