## Mathematics 690, Fall 2014

### Knots and Three-Manifolds

Instructor: Lenny Ng

Tuesdays, Thursdays 1:25-2:40pm, Physics 227

Office hours: Mondays 3:00-4:00, Wednesdays 2:30-4:00,
or by appointment.

**Please note:** it looks like you need a permission number to enroll in
the course. This is no problem; please just send me an email and I'll
be happy to give you a permission number. Thanks!

**Alexander polynomial handout**

### Course information

Knots play a central role in the modern study of the topology of
three-dimensional manifolds. In this graduate topics class, we'll
see how knots can be used to construct and understand
three-manifolds, and we'll also study the rich algebraic theory of
knot invariants, including recent developments that have shaped the
field in the past decade.

The course should be accessible to anyone with a basic background
in algebraic topology (along the lines of Math 611). I'm hoping that
it will be interesting and useful to anyone of an algebraic or
topological bent.

Topics I plan to cover include:

- polynomial knot invariants (Alexander, Jones, HOMFLY), skein relations, Vassiliev (finite type) invariants
- braids and their relation to knots: Alexander and Markov theorems, Burau representation, Hecke algebra and the Jones polynomial
- constructing 3-manifolds via knots and Kirby calculus: Heegaard diagrams, Dehn surgery, Kirby moves, and examples
- the Temperley-Lieb algebra and Witten's quantum invariants of 3-manifolds
- Khovanov homology and categorification
- knot Heegaard Floer homology.

I will begin the class loosely following the book *Knots, Links, Braids, and 3-Manifolds* by Prasolov and Sossinsky, but will frequently diverge from it. I'll provide additional sources for further reading, as appropriate.

**Course syllabus** containing all of
the above information and more.