583 research outputs found
Heegaard Splittings with Boundary and Almost Normal Surfaces
This paper generalizes the definition of a Heegaard splitting to unify
Scharlemann and Thomspon's concept of thin position for 3-manifolds, Gabai's
thin position for knots, and Rubinstein's almost normal surface theory. This
gives generalizations of theorems of Scharlemann, Thompson, Rubinstein, and
Stocking. In the final section, we use this machinery to produce an algorithm
to determine the bridge number of a knot, provided thin position for the knot
coincides with bridge position. We also present several finiteness and
algorithmic results about Dehn fillings with "small" Heegaard genus.Comment: 33 pages, 13 figures; New 13 page erratum giving a complete proof of
Theorem 6.3 has been adde
Normalizing Topologically Minimal Surfaces II: Disks
We show that a topologically minimal disk in a tetrahedron with index is
either a normal triangle, a normal quadrilateral, or a normal helicoid with
boundary length 4(n+1). This mirrors geometric results of Colding and
Minicozzi
Barriers to Topