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Video AKT-091001-1

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Notes on AKT-091001-1:    [edit, refresh]

Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.

# 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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0:00:20 [edit] Locating where we are (moving from combinatorics to low algebra of usual knots)
0:01:51 [edit] Reminder from last time:

Space of knots \mathcal{K}; space of finite type knot invariants \mathcal{V} ; weight systems of finite type invariants \mathcal{W} = \mathcal{A}^* where \mathcal{A} = \mathcal{D}/4T

We proved that \mathcal{A} is a graded commutative co-commutative bi-algebra.

0:03:45 [edit] Goal: Describe the relationship between \mathcal{A} and Lie algebras.
0:04:11 [edit] Introducing a new pictorial presentation \mathcal{A}^t of \mathcal{A} in order to introduce the relationship between \mathcal{A} and Lie algebras (t stands for trivalent vertex).
0:11:12 [edit] Degree of a trivalent diagram = total number of vertices / 2

\mathcal{A}^t = set generated by trivalent diagrams / (AS, STU)

definition of IHX relation

0:15:44 [edit] Claim: \mathcal{A}^t(O) = \mathcal{A}^t(|)

Pf: essentially same as \mathcal{A}(O) = \mathcal{A}(|) (close/open the loop, apply the relations to prove 'open' is well defined)

0:22:59 [edit] alternative view for the relations (in terms of connecting to hooks around a vertex).
0:23:38 [edit] Prop: \mathcal{D}^c \rightarrow \mathcal{D}^t under inclusion. The inclusion descends to an isomorphism from \mathcal{A} to \mathcal{A}^t (in particular the map is well defined under the equivalence relations modded out)
0:28:10 [edit] the descended map is onto
0:29:58 [edit] The descended map has a well-defined inverse.

(By induction on the number of internal vertices)

0:42:04 [edit] Claim: STU \Rightarrow IHX.
0:44:23 [edit] The ugly way of proving the claim
0:45:32 [edit] The pretty way (through free associative algebra and commutators)

- i.e. the STU relation translates to the definition of [ , ] and the IHX relation translates to the Jacobi identity.