||Videos, Notes, and Links
||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.
|| 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.
|| 090922: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.|
090924-1: Some dimensions of , is a commutative algebra, .
090924-2: is a co-commutative algebra, the relation with products of invariants, is a bi-algebra.
||Homework Assignment 1|
Homework Assignment 1 Solutions
090929: The Milnor-Moore theorem, primitives, the map .
091001-1: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
091001-2: The very basics on Lie algebras.
|| 091006: Lie algebraic weight systems, .|
091008-1: More on , Lie algebras and the four colour theorem.
091008-2: The "abstract tenssor" approach to weight systems, and PBW, the map .
|| 091013: Algebraic properties of vs. algebraic properties of .|
Thursday's class canceled.
|| 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.
|| 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).
|| 091103: The details of .|
091105-1: Three basic problems: genus, unknotting numbers, ribbon knots.
091105-2: The three basic problems and algebraic knot theory.
|| 091110: Tangles and planar algebras, shielding and the generators of KTG.|
Homework Assignment 3
No Thursday class.
||Local Khovanov Homology|
091119-1: Local Khovanov homology, I.
091119-2: Local Khovanov homology, II.
|| 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.
|| 091201: The relations in KTG.|
091203-1: The Existence of the Exponential Function.
091203-2: The Final Exam, Dror's failures.
||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 and then (as what one would expect by looking at degrees of polynomials) and where and is the multiplication of rationals. (See 090924-2, minute 36:01).
Problem 2. Let be the multiplication operator by the 1-chord 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 co-Leibnitz law: (why does this deserve the name "the co-Leibnitz law"?).
- is a co-algebra morphism: (where is the co-unit 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.