Dror Bar-Natan: Classes: 2000-01:

Knot Theory

Instructor: Dror Bar-Natan, drorbn@math.huji.ac.il, 02-658-4187.

Classes: Tuesdays 14:00-16:00 at Sprintzak 215.

Office hours: Sundays 14:00-15:00 in my office, Mathematics 309.

Agenda: To learn about finite type knot invariants, and especially about their multiply-proven but not-sufficiently-well understood fundamental theorem, whose different proofs relate to almost everything in mathematics.

Further resources:

Advertisement Poster (with some extra syllabus details).
Pathologies - strange things happen in the continuous category.
A very simple Mathematica program that can verify the invariance of the Kauffman bracket and compute it on small knots:
- Mathematica Notebook (requires Mathematica or MathReader).
- PDF (requires Acrobat Reader or GhostView.
A handout on "Some Dimensions of A":
- ChordDiagrams.pdf.
- The files needed to make ChordDiagrams.ps: ChordDiagrams.zip.
Class notes for April 17th, 2001. (Reminders, the algebra structure of $\mathcal A$ , algebras and co-algebras, the co-algebra structure of $\mathcal A$ )
Class notes for April 24th, 2001. (The co-algebra structure of $\mathcal A$ , tensor products of representations of algebras and the notion of "a bialgebra", $\mathcal A$ is a bialgebra, preview of the relation between $\mathcal A$ , diagrams and Lie algebras).
Class notes for May 1st, 2001. (Proof that STU is equivalent to 4T, Lie algebras, representations and weight systems).
Class notes for May 22nd, 2001. (Universal invariants, cabling, knotted trivalent graphs and their finite type invariants, the basic operations).
Class notes for May 29th, 2001. (Finite generation of knotted trivalent graphs, diagrammatic versions of knot-theoretic operations).
Class notes for June 5th, 2001. (More on the diagrammatic versions of knot-theoretic operations, gauge transformations on universal invariants).
Class notes for June 12th, 2001. (Universality, more on gauge transformations).
Class notes for June 19th, 2001. (Getting rid of orientations, more on gauge transformations, the most general general invariance argument, identifying $\mathcal A$ (graph) in simpler terms)
Class notes for June 26th, 2001. (The Pachner move, linearization, beginning of symmetrization).
If you're taking this class for credit, you must write a project.

Class Photo, April 17, 2001

Click on the image to get the full size version

Background image: A table of knots and links by Rob Scharein, taken from http://www.cs.ubc.ca/nest/imager/contributions/scharein/zoo/knotzoo.html.