#

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 , is a commutative algebra, . Class Photo 0909242: is a cocommutative algebra, the relation with products of invariants, is a bialgebra.

4

Sep 28

Homework Assignment 1 Homework Assignment 1 Solutions 090929: The MilnorMoore theorem, primitives, the map . 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, . 0910081: More on , Lie algebras and the four colour theorem. 0910082: The "abstract tenssor" approach to weight systems, and PBW, the map .

6

Oct 12

091013: Algebraic properties of vs. algebraic properties of . 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 . 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

Add your name / see who's in!



Solve the following problems and submit them in class by October 13, 2009:
Problem 1. If and then (as what one would expect by looking at degrees of polynomials) and where and is the multiplication of rationals. (See 0909242, minute 36:01).
Problem 2. Let be the multiplication operator by the 1chord diagram , and let be the adjoint of multiplication by on , where is the obvious dual of in . Let be defined by
Verify the following assertions, but submit only your work on assertions 4,5,7,11:
 , where is the identity map and where for any two operators.
 is a degree operator; that is, for all .
 satisfies Leibnitz' law: for any .
 is an algebra morphism: and .
 satisfies the coLeibnitz law: (why does this deserve the name "the coLeibnitz law"?).
 is a coalgebra morphism: (where is the counit of ) and .
 and hence , where is the ideal generated by in the algebra .
 If is defined by then for all .
 .
 descends to a Hopf algebra morphism , and if is the obvious projection, then is the identity of . (Recall that ).
 .
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.