Compressing thin spheres in the complement of a link

Abstract

Let L be a link in S 3 that is in thin position but not in bridge position and let P be a thin level sphere. We generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks P can have, including identifying thin spheres with unique compressing disks. We also give conditions under which P must be incompressible on a particular side or be weakly incompressible. If P is strongly compressible we describe how a pair of compressing disks must lie relative to the link.

Waist and trunk of knots

“…Following [20, Lemma 2] or [19,Lemma 3.2], by "shrinking and straightening" α or β, we may assume that D has only one maximal point with respect to h. Since P is a thin level 2-sphere, we may assume without loss of generality that β has a minimal point. Let P be a "thinnest" level 2-sphere for β, that is, either a thin level 2-sphere between a maximal point p and a minimal point q of β or possibly parallel to P such that it has the minimal number of intersection with β in the region…”

Thin position for knots and 3–manifolds: a unified approach

“…Maggy Tomova continued this investigation in [33]. She proved more refined results about compressing disks for thin levels of links in thin position.…”

“…This fact is used to advantage by D Heath and T Kobayashi in [5] to produce a canonical tangle decomposition of a knot and in [7] to produce a method to search for thin presentations of a knot. M Tomova has made strides in understanding this phenomenon, see [33]. We discuss these theories below.…”

“…Her results rely on a number of concepts, observations and lemmas. We give a very brief overview, for details see [33]. In particular, note that the description below relies on many technical lemmas.…”

“…Figure 16: Two disks of the same height Theorem 4 (Tomova [33]) There exists a collection of disjoint irreducible compressing disks for P that contains one representative from each possible height.…”