#### Volume 6, issue 3 (2006)

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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 $M\left(‘K\right)$ in a closed, connected, orientable 3–manifold $\Sigma$ such that ${\pi }_{1}\left(\Sigma \right)$ 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) ${d}_{K}\ge 2$ 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