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Week of...

Videos, Notes, and Links

1

Sep 7

About This Class 0909101: 3colourings, Reidemeister's theorem, invariance, the Kauffman bracket. 0909102: 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. 0909171: The definition of finite type, weight systems, Jones is a finite type series. 0909172: The skein relation for Jones; HOMFLYPT 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. 0909241: Some dimensions of ${\mathcal {A}}_{n}$, ${\mathcal {A}}$ is a commutative algebra, ${\mathcal {A}}(\bigcirc )\equiv {\mathcal {A}}(\uparrow )$. Class Photo 0909242: ${\mathcal {A}}$ is a cocommutative algebra, the relation with products of invariants, ${\mathcal {A}}$ is a bialgebra.

4

Sep 28

Homework Assignment 1 Homework Assignment 1 Solutions 090929: The MilnorMoore theorem, primitives, the map ${\mathcal {A}}^{r}\to {\mathcal {A}}$. 0910011: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T. 0910012: The very basics on Lie algebras.

5

Oct 5

091006: Lie algebraic weight systems, $gl_{N}$. 0910081: More on $gl_{N}$, Lie algebras and the four colour theorem. 0910082: 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 0910221: The Stonehenge Story to IHX and STU. 0910222: The Stonhenge Story: anomalies, framings, relation with physics.

8

Oct 26

091027: Knotted trivalent graphs and their chord diagrams. 0910291: Zsuzsi Dancso on the Kontsevich Integral (1). 0910292: Zsuzsi Dancso on the Kontsevich Integral (2).

9

Nov 2

091103: The details of ${\mathcal {A}}^{TG}$. 0911051: Three basic problems: genus, unknotting numbers, ribbon knots. 0911052: 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 0911191: Local Khovanov homology, I. 0911192: Local Khovanov homology, II.

12

Nov 23

091124: Emulation of one structure inside another, deriving the pentagon. 0911261: Peter Lee on braided monoidal categories, I. 0911262: Peter Lee on braided monoidal categories, II.

13

Nov 30

091201: The relations in KTG. 0912031: The Existence of the Exponential Function. 0912032: The Final Exam, Dror's failures.

F

Dec 7

The Final Exam on Thu Dec 10, 911, Bahen 6183.

Register of Good Deeds / To Do List

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Solve the following problems and submit them in class by October 13, 2009:
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 $W_{f\cdot g}=m_{\mathbb {Q} }\circ (W_{f}\otimes W_{g})\circ \Box$ where $(W_{f}\otimes W_{g})\circ \Box :{\mathcal {A}}\rightarrow \mathbb {Q} \otimes \mathbb {Q}$ and $m_{\mathbb {Q} }$ is the multiplication of rationals. (See 0909242, minute 36:01).
Problem 2. Let $\Theta :{\mathcal {A}}\to {\mathcal {A}}$ be the multiplication operator by the 1chord 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 $[A,B]:=ABBA$ 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 coLeibnitz law: $\Box \circ \Theta =(\Theta \otimes 1+1\otimes \Theta )\circ \Box$ (why does this deserve the name "the coLeibnitz law"?).
 $P$ is a coalgebra morphism: $\eta \circ P=\eta$ (where $\eta$ is the counit 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 ${\mathcal {A}}^{r}={\mathcal {A}}/\langle \theta \rangle$).
 $P^{2}=P$.
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.