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

# Video AKT-090924-1

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Notes on AKT-090924-1:    [edit, refresh]

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
: 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  The handout "Some Dimensions of A" is from DBN: Classes: 2000-01: Knot Theory: ChordDiagrams.pdf.
0:00:44  Review of the 3 by 4 grid and explaining where we are (topology and combinatorics of usual knots)
0:02:41  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  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  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  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  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  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  Remark: We do not have a generating function for the dimension of $\mathcal{A}_n$.
0:25:49  $\mathcal{A} = \hat{\bigoplus} \mathcal{A}_n$ 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  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  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.)