The Junkyard
The following papers were all submitted for publication at one time or
another. In each case they were rejected because they did not represent
a significant enough contribution to the field. I did not resubmit,
because in each case I either agreed with this assessment, or included
the results in a subsequent paper. However, modulo minor errors I think
they are all essentially correct, and am posting them here because
someone might find them interesting. Feel free to contact me about any
of these.
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)