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. 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, 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:

1. $\left[\partial _{\theta },\Theta \right]=1$, where $1:{\mathcal {A}}\to {\mathcal {A}}$ is the identity map and where $[A,B]:=AB-BA$ for any two operators.
2. $P$ is a degree $0$ operator; that is, $\deg Pa=\deg a$ for all $a\in {\mathcal {A}}$.
3. $\partial _{\theta }$ satisfies Leibnitz' law: $\partial _{\theta }(ab)=(\partial _{\theta }a)b+a(\partial _{\theta }b)$ for any $a,b\in {\mathcal {A}}$.
4. $P$ is an algebra morphism: $P1=1$ and $P(ab)=(Pa)(Pb)$.
5. $\Theta$ 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"?).
6. $P$ is a co-algebra morphism: $\eta \circ P=\eta$ (where $\eta$ is the co-unit of ${\mathcal {A}}$) and $\Box \circ P=(P\otimes P)\circ \Box$.
7. $P\theta =0$ and hence $P\langle \theta \rangle =0$, where $\langle \theta \rangle$ is the ideal generated by $\theta$ in the algebra ${\mathcal {A}}$.
8. 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 $a\in {\mathcal {A}}$.
9. $\ker P=\langle \theta \rangle$.
10. $P$ 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 ${\mathcal {A}}^{r}={\mathcal {A}}/\langle \theta \rangle$).
11. $P^{2}=P$.

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