Math 221b: Knots and Low-Dimensional TopologyThis spring, Math 221b will be a course in knot theory and some related low-dimensional topology. The course will have roughly two parts. The first will be centered around some topics in classical knot theory, with some basic 3-manifold topology thrown in to prove the big theorems. I plan to use as basic text Rolfsen's book, Knots and Links, with supplementary material from other sources. (The link is to the AMS bookstore, which has a sale price of $41 for members.) The second part of the course will be an introduction to a more advanced topic, to be agreed upon with the participants.
Time: Monday, Wednesday, and Thursday 11-12, Room 226 Goldsmith.
Part I Outline:
- Basic material: knot group, Seifert matrix, covering spaces, Alexander polynomial.
- Some 3-manifold topology: Dehn's lemma and unknotting criterion.
- Surgery on knots; three-dimensional and four-dimensional aspects. Kirby calculus.
- Knot cobordism: Signatures and other classical cobordism invariants. Twisted Alexander polynomials and torsion.
Part II possible topics:
- Basic 4-manifold topology and Kirby calculus for 4-manifolds.
- Introduction to Heegaard Floer homology and/or knot homology.
- Knot theory and contact geometry: Legendrian knots, Bennequin invariant, open books.
- More on knot cobordism: Casson-Gordon invariants and current work of Cochran-Orr-Teichner.