AKT-09/HW1

DBN: Classes: 2009-10: AKT-09/Navigation # Week of... Videos, Notes, and Links 1 Sep 7 About This Class  090910-1: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.  090910-2: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial. Tricolourability 2 Sep 14  090915: More on Jones, some pathologies and more on Reidemeister, our overall agenda.  090917-1: The definition of finite type, weight systems, Jones is a finite type series.  090917-2: The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones. 3 Sep 21  090922: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.  090924-1: Some dimensions of [math]\displaystyle{ {\mathcal A}_n }[/math], [math]\displaystyle{ {\mathcal A} }[/math] is a commutative algebra, [math]\displaystyle{ {\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow) }[/math]. Class Photo  090924-2: [math]\displaystyle{ {\mathcal A} }[/math] is a co-commutative algebra, the relation with products of invariants, [math]\displaystyle{ {\mathcal A} }[/math] is a bi-algebra. 4 Sep 28 Homework Assignment 1 Homework Assignment 1 Solutions  090929: The Milnor-Moore theorem, primitives, the map [math]\displaystyle{ {\mathcal A}^r\to{\mathcal A} }[/math].  091001-1: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.  091001-2: The very basics on Lie algebras. 5 Oct 5  091006: Lie algebraic weight systems, [math]\displaystyle{ gl_N }[/math].  091008-1: More on [math]\displaystyle{ gl_N }[/math], Lie algebras and the four colour theorem.  091008-2: The "abstract tenssor" approach to weight systems, [math]\displaystyle{ {\mathcal U}({\mathfrak g}) }[/math] and PBW, the map [math]\displaystyle{ {\mathcal T}_{\mathfrak g} }[/math]. 6 Oct 12  091013: Algebraic properties of [math]\displaystyle{ {\mathcal U}({\mathfrak g}) }[/math] vs. algebraic properties of [math]\displaystyle{ {\mathcal A} }[/math]. Thursday's class canceled. 7 Oct 19  091020: Universal finite type invariants, filtered and graded spaces, expansions. Homework Assignment 2 The Stonehenge Story  091022-1: The Stonehenge Story to IHX and STU.  091022-2: The Stonhenge Story: anomalies, framings, relation with physics. 8 Oct 26  091027: Knotted trivalent graphs and their chord diagrams.  091029-1: Zsuzsi Dancso on the Kontsevich Integral (1).  091029-2: Zsuzsi Dancso on the Kontsevich Integral (2). 9 Nov 2  091103: The details of [math]\displaystyle{ {\mathcal A}^{TG} }[/math].  091105-1: Three basic problems: genus, unknotting numbers, ribbon knots.  091105-2: The three basic problems and algebraic knot theory. 10 Nov 9  091110: Tangles and planar algebras, shielding and the generators of KTG. Homework Assignment 3 No Thursday class. 11 Nov 16 Local Khovanov Homology  091119-1: Local Khovanov homology, I.  091119-2: Local Khovanov homology, II. 12 Nov 23  091124: Emulation of one structure inside another, deriving the pentagon.  091126-1: Peter Lee on braided monoidal categories, I.  091126-2: Peter Lee on braided monoidal categories, II. 13 Nov 30  091201: The relations in KTG.  091203-1: The Existence of the Exponential Function.  091203-2: 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 Add your name / see who's in!

Solve the following problems and submit them in class by October 13, 2009:

Problem 1. If [math]\displaystyle{ f \in {\mathcal V}_n }[/math] and [math]\displaystyle{ g \in {\mathcal V}_m }[/math] then [math]\displaystyle{ f \cdot g \in {\mathcal V}_{n+m} }[/math] (as what one would expect by looking at degrees of polynomials) and [math]\displaystyle{ W_{f \cdot g} = m_\mathbb{Q} \circ (W_f \otimes W_g) \circ \Delta }[/math] where [math]\displaystyle{ (W_f \otimes W_g) \circ \Delta: {\mathcal A} \rightarrow \mathbb{Q} \otimes \mathbb{Q} }[/math] and [math]\displaystyle{ m_\mathbb{Q} }[/math] is the multiplication of rationals. (See 090924-2, minute 36:01).

