The diameter of the set of boundary slopes of a knot

Abstract

Let K be a tame knot with irreducible exterior M.K/ in a closed, connected, orientable 3-manifold † such that 1 . †/ is cyclic. If 1 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 [3] with a result about the effect of cabling on boundary slopes. 57M15, 57M25; 57M50

The slope conjecture for graph knots

“…The next result was essentially shown by Klaff and Shalen [11], but we give a modified proof here. See also [10,…”

“…Since the Jones slope pq is the boundary slope of the cabling annulus of C p,q (K), pq ∈ bs(C p,q (K)). The next result was essentially shown by Klaff and Shalen [12], but we give a modified proof here. See also [10…”

Knots with only two strict essential surfaces

“…We restate them here for completeness. Proposition 2.1 (Proposition 1.1 of [9]) Suppose that F is a bounded essential surface in an irreducible knot manifold M , and suppose that α is a path in F which has its endpoints in ∂F and is fixed-endpoint homotopic in M to a path in ∂M . Then α is fixed-endpoint homotopic in F to a path in ∂F .…”

“…Definition 6. 9 We will say that M is a two-surface knot manifold provided that M is an irreducible knot manifold and that M has at most two distinct isotopy classes of strict essential surfaces. We say that a two-surface knot manifold M is an exceptional two-surface knot manifold if M is Seifert fibered or if M is an exceptional graph manifold.…”

“…Then α is fixed-endpoint homotopic in F to a path in ∂F . Proposition 2.2 (Proposition 1.3 of [9]) Let M be a compact orientable irreducible 3-manifold containing an essential torus T and let M ′ be the manifold obtained by splitting M along T and let q : M ′ → M denote the quotient map. Let F be a connected properly embedded surface in M which is not isotopic to T .…”