© | Dror Bar-Natan: Classes: 2009-10: AKT:

# Video AKT-091020

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Videography by Karene Chu

Notes on AKT-091020:    [edit, refresh]

Universal finite type invariants, filtered and graded spaces, expansions.

# Week of... Videos, Notes, and Links
: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.
: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.
Tricolourability
2 Sep 14 : More on Jones, some pathologies and more on Reidemeister, our overall agenda.
: The definition of finite type, weight systems, Jones is a finite type series.
: The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones.
3 Sep 21 : FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.
: Some dimensions of ${\mathcal A}_n$, ${\mathcal A}$ is a commutative algebra, ${\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow)$.
Class Photo
: ${\mathcal A}$ is a co-commutative algebra, the relation with products of invariants, ${\mathcal A}$ is a bi-algebra.
4 Sep 28 Homework Assignment 1
Homework Assignment 1 Solutions
: The Milnor-Moore theorem, primitives, the map ${\mathcal A}^r\to{\mathcal A}$.
: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
: The very basics on Lie algebras.
5 Oct 5 : Lie algebraic weight systems, $gl_N$.
: More on $gl_N$, Lie algebras and the four colour theorem.
: The "abstract tenssor" approach to weight systems, ${\mathcal U}({\mathfrak g})$ and PBW, the map ${\mathcal T}_{\mathfrak g}$.
6 Oct 12 : Algebraic properties of ${\mathcal U}({\mathfrak g})$ vs. algebraic properties of ${\mathcal A}$.
Thursday's class canceled.
7 Oct 19 : Universal finite type invariants, filtered and graded spaces, expansions.
Homework Assignment 2
The Stonehenge Story
: The Stonehenge Story to IHX and STU.
: The Stonhenge Story: anomalies, framings, relation with physics.
8 Oct 26 : Knotted trivalent graphs and their chord diagrams.
: Zsuzsi Dancso on the Kontsevich Integral (1).
: Zsuzsi Dancso on the Kontsevich Integral (2).
9 Nov 2 : The details of ${\mathcal A}^{TG}$.
: Three basic problems: genus, unknotting numbers, ribbon knots.
: The three basic problems and algebraic knot theory.
10 Nov 9 : Tangles and planar algebras, shielding and the generators of KTG.
Homework Assignment 3
No Thursday class.
11 Nov 16 Local Khovanov Homology
: Local Khovanov homology, I.
: Local Khovanov homology, II.
12 Nov 23 : Emulation of one structure inside another, deriving the pentagon.
: Peter Lee on braided monoidal categories, I.
: Peter Lee on braided monoidal categories, II.
13 Nov 30 : The relations in KTG.
: The Existence of the Exponential Function.
: The Final Exam, Dror's failures.
F Dec 7 The Final Exam on Thu Dec 10, 9-11, Bahen 6183.
Register of Good Deeds / To Do List

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0:00:00  Dror's notes are at AKT-091020, Hour 16.
0:01:05  Warning: the class is going to be extremely boring
0:02:23  Reminder:

Fundamental theorem: Every weight system integrates.

0:05:39  Def: a universal finite type invariant is an invariant $Z: \mathcal{K} \rightarrow \mathcal{A}$ s.t. for $n$-singular $K$, $Z(K) = D_K +$ diagrams of higher degree. (Note that a universal finite type invariant is not an finite type invariant)
0:08:39  Claim: the fundamental theorem holds iff universal finite type exists.
0:10:00  Proof: 1) $\ \exists$ UFTI $Z \ \ \Rightarrow \$ fundamental theorem

Given $W \in \mathcal{A}_n^*$, set $V = W \circ Z$

0:11:41  $V$ is of type $n$ and $W$ is the weight system of $V$
0:15:25  Karnen's question: Does $V$ also vanish on knots of type $< n$? (No)
0:19:13  Fundamental theorem $\Rightarrow \ \exists$ UFTI $Z$

For all $n$, pick basis $W_{n, i}$ of $\mathcal{A}_n^*$, by fundamental theorem, there are type $n$ invariants $V_{n, i}$ where $V_{n, i}^{(n)}=W_{n,i}$

Let $D_{n,i}$ be the dual basis of $W_{n, i}$

0:22:01  Let $Z(K)=\Sigma_n \Sigma_{i=1}^{\dim(\mathcal{A}_n)} D_{n,i}V_{n,i}(K)$
0:29:55  F-vect: category of filtered vector spaces
0:32:25  g-vect: category of graded vector spaces
0:34:31  Define funtor Fil: g-vect $\rightarrow$ F-vect

and funtor gr: F-vect $\rightarrow$ g-vect

0:37:29  gr $\circ$ fil $\cong$(naturally equivalent) Id,

but fil $\circ$ gr is not naturally equivalent to identity.

0:38:54  Definition of expansion
0:42:10  Claim: Expansions always exist.