Summary: THINNING GENUS TWO HEEGAARD SPINES IN S3
MARTIN SCHARLEMANN AND ABIGAIL THOMPSON
ABSTRACT. We study trivalent graphs in S3 whose closed complement
is a genus two handlebody. We show that such a graph, when put in thin
position, has a level edge connecting two vertices.
We briefly review the terminology of Heegaard splittings, referring the
reader to [Sc] for a more complete description. A Heegaard splitting of a
closed 3-manifold M is a division of M into two handlebodies by a con-
nected closed surface, called the Heegaard surface or the splitting surface.
A spine for a handlebody H is a graph interior´Hµ so that H is a regular
neighborhood of . A Heegaard spine in M is a graph M whose regu-
lar neighborhood ´µ has closed complement a handlebody. Equivalently,
´µ is a Heegaard surface for M. We say that is of genus g if ´µ is
a surface of genus g.
Any two spines of the same handlebody are equivalent under edge slides
(see [ST1]). It's a theorem of Waldhausen [Wa] (see also [ST2]) that any
Heegaard splitting of S3 can be isotoped to a standard one of the same genus.
An equivalent statement, then, is that any Heegaard spine for S3 can be made
planar by a series of edge slides.