Thin position of a pair (3-manifold, 1-submanifold)

Abstract

We introduce the notions of Heegaard splittings and thin multiple Heegaard splittings of 1-submanifolds in compact orientable 3-manifolds, which are generalizations of those of bridge decompositions and thin positions. We show that either a thin multiple Heegaard splitting of 1-submanifold T is also a Heegaard splitting with minimal complexity or the exterior of T contains an essential surface with meridional boundary other than the boundary parallel annulus.

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

“…However, their approach is different from ours and is described below. An similar approach to the one presented here can be found in Hayashi and Shimokawa [4].…”

Heegaard Splittings of Trivial Arcs in Compression Bodies

“…By Lemma 2.1, we have only to consider the case where genus (C) > 1 or T contains two or more components. At this time (C, T) is separable, that is, there is a T-compressing disc D of d+C in C. Then there is a T-compressing disc D f of d+C in C such that D' fl H is a single simple closed curve c by Theorem 1.3 in [HS1]. We compress H along the disc D f into H' and then cut C along D' to obtain a handlebody (or handlebodies) O'.…”

“…It is sufficient to prove that H satisfies the conclusion of the theorem as a Heegaard splitting of (C, T). Hence we can assume that d-C does not have a sphere component disjoint from T. When (C, T) is inseparable, by Lemma 5.1 in [HS1], either C is a ball and T is a single $_|_-parallel arc, or C is surface x I and T is a union of vertical arcs. In the former case, Theorem 1.1 follows from Lemma 2.1.…”

“…When T = 0, we take a vertical arc t of F x J, and isotope t so that H is a Heegaard splitting of (F x I, t) as in Lemma 2.1 in [HS1], and so that |U H t\ is minimal among all positions of Heegaard splittings. Note that H is not cancellable.…”

“…M = C\ U# C2, where C\ and C2 are compression bodies and H = d + Ci = d+CV Let T be a properly imbedded 1-manifold in M. The Heegaard splitting H of M is a Heegaard splitting of (M, T) if H is transverse to T and T* = TflCi is trivial in Ci for i = 1 and 2. We say also that T is in a position of a Heegaard splitting with respect to H in M. Every properly imbedded 1-manifold has a Heegaard splitting as shown in [Lemma 2.1,HS1].…”