Dror Bar-Natan: Classes: 2003-04: Math 1350F - Knot Theory: (2) Next: Some Non Obvious Examples
Previous: FEEDBACK

About This Class

URL: http://www.math.toronto.edu/~drorbn/classes/0304/KnotTheory/.

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.

Textbook: W.B. Raymond Lickorish, An Introduction to Knot Theory, Springer Graduate Texts in Mathematics vol. 175, New York 1997. Also useful is G. Burde and H. Zieschang, Knots, de Gruyter Studies in Mathematics vol. 5, Berlin 2003. In addition we will follow several current research articles as will be discussed in class.

Lecture Notes: I'll be happy to scan the lecture notes of one of the students after every class and post them on the web. We need a volunteer with a good handwriting!

The Final Grade: Around the third week of classes (after I'll know you a little better) I will decide on the grading scheme for this class. While the specifics are still open, your final grade is sure to depend on your homework grade and on some final test/report/presentation.

Problem Sets: There will be about 10 problem sets. These will be handed out Thursday in lecture, and will be due in class on the following Thursday (7 days later). Late submissions will not be graded. I encourage you to discuss the homeworks with other students or even browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions.

Feedback: I'd be very happy to hear from you. There's a link to a feedback form at the top of this class' web site (and here). Anonymous messages are fine, provided they are written with good intent. Though remember that if I don't know who you are I may not be able to address your concern. You will each be required to use this feedback form at least once, on the third week of classes (see below).

Class Photo: To help me learn your names, I will take a class photo on Thursday of the third week of classes. I will post the picture on the class' web site and you will be required to use the feedback form to identify yourself in the picture.

Finally, here's our entry at the departmental Graduate Course Descriptions web page:

MAT 1350F
D. Bar-Natan

Synopsis. At first sight the natural numbers are dull. You can add and multiply and everybody knows how. But number theory is anything but dull - it is rich and deep and relates to almost everything in mathematics. Likewise knots seem dull until you start playing with them and realize that their theory too is rich and deep and relates to almost everything in mathematics - even to number theory! Our goal in this class will be to taste just a bit of that as follows:

Part I. Introduction to knots and links: Reidemeister moves, colorings, linking numbers, the fundamental group, Seifert surfaces, knot factorization, the Kauffman bracket and the Jones polynomial, briefly on the Alexander, HOMFLY and Kauffman polynomials, alternating links.

Part II. Introduction to finite type (Vassiliev) invariants: Basics, the classical polynomials, the bialgebra structure, framed and unframed knots, the relation with Lie algebras, universal finite type invariants, the Kontsevich integral, briefly on Chern-Simons theory, configuration spaces and parenthesized tangles.

For part I the primary reference will be Lickorish's book "An Introduction to Knot Theory". For part II the primary references will be research articles that will be distributed in class.