In recent years, symplectic geometry has emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to study the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe how one can use this approach to construct an invariant of knots called "knot contact homology". This invariant is still pretty mysterious 10 years on, but I'll outline some surprising relations to representations of the knot group and to mirror symmetry.