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Foliation cones

John Cantwell and Lawrence Conlon

Geometry & Topology Monographs 2 (1999) 35–86

DOI: 10.2140/gtm.1999.2.35

Erratum: Geometry & Topology Monographs 2 (1999) 571–575

arXiv: math.GT/9809105


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.


foliation, depth one, foliated form, foliation cycle, endperiodic, pseudo-Anosov

Mathematical Subject Classification

Primary: 57R30

Secondary: 57M25, 58F15


Received: 18 September 1998
Revised: 13 April 1999
Published: 17 November 1999

John Cantwell
Department of Mathematics
St. Louis University
St. Louis MO 63103
Lawrence Conlon
Department of Mathematics
Washington University
St. Louis MO 63130