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

# Video AKT-090924-2

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Videography by Karene Chu

Notes on AKT-090924-2:    [edit, refresh]

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

# Week of... Videos, Notes, and Links
: 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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0:00:18  Deduce (from the two pictures given last time) that $\mathcal{A}$ is a commutative algebra.
0:02:44  Definition of (associative) algebra: vector space with multiplication and unit map s.t. associativity and unit property holds
0:09:02  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  Definition of co-algebra (co-commutative co-algebra with a co-unit) all commutative diagrams holds with the arrows reversed.
0:15:13  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  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  Claim: The co-multiplication is well-defined (check that it is constant for diagrams differing by a $4T$ relation).
0:36:01  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  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  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  Claim: $\mathcal{A}$ is a bi-algebra.