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 ${\mathcal {A}}_{n}$ , ${\mathcal {A}}$ is a commutative algebra, ${\mathcal {A}}(\bigcirc )\equiv {\mathcal {A}}(\uparrow )$ . Class Photo 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 090929: The Milnor-Moore theorem, primitives, the map ${\mathcal {A}}^{r}\to {\mathcal {A}}$ . 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, $gl_{N}$ . 091008-1: More on $gl_{N}$ , Lie algebras and the four colour theorem. 091008-2: The "abstract tenssor" approach to weight systems, ${\mathcal {U}}({\mathfrak {g}})$ and PBW, the map ${\mathcal {T}}_{\mathfrak {g}}$ .
6	Oct 12	091013: Algebraic properties of ${\mathcal {U}}({\mathfrak {g}})$ vs. algebraic properties of ${\mathcal {A}}$ . 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 ${\mathcal {A}}^{TG}$ . 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.
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Solve the following problems and submit them in class by October 13, 2006:

Problem 1. 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. (See 090924-2, minute 36:01).

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

P=\sum _{n=0}^{\infty }{\frac {(-\Theta )^{n}}{n!}}\partial _{\theta }^{n}.

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

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