06-1350/Class Notes for Tuesday November 7

06-1350/Navigation Panel   # Week of... Links 1 Sep 11 About, Tue, Thu 2 Sep 18 Tue, Kurlin(P), Thu 3 Sep 25 Tue, Photo, Thu 4 Oct 2 HW1, Tue, Thu 5 Oct 9 Tue(P), Thu(P) 6 Oct 16 HW2, Tue(P), Thu(P) 7 Oct 23 Tue(P), Thu 8 Oct 30 HW3, Tue, Thu 9 Nov 6 Tue ($\nu$), Thu 10 Nov 13 Tue, Thu 11 Nov 20 HW4(P), Thu 12 Nov 27 Thu 13 Dec 4 Syzygies in Asymptote, Final Jan 8 Grades Note. (P) means "contains a problem that Dror cares about". Add your name / see who's in! On to 07-1352

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

in ${\mathcal A}(\bigcirc)$, where 
denotes the positively-twisted Möbius band and where 
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 $u^\nu_e$ is the corresponding operation in ${\mathcal A}^\nu$, the two operations are related by $u^\nu_e=\nu^{1/2}_{e'}\nu^{1/2}_{e''}u_e\nu^{-1/2}_e$. Here "$\nu^{-1/2}_e$" means "inject a copy of $\nu^{-1/2}$ on the edge $e$ of $\Gamma$, and likewise, "$\nu^{1/2}_{e'}\nu^{1/2}_{e''}$" 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.