||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 November 3, 2009:
Problem 1. Let and be finite dimensional metrized Lie algebras, let denote their direct sum with the obvious "orthogonal" bracket and metric, and let be the canonical isomorphism . Prove that
where is the co-product and denotes the -valued "tensor map" on . Can you relate this with the first problem of HW1?
- Find a concise algorithm to compute the weight system associated with the Lie algebra in its defining representation.
- Verify that your algorithm indeed satisfies the relation.
Problem 3. The Kauffman polynomial (see [Kauffman]) of a knot or link is where is the writhe of and where is the regular isotopy invariant defined by the skein relations
(here is a strand and is the same strand with a kink added) and
Failed to parse (unknown function\backoverslash): L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)
and by the initial condition . State and prove the relationship between and .
Mandatory but unenforced. Find yourself in the class photo and identify yourself as explained in the photo page.
[Kauffman] ^ L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 312 (1990) 417-471.