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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 , 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.

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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 coproduct and denotes the valued "tensor map" on . Can you relate this with the first problem of HW1?
Problem 2.
 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) 417471.