Abstract: I will explain the combinatorics behind the cluster algebra structure on braid and Richardson varieties. This generalizes the previously known constructions for positroid varieties and double Bruhat cells. The cluster algebra is described in terms of a 3-dimensional generalization of Postnikov’s plabic graphs, and the underlying quiver is induced by the intersection form of the Goncharov–Kenyon conjugate surface associated to a 3D plabic graph. Joint work with T. Lam, M. Sherman-Bennett, and D. Speyer.