Branched two-manifolds supporting all links

We resolve several conjectures of J. Birman and R. F. Williams concerning the knotting and linking of closed orbits of flows on 3-manifolds. Our methods center on the symbolic dynamics of semiflows on branched 2-manifolds, or templates. By proving the existence of “universal templates”, or embedded branched 2-manifolds supporting all finite links, we conclude that the set of closed orbits of any flow transverse to the fibration of the figure-eight knot complement in S3 contains representatives of every (tame) knot and link isotopy class.