Problem 2. Let [math]\displaystyle{ \Theta:{\mathcal A}\to{\mathcal A} }[/math] be the multiplication operator by the 1-chord diagram [math]\displaystyle{ \theta }[/math], and let [math]\displaystyle{ \partial_\theta=\frac{d}{d\theta} }[/math] be the adjoint of multiplication by [math]\displaystyle{ W_\theta }[/math] on [math]\displaystyle{ {\mathcal A}^\star }[/math], where [math]\displaystyle{ W_\theta }[/math] is the obvious dual of [math]\displaystyle{ \theta }[/math] in [math]\displaystyle{ {\mathcal A}^\star }[/math]. Let [math]\displaystyle{ P:{\mathcal A}\to{\mathcal A} }[/math] be defined by

[math]\displaystyle{ P = \sum_{n=0}^\infty \frac{(-\Theta)^n}{n!}\partial_\theta^n. }[/math]

Verify the following assertions, but submit only your work on assertions 4,5,7,11:

1. [math]\displaystyle{ \left[\partial_\theta,\Theta\right]=1 }[/math], where [math]\displaystyle{ 1:{\mathcal A}\to{\mathcal A} }[/math] is the identity map and where [math]\displaystyle{ [A,B]:=AB-BA }[/math] for any two operators.
2. [math]\displaystyle{ P }[/math] is a degree [math]\displaystyle{ 0 }[/math] operator; that is, [math]\displaystyle{ \deg Pa=\deg a }[/math] for all [math]\displaystyle{ a\in{\mathcal A} }[/math].
3. [math]\displaystyle{ \partial_\theta }[/math] satisfies Leibnitz' law: [math]\displaystyle{ \partial_\theta(ab)=(\partial_\theta a)b+a(\partial_\theta b) }[/math] for any [math]\displaystyle{ a,b\in{\mathcal A} }[/math].
4. [math]\displaystyle{ P }[/math] is an algebra morphism: [math]\displaystyle{ P1=1 }[/math] and [math]\displaystyle{ P(ab)=(Pa)(Pb) }[/math].
5. [math]\displaystyle{ \Theta }[/math] satisfies the co-Leibnitz law: [math]\displaystyle{ \Box\circ\Theta=(\Theta\otimes 1+1\otimes\Theta)\circ\Box }[/math] (why does this deserve the name "the co-Leibnitz law"?).
6. [math]\displaystyle{ P }[/math] is a co-algebra morphism: [math]\displaystyle{ \eta\circ P=\eta }[/math] (where [math]\displaystyle{ \eta }[/math] is the co-unit of [math]\displaystyle{ {\mathcal A} }[/math]) and [math]\displaystyle{ \Box\circ P=(P\otimes P)\circ\Box }[/math].
7. [math]\displaystyle{ P\theta=0 }[/math] and hence [math]\displaystyle{ P\langle\theta\rangle=0 }[/math], where [math]\displaystyle{ \langle\theta\rangle }[/math] is the ideal generated by [math]\displaystyle{ \theta }[/math] in the algebra [math]\displaystyle{ {\mathcal A} }[/math].
8. If [math]\displaystyle{ Q:{\mathcal A}\to{\mathcal A} }[/math] is defined by [math]\displaystyle{ Q = \sum_{n=0}^\infty \frac{(-\Theta)^n}{(n+1)!}\partial_\theta^{(n+1)} }[/math] then [math]\displaystyle{ a=\theta Qa+Pa }[/math] for all [math]\displaystyle{ a\in{\mathcal A} }[/math].
9. [math]\displaystyle{ \ker P=\langle\theta\rangle }[/math].
10. [math]\displaystyle{ P }[/math] descends to a Hopf algebra morphism [math]\displaystyle{ {\mathcal A}^r\to{\mathcal A} }[/math], and if [math]\displaystyle{ \pi:{\mathcal A}\to{\mathcal A}^r }[/math] is the obvious projection, then [math]\displaystyle{ \pi\circ P }[/math] is the identity of [math]\displaystyle{ {\mathcal A}^r }[/math]. (Recall that [math]\displaystyle{ {\mathcal A}^r={\mathcal A}/\langle\theta\rangle }[/math]).
11. [math]\displaystyle{ P^2=P }[/math].

Idea for a good deed. Later than October 13, prepare a beautiful TeX writeup (including the motivation and all the details) of the solution of this assignment for publication on the web. For all I know this information in this form is not available elsewhere.

Mandatory but unenforced. Find yourself in the class photo and identify yourself as explained in the photo page.