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Knots with only two strict essential surfaces

Marc Culler and Peter B Shalen

Geometry & Topology Monographs 7 (2004) 335–430

DOI: 10.2140/gtm.2004.7.335

Abstract

We consider irreducible 3–manifolds M that arise as knot complements in closed 3–manifolds and that contain at most two connected strict essential surfaces. The results in the paper relate the boundary slopes of the two surfaces to their genera and numbers of boundary components. Explicit quantitative relationships, with interesting asymptotic properties, are obtained in the case that M is a knot complement in a closed manifold with cyclic fundamental group.

Keywords

knot complement, hyperbolic 3–manifold, boundary slope, strict essential surface, essential homotopy, cyclic fundamental group, character variety

Mathematical Subject Classification

Primary: 57M15

Secondary: 57M25, 57M50

Publication

Received: 9 April 2004
Revised: 8 December 2004
Accepted: 30 August 2004
Published: 11 December 2004

Authors
Marc Culler
Department of Mathematics, Statistics, and Computer Science (M/C 249)
University of Illinois at Chicago
851 S Morgan St
Chicago IL 60607-7045
USA
Peter B Shalen
Department of Mathematics, Statistics, and Computer Science (M/C 249)
University of Illinois at Chicago
851 S Morgan St
Chicago IL 60607-7045
USA