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The diameter of the set of boundary slopes of a knot

Ben Klaff and Peter B Shalen

Algebraic & Geometric Topology 6 (2006) 1095–1112

arXiv: math.GT/0412147

Abstract

Let K be a tame knot with irreducible exterior M(K) in a closed, connected, orientable 3–manifold Σ such that π1(Σ) is cyclic. If is not a strict boundary slope, then the diameter of the set of strict boundary slopes of K, denoted dK, is a numerical invariant of K. We show that either (i) dK 2 or (ii) K is a generalized iterated torus knot. The proof combines results from Culler and Shalen [Comment. Math. Helv. 74 (1999) 530-547] with a result about the effect of cabling on boundary slopes.

Keywords
knot exterior, strict essential surface, strict boundary slope, diameter, $3$–manifold, cyclic fundamental group, cable knot, generalized iterated torus knot
Mathematical Subject Classification 2000
Primary: 57M15, 57M25
Secondary: 57M50
References
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Publication
Received: 12 November 2005
Accepted: 14 March 2006
Published: 29 August 2006
Authors
Ben Klaff
Department of Mathematics
University of Texas at Austin
1 University Station
Austin, TX 78741
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