Thin Position and Planar Surfaces for Graphs in the 3-sphere

We show that given a trivalent graph in S, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful planar surface. Thin position for a knot is a powerful tool developed by Gabai [1]. Given a knot in thin position, there is a useful planar surface in the knot complement and this planar surface plays an important role in Gabai’s proof of property R [1] and Gordon-Luecke’s solution of the knot complement problem [2]. Scharlemann and Thompson later generalized thin position to graphs in S and used it in a new proof of Waldhausen’s theorem that any Heegaard splitting of S is standard, see [5] and [4, section 5]. In [6], Thompson proved that either thin position for a knot is also bridge position or the knot complement contains an essential meridional planar surface. Wu improved this result by showing that the thinest level surface is an essential planar surface [7]. In this paper, we generalize Thompson’s theorem to graphs in S and show that either thin position is also bridge position or there is an essential planar surface in the graph complement and all but at most one of the boundary components of this planar surface are meridians for the graph. A consequence of this theorem is that there is always a nice planar surface in any graph complement. The existence of such a nice planar surface is a key in the proof of a theorem in [3] which says that given a graph Γ in S, if one glues back a handlebody N(Γ) to S −N(Γ) via a sufficiently complicated map, then the resulting closed 3-manifold cannot be S. Definition 1. Let N be a compact orientable 3-manifold with boundary and let P be an orientable surface properly embedded in N . We say P is essential if either P is a compressing disk for N or P is incompressible and ∂-incompressible. We say P is strongly irreducible if P is separating, P has compressing disks on both sides, and each compressing disk on one side meets each compressing disk on the other side. P is ∂-strongly irreducible if (1) every compressing and ∂-compressing disk on one side meets every compressing and ∂-compressing disk on the other side, and (2) there is at least one compressing or ∂-compressing disk on each side. Let Γ be a graph in S, N(Γ) an open regular neighborhood of Γ, and P a planar surface properly embedded in S−N(Γ). We say P is meridional if every component of ∂P bounds a compressing disk for the handlebody N(Γ) and we say P is an Partially supported by an NSF grant.