Dror Bar-Natan: Classes: 2003-04: | FEEDBACK |

**Agenda:** Use knot theory as an excuse to learning deep and
beautiful mathematics.

**Instructor:** Dror Bar-Natan, drorbn@math.toronto.edu,
Sidney Smith 5016G, 416-946-5438. Office hours: Thursdays 12:30-1:30.

**Classes:** Tuesdays 1-3 at Sidney Smith 5017A and Thursdays 2-3 at
Sidney Smith 2128.

# | Week of ... | |

1 | September 8 |
Handout:
About This Class
Handout: Some Non Obvious Examples Handout: Pathologies Class notes for Tuesday September 9, 2003 (non obvious examples, pathologies, Reidemeister's theorem, 3-colorings, the Kauffman bracket) Homework Assignment 1 Class notes for Thursday September 11, 2003 (R-II and R-III for the Kauffman bracket, R-I, the writhe, the Jones polynomial, a little programming) |

2 | September 15 |
Handouts:
The Rolfsen Knot Table,
Torus Knots,
The Knot 9-32
Class notes for Tuesday September 16, 2003 (The reverse and the mirror of a knot, sketch of the proof of Redemeister's theorem, linking numbers, connected sum, classification of surfaces and the genus of a knot) Homework Assignment 2 Class notes for Thursday September 18, 2003 (the genus of a knot, Seifert's algorithm, subadditivity) Emily Redelmeier's drawings of Seifert's Algorithm |

3 | September 22 |
Class notes for Tuesday September 23, 2003 (additivity of the genus and part of the proof of the uniqueness of decomposition into primes)
Homework Assignment 3 (includes our overall grading policy!) Class notes for Thursday September 25, 2003 (conclusion of the uniqueness proof for prime decomposition) We took our Class Photo this week! |

3 | September 29 |
Dror: This week's office hours are on Friday 9:30 - 10:30.
Handout: A Model by Tom Erez Class notes for Tuesday September 30, 2003 (an alternating link is split iff it looks split) Handout: A Topological Disk Homework Assignment 4 Class notes for Thursday October 2, 2003 (an alternating link is split iff it looks split) |

5 | October 6 |
Class notes for Tuesday October 7, 2003 (end of discussion of alternating links and beginning of finite type invariants)
Class notes for Thursday October 9, 2003 (The FI and 4T relations) |

6 | October 13 |
Monday is Thanksgiving.
Some Dimensions of A Class notes for Tuesday October 14, 2003 (computer study of A, some simple weight systems and the weight system of the Jones polynomial) Homework Assignment 5 Class notes for Thursday October 16, 2003 (A is associative commutative, coalgebras, A is coassociative and cocommutative) |

7 | October 20 |
Class notes for Tuesday October 21, 2003 (coalgebras and duals of algebras, Hopf algebras, A is Hopf, Milnor-Moore and primitives, some primitives of A, trivalent vertices, the product of FT invariants is FT)
Homework Assignment 6: Deframing Class notes for Thursday October 23, 2003 (the deframing projection, STU is the same as 4T) |

8 | October 27 |
Class notes for Tuesday October 28, 2003 (Lie groups and Lie algebras, representations and metrics)
Homework Assignment 7: so(N) Class notes for Thursday October 30, 2003 (structure constants and weight systems from structure constants, gl(N)) |

8 | November 3 |
Class notes for Tuesday November 4, 2003 (the four colour theorem)
Homework Assignment 8: Primitives; W(L) is well defined Class notes for Thursday November 6, 2003 (more on 4CT, well-defindedness of W(L), primitives, the three proofs of the fundamental theorem) |

10 | November 10 |
The Algebraic Approach
Knotted Trivalent Graphs Class notes for Tuesday November 11, 2003 (universal finite type invariants, the algebraic approach) Homework Assignment 9: Gaussian integration in R^{n}
Class notes for Thursday November 13, 2003 (a general introduction to Chern-Simons theory, Gaussian integration) |

11 | November 17 |
Dror was away on Tuesday, in Banff. Class was cancelled.
Class notes for Thursday November 20, 2003 (more on Gaussian integration, inverting the laplacian on R and integral kernels) |

12 | November 24 |
Class notes for Tuesday November 25, 2003 (integral kernels and Green functions for the Laplacian, an integral form for the linking number)
Class notes for Thursday November 27, 2003 (more on the integral formula for the linking number) |

13 | December 1 |
The Emergence of Feynman Diagrams
From Stonehenge to Witten Skipping all the Details Class notes for Tuesday December 2, 2003 (a very quick overview of Feynman diagrams and Chern-Simons theory) Class notes for Thursday December 4, 2003 (the IHX relation from a stonehenge perspective) |

Exam | December 8 | The Final Exam, posted December 9, 2003. |

**Further resources:**

- The Knot Atlas.
- Graduate Studies at the UofT Math Department. In particular, 2003-2004 Graduate Course Descriptions.

September 25, 2003