The diameter of the set of boundary slopes of a knot

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.