The diameter of the set of boundary slopes of a knot

Klaff, Ben; Shalen, Peter B

doi:10.2140/agt.2006.6.1095

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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

DOI: 10.2140/agt.2006.6.1095

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 $π_{1} (Σ)$ is cyclic. If $\infty$ is not a strict boundary slope, then the diameter of the set of strict boundary slopes of $K$ , denoted $d_{K}$ , is a numerical invariant of $K$ . We show that either (i) $d_{K} \geq 2$ 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

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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

Volume 6, issue 3 (2006)