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Video AKT-090924-2

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{\mathcal A} is a co-commutative algebra, the relation with products of invariants, {\mathcal A} is a bi-algebra.

# 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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0:00:18 [edit] Deduce (from the two pictures given last time) that \mathcal{A} is a commutative algebra.
0:02:44 [edit] Definition of (associative) algebra: vector space with multiplication and unit map s.t. associativity and unit property holds
0:09:02 [edit] An algebra is called a commutative algebra if m \circ \sigma = m where \sigma is the transpose map on B \otimes B and m is the multiplication.
0:10:17 [edit] Definition of co-algebra (co-commutative co-algebra with a co-unit) all commutative diagrams holds with the arrows reversed.
0:15:13 [edit] Example: Space of all polynomial with two variables with rational coefficient, with co-multiplication \Delta: \mathbb{Q}[x,y] \rightarrow \mathbb{Q}[x,y] \otimes \mathbb{Q}[x,y] be \Delta(f(x,y)) = f(x_1+x_2, y_1+y_2) where x_1 corresponds to an x (or y) before the tensor sign and x_2 being after the tensor sign, similarly for y. Co-unit being the map sending each polynomial to its constant term. Check that this is a co-algebra (together with the usual polynomial multiplication, this space is a bi-algebra)
0:24:38 [edit] Claim: Our space \mathcal{A} is a co-commutative co-algebra (under the co-multiplication and co-identity as defined, both of which are analogous to the polynomial example).
0:31:10 [edit] Claim: The co-multiplication is well-defined (check that it is constant for diagrams differing by a 4T relation).
0:36:01 [edit] Exercise. If f \in {\mathcal V}_n and g \in {\mathcal V}_m then f \cdot g \in {\mathcal V}_{n+m} (as what one would expect by looking at degrees of polynomials) and W_{f \cdot g} = m_\mathbb{Q} \circ (W_f \otimes W_g) \circ \Delta where (W_f \otimes W_g) \circ \Delta: {\mathcal A} \rightarrow \mathbb{Q} \otimes \mathbb{Q} and m_\mathbb{Q} is the multiplication of rationals.
0:39:41 [edit] Hints:

1. Leibniz rule for derivative of products:

\partial_x(f.g)=(\partial_x f) g +f (\partial_x g)

2. Iterated Leibniz rule (for expressions like (\partial_x \partial_y \partial_z) (f.g))

3. Leibniz rule in a discrete (combinatorial) setting:

Failed to parse (unknown function\doublepoint): (f.g)(\doublepoint)=f(\doublepoint)g(\overcrossing) + f(\undercrossing)g(\doublepoint)

0:46:01 [edit] Definition: A bialgebra B is a  B with an algebra and a co-algebra structure such that the co-algebra operations (comultiplication and counit) are morphisms of algebras.
0:50:15 [edit] Claim: \mathcal{A} is a bi-algebra.