Essential tangle decomposition from thin position of a link

Abstract

In this paper, we develop the idea of Thompson which treats the relationship between bridge position, incompressible meridianal planar surfaces, and thin position. We show that for a link in thin position there exits a canonical depth 1 nested tangle decomposition with incompressible 2-spheres arising from the thin position (Proposition 3.7), and we show that there is a maximal essential tangle decomposition of the link that is closely related to the thin position (Theorem 4.3).

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

“…In this section we briefly summarize their results. Details on these results may be found in three of their joint papers [5,6,7]. The illustrations alone are each worth a thousand words.…”

“…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. In [5], D Heath and T Kobayashi also exhibit a knot containing a meridional incompressible surface that is not realized as a thin level in a thin presentation of the knot. This propounds the idea that a decomposing sphere for a connected sum need not be realized as a thin level in a thin presentation of a composite knot.…”

“…Proposition 7.2 (Heath-Kobayashi [5,Proposition 3.7]) If a link L has the property that thin position differs from bridge position, then there exists an ambient isotopy f s , such that L = f 1 (L) is h-equivalent to L and L has a tangle decomposition by a finite number of non-trivial, non-nested, flat face up, bowl like 2-spheres, each of which is incompressible in the link complement. In this decomposition we have a tangle "on top" (above P) with all of the incompressible 2-spheres below it connected by vertical strands.…”

A search method for thin positions of links

“…Moreover Ying-Qing Wu shows that a thinnest level sphere in a thin position of a link gives an essential meridional planar surface [11]. The purpose of this paper is, by using the authors' previous result [5], to give a search method for finding thin positions of a given link, up to the determination of the bridge positions of given signed graphs and meridional essential planar surfaces, which gives a natural generalization of the result of Rieck-Sedgwick.…”