#### 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.

**Correction:**
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

Email: cantwelljc@slu.edu, lc@math.wustl.edu

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