  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. (Based on lectures
presented at the Clay Research Conference at Oxford University, and also at the
Galileo Galilei Institute in Florence, the University of Milan, Harvard
University, and the University of Pennsylvania.)
