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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 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.)