Thin Position with respect to
a Heegaard surface
We present a definition of thin position for a knot in a
3-manifold, with respect to a Heegaard surface, motivated by
Scharlamenn and Thompson's definition of thin position for 3-manifolds,
and Gabai's definition of thin position for knots in S 3 . We then show that if a knot, K , in a 3-manifold, M , whose exterior contains no meridional, planar, essential
surfaces, is put in thin position then all thin levels of K are essential in the
complement of K
. A corollary of this is that if a small knot,
K , is put in thin position with respect to
a strongly irreducible Heegaard surface, F , then K is in bridge position with
respect to F ,
or it is a core of one of the handlebodies bounded by F . This generalizes a
Theorem of Thompson for knots in S
3 . (Updated 10/10/01)
Normalizing
Heegaard-Scharlemann-Thompson Splittings
We define a
Heegaard-Scharlemann-Thompson (HST) splitting of a 3-manifold $M$ to be
a sequence of pairwise-disjoint, embedded surfaces, $\{F_i\}$, such
that for each odd value of $i$,
$F_i$ is a Heegaard splitting of the submanifold of $M$ cobounded by
$F_{i-1}$ and $F_{i+1}$. Our main result is the following: Suppose $M$
($\neq B^3$ or $S^3$) is an irreducible submanifold of a triangulated
3-manifold, bounded by a normal or almost normal surface, and
containing
at most one maximal normal 2-sphere. If $\{F_i\}$ is a strongly
irreducible
HST splitting of $M$ then we may isotope it so that for each even value
of $i$ the surface $F_i$ is normal and for each odd value of $i$ the
surface
$F_i$ is almost normal.
We then show how various theorems of Rubinstein,
Thompson, Stocking and Schleimer follow from this result. We also show
how our results imply the following: (1) a manifold that contains a
non-separating surface contains an almost normal one, and (2) if a
manifold
contains a normal Heegaard surface then it contains two almost normal
ones that are topologically parallel to it. (Updated 5/2/03)
2-Normal
Surfaces
We define a 2-normal
surface to be one which intersects every 3-simplex of a triangulated
3-manifold in normal triangles and quadrilaterals, with
one or two exceptions. The possible exceptions are a pair of octagons,
a pair of unknotted tubes, an octagon and a tube, or a 12-gon.
In this paper we use the theory of critical surfaces
developed in "
critical Heegaard surfaces " to prove the existence of
topologically interesting 2-normal surfaces. Our main results are (1)
if a ball with normal boundary in a triangulated 3-manifold contains
two almost normal 2-spheres then it contains a 2-normal 2-sphere and
(2) in a non-Haken 3-manifold with a given triangulation the minimal
genus common stabilization of any pair of strongly irreducible Heegaard
splittings can be isotoped to an almost normal or a 2-normal surface.
(Updated 7/25/02)
Connected sums
of unstabilized Heegaard splittings are unstabilized,
submitted for publication April 2004
Let M_1 and M_2 be closed, orientable
3-manifolds. Let H_i denote a Heegaard surface in M_i. We prove that if
H_1 # H_2 comes from stabilizing a lower genus splitting of M_1 # M_2
then either H_1 or H_2 comes from stabilizing a lower genus splitting.
If H_i and G_i are non-isotopic Heegaard surfaces in M_i, and H_i is
unstabilized, then we show H_1 # H_2 is not isotopic to G_1 # G_2 in
M_1 # M_2. The former result answers a question of C. Gordon (Problem
3.91 from Kirby's problem list). (Updated 9/13/06)