A search method for thin positions of links

Abstract

We give a method for searching for thin positions of a given link.

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

“…Theorem 1 (Heath-Kobayashi [7,Theorem 2]) There is a thin position of L that is realized through the process described above.…”

“…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].…”

“…The greatest challenge in this and many other investigations of thin position for knots is that thin levels need not be incompressible. 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].…”

Width is not additive

“…Definition 3.3 Following Heath and Kobayashi [8], a sphere P in S 3 is called bowllike with respect to a height function h if it can be decomposed into two disks, E 1 and E 2 , glued along their boundary such that E 1 is contained in a level surface for h and E 2 is a monotone disk disjoint from K .…”