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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.

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Algebraic Knot Theory
Department of Mathematics, University of Toronto, Fall 2009
Agenda: Understand "(u, v, and w knots) x (topology, combinatorics, low algebra, and high algebra)". Understand the promise and the difficulty of the notyetexistant "Algebraic Knot Theory".
Instructor: Dror BarNatan, drorbn@math.toronto.edu, Bahen 6178, 4169465438. Office hours: by appointment.
Classes: Tuesdays 1011 in 215 Huron room 1018 and Thursdays 911 in Bahen 6183.
Further Resources