Foliation Cones

Abstract

David Gabai showed that disk decomposable knot and link complements carry taut foliations of depth one. In an arbitrary sutured 3-manifold M, such foliations F, if they exist at all, are determined up to isotopy by an associated ray [F] issuing from the origin in H^1(M;R) and meeting points of the integer lattice H^1(M;Z). Here we show that there is a finite family of nonoverlapping, convex, polyhedral cones in H^1(M;R) such that the rays meeting integer lattice points in the interiors of these cones are exactly the rays [F]. In the irreducible case, each of these cones corresponds to a pseudo-Anosov flow and can be computed by a Markov matrix associated to the flow. Examples show that, in disk decomposable cases, these are effectively computable. Our result extends to depth one a well known theorem of Thurston for fibered 3-manifolds. The depth one theory applies to higher depth as well.