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2006 The diameter of the set of boundary slopes of a knot
Ben Klaff, Peter B Shalen
Algebr. Geom. Topol. 6(3): 1095-1112 (2006). DOI: 10.2140/agt.2006.6.1095


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) dK2 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.


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Ben Klaff. Peter B Shalen. "The diameter of the set of boundary slopes of a knot." Algebr. Geom. Topol. 6 (3) 1095 - 1112, 2006.


Received: 12 November 2005; Accepted: 14 March 2006; Published: 2006
First available in Project Euclid: 20 December 2017

zbMATH: 1134.57001
MathSciNet: MR2253442
Digital Object Identifier: 10.2140/agt.2006.6.1095

Primary: 57M15 , 57M25
Secondary: 57M50

Keywords: $3$–manifold , cable knot , cyclic fundamental group , diameter , generalized iterated torus knot , knot exterior , strict boundary slope , strict essential surface

Rights: Copyright © 2006 Mathematical Sciences Publishers


Vol.6 • No. 3 • 2006
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