Geometry & Topology Monographs 2 (1999), Proceedings of the Kirbyfest, paper no. 3, pages 35-86.

Foliation Cones

John Cantwell, Lawrence Conlon

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.

Note: there is a correction to this paper, which should be read alongside the paper.

Keywords. Foliation, depth one, foliated form, foliation cycle, endperiodic, pseudo-Anosov

AMS subject classification. Primary: 57R30. Secondary: 57M25, 58F15.

E-print: arXiv:math.GT/9809105

Submitted: 18 September 1998. (Revised: 13 April 1999.) Published: 17 November 1999.


Notes on file formats

John Cantwell, Lawrence Conlon

Department of Mathematics, St. Louis University
St. Louis, MO 63103

Department of Mathematics, Washington University
St. Louis, MO 63130


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