The study of transport in Hamiltonian and related systems is greatly illuminated if one can construct a framework of "almost invariant" surfaces to organized the dynamics. This can be done in the case of area-preserving twist maps, using pieces of table and unstable manifold of periodic orbits or cantori, as shown by MacKay, Meiss and Percival. The resulting surfaces, however, are not necessarily the most appropriate ones, as they need not be graphs, nor is it clear that they can always be chosen mutually disjoint.