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 π₁(Σ) is cyclic. If ∞ 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≥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

Primary: 57M15, 57M25

Secondary: 57M50

References

Publication

Received: 12 November 2005
Accepted: 14 March 2006
Published: 29 August 2006

Authors