Two Lectures On The Jones Polynomial And Khovanov Homology

Understand

In the first of these two lectures, I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q.

• The two main steps are to reinterpret three-dimensional Chern-Simons gauge theory in four dimensional terms and then to apply electric-magnetic duality.
• The variable q is associated to instanton number in the dual description in four dimensions.
• In the second lecture, I describe how Khovanov homology can emerge upon adding a fifth dimension.