Math 195: Algebraic Methods in Knot Theory

Spring 2012

Instructor: David Rose

Credits:

Time: MWF, 11:55-12:45

Location: Physics 205

Synopsis

This course will be an introduction to knot theory, the Jones polynomial, Khovanov homology, and categorification. In the mid-80's, Vaughan Jones introduced a polynomial invariant of knots which is both powerful and readily computable. This discovery birthed a new area of mathematics, quantum topology, a field which lies at the intersection of many areas of modern mathematics. In the late 90's, Mikhail Khovanov showed that Jones's invariant was a shadow of a higher invariant now called Khovanov homology. This sparked the program of categorification - the study of finding `higher categorical' versions of known mathematical structures. We will explore these constructions and related topics, taking an elementary and hands-on approach. No background in knot theory or topology is assumed.

Topics to be covered include:

Knots and links, isotopy, Reidemeister moves
Braid group, Alexander and Markov theorems
Kauffman bracket, Jones polynomial, skein relations, sl(n) polynomials, HOMFLY-PT polynomial
Graded vector spaces, tensor product, chain complexes of vector spaces, homology
Khovanov homology for knots and links
Tangles, Categories, Temperley-Lieb algebra
Khovanov homology for tangles and cobordisms, categorification
Quantum groups, knot polynomials as quantum invariants (time permitting)

Homework

Homework will be assigned (bi-)weekly:
- Homework 1 due 2/1/12; Solutions.
- Homework 2 due 2/17/12; Solutions.
- Homework 3 due 3/2/12; Solutions.
- Homework 4 due 3/23/12; Solutions.
- Homework 5 due 4/4/12; Solutions.
- Homework 6 due 4/25/12.

Prerequisites

One course in abstract algebra is required (either Math 200 or Math 121).
Math 205 would add perspective to the course, but is not required.

Exams

There will be a cumulative final exam.

Textbooks

There will be no offical text for the course. The following books and articles may be useful as references, or for additional reading:

Knots knotes by Justin Roberts, found here
Knots, links, braids, and 3-manifolds: an introduction to the new invariants in low-dimensional topology by Prasolov and Sossinsky
The knot book: an elementary introduction to the mathematical theory of knots by Colin Conrad Adams
On Khovanov's categorification of the Jones polynomial by Dror Bar-Natan, found here

Items 2. and 3. have been placed on reserve at Perkins Library.