Two Lectures on the Jones Polynomial and Khovanov Homology

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 <italic>q</italic>. 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 <italic>q</italic> 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.