Mathematics 797KI - ST-Knot Invariants

We will discuss various aspects of knot theory, concentrating on polynomial invariants of knots. Such invariants associate a polynomial to an embedded knot which does not depend on the particular presentation of it. The Jones invariant of knots and the categorical version of it, known as the Khovanov invariant, will be the main topic of the course. Khovanov?s invariant is a very strong invariant of the knot: it is shown by Mrowka and Kronheimer that if the Khovanov invariant of a knot is 1, then the knot is the unknot. The ultimate goal of the class will be a proof of Ramussen?s theorem (Milnor?s conjecture) on the slice genus of the torus knots. The representation theory of quantum sl(2) and Hecke algebras is the main tool in defining the above invariant. We will discuss the related topics in representation theory and discuss how knot theory motivates further development of the representation theory.

PreReq: MATH 611 Prerequisites for this course are: Undergraduate group theory, Linear algebra, Vector calculus, and Math 611.