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 H1 (M;R) and meeting points of the integer lattice
H1 (M;Z). Here we show that there is a finite family
of nonoverlapping, convex, polyhedral cones in H1 (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.
