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

Video AKT-091020

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Notes on AKT-091020:    [edit, refresh]

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

# Week of... Videos, Notes, and Links
1 Sep 7 About This Class
dbnvp 090910-1: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.
dbnvp 090910-2: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.
2 Sep 14 dbnvp 090915: More on Jones, some pathologies and more on Reidemeister, our overall agenda.
dbnvp 090917-1: The definition of finite type, weight systems, Jones is a finite type series.
dbnvp 090917-2: The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones.
3 Sep 21 dbnvp 090922: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.
dbnvp 090924-1: Some dimensions of {\mathcal A}_n, {\mathcal A} is a commutative algebra, {\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow).
Class Photo
dbnvp 090924-2: {\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
dbnvp 090929: The Milnor-Moore theorem, primitives, the map {\mathcal A}^r\to{\mathcal A}.
dbnvp 091001-1: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
dbnvp 091001-2: The very basics on Lie algebras.
5 Oct 5 dbnvp 091006: Lie algebraic weight systems, gl_N.
dbnvp 091008-1: More on gl_N, Lie algebras and the four colour theorem.
dbnvp 091008-2: The "abstract tenssor" approach to weight systems, {\mathcal U}({\mathfrak g}) and PBW, the map {\mathcal T}_{\mathfrak g}.
6 Oct 12 dbnvp 091013: Algebraic properties of {\mathcal U}({\mathfrak g}) vs. algebraic properties of {\mathcal A}.
Thursday's class canceled.
7 Oct 19 dbnvp 091020: Universal finite type invariants, filtered and graded spaces, expansions.
Homework Assignment 2
The Stonehenge Story
dbnvp 091022-1: The Stonehenge Story to IHX and STU.
dbnvp 091022-2: The Stonhenge Story: anomalies, framings, relation with physics.
8 Oct 26 dbnvp 091027: Knotted trivalent graphs and their chord diagrams.
dbnvp 091029-1: Zsuzsi Dancso on the Kontsevich Integral (1).
dbnvp 091029-2: Zsuzsi Dancso on the Kontsevich Integral (2).
9 Nov 2 dbnvp 091103: The details of {\mathcal A}^{TG}.
dbnvp 091105-1: Three basic problems: genus, unknotting numbers, ribbon knots.
dbnvp 091105-2: The three basic problems and algebraic knot theory.
10 Nov 9 dbnvp 091110: Tangles and planar algebras, shielding and the generators of KTG.
Homework Assignment 3
No Thursday class.
11 Nov 16 Local Khovanov Homology
dbnvp 091119-1: Local Khovanov homology, I.
dbnvp 091119-2: Local Khovanov homology, II.
12 Nov 23 dbnvp 091124: Emulation of one structure inside another, deriving the pentagon.
dbnvp 091126-1: Peter Lee on braided monoidal categories, I.
dbnvp 091126-2: Peter Lee on braided monoidal categories, II.
13 Nov 30 dbnvp 091201: The relations in KTG.
dbnvp 091203-1: The Existence of the Exponential Function.
dbnvp 091203-2: The Final Exam, Dror's failures.
F Dec 7 The Final Exam on Thu Dec 10, 9-11, Bahen 6183.
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0:00:00 [edit] Dror's notes are at AKT-091020, Hour 16.
0:01:05 [edit] Warning: the class is going to be extremely boring
0:02:23 [edit] Reminder:

Fundamental theorem: Every weight system integrates.

0:05:39 [edit] 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 [edit] Claim: the fundamental theorem holds iff universal finite type exists.
0:10:00 [edit] Proof: 1) \ \exists UFTI Z \ \ \Rightarrow \ fundamental theorem

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

0:11:41 [edit] V is of type n and W is the weight system of V
0:15:25 [edit] Karnen's question: Does V also vanish on knots of type  < n? (No)
0:19:13 [edit] 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 [edit] Let Z(K)=\Sigma_n \Sigma_{i=1}^{\dim(\mathcal{A}_n)} D_{n,i}V_{n,i}(K)
0:29:55 [edit] F-vect: category of filtered vector spaces
0:32:25 [edit] g-vect: category of graded vector spaces
0:34:31 [edit] Define funtor Fil: g-vect \rightarrow F-vect

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

0:37:29 [edit] gr \circ fil \cong(naturally equivalent) Id,

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

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