Geometry & Topology Monographs, Vol. 7 (2004),
Proceedings of the Casson Fest,
Paper no. 14, pages 335--430.

Knots with only two strict essential surfaces

Marc Culler, Peter B Shalen

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

AMS subject classification. Primary: 57M15. Secondary: 57M25, 57M50.

E-print: arXiv:math.GT/0404195

Submitted to GT on 9 April 2004. (Revised 8 December 2004.) Paper accepted 30 August 2004. Paper published 11 December 2004.

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Marc Culler, 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

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