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Video AKT-090924-1

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Some dimensions of {\mathcal A}_n, {\mathcal A} is a commutative algebra, {\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow).

# 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.
Tricolourability
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] The handout "Some Dimensions of A" is from DBN: Classes: 2000-01: Knot Theory: ChordDiagrams.pdf.
0:00:44 [edit] Review of the 3 by 4 grid and explaining where we are (topology and combinatorics of usual knots)
0:02:41 [edit] Let V_n be the space of invariants of type n (for framed knots) Consider \overline{0} \rightarrow V_{n-1} \rightarrow V_n \rightarrow W_n \rightarrow \overline{0} where V_{n-1} \rightarrow V_n under the inclusion map and V_n \rightarrow W_n being the map assigning each invariant its weight system. This is an exact sequence.
0:05:49 [edit] Correction: let \mathcal{D}_n be the space freely generated by all chord diagrams with n chords (instead of being just the set of all such diagrams)

Restating of the fundamental theorem: W_n = {\mathcal{A}^\star_n} where \mathcal{A}_n = \mathcal{D}_n /{4T}

0:07:21 [edit] The line \mathcal{A}_n^\star = \mathcal{A}_n /{4T} is a misprint and should in fact be \mathcal{A}_n = \mathcal{D}_n /{4T}.
0:07:33 [edit] We can also express \mathcal{A}_n=\mathcal{D}^0_n/b(\mathcal{D}^1_n) where \mathcal{D}^0_n is the v.s. of chord diagrams with n chords, \mathcal{D}^1_n is the v.s. of chord diagrams with one T shape and (n-2) chords and b is the map that takes the T shape to the signed sum of its four resolutions as in the 4T relation.
0:10:30 [edit] Going over the handout (computation dimension of \mathcal{A}_n)

unique (up to multiplication by group elements) type 0 invariant: constant 1 map type 1: Writhe ...

0:21:04 [edit] Type n invariants for n=0,1,2:

  1. Type 0: Constant
  2. Type 1: Writhe=W
  3. Type 2: W^2, J_2 (The second coefficient in the expansion of J(e^x) where J is the Jones polynomial.)
0:24:07 [edit] Remark: We do not have a generating function for the dimension of \mathcal{A}_n.
0:25:49 [edit] \mathcal{A} = \hat{\bigoplus} \mathcal{A}_n is (in some sense) the double dual of the space of knots

Theorem: \mathcal{A} is a commutative co-commutative bi-algebra. (For now, we prove \mathcal{A} is a commutative algebra.)

0:30:11 [edit] Claim 1: There is a bijection between round knots (i.e. knots on a circle) and long knots (i.e. knots on a long line):

\mathcal{K}(\bigcirc)=\mathcal{K}(|)

Claim 2: \mathcal{K}(|) is an abelian monoid.

Remark: I think it's worth checking that the map from circle to line is independent of the choice of point to 'open up' and the path we 'pull out' the two ends after cutting. However it is indeed independent.

0:35:01 [edit] Analog for \mathcal{A}: \mathcal{A}(\bigcirc) is isomorphic to \mathcal{A}(|).

(Checking the map of 'breaking the circle' is independent of the choice of breaking point is interesting.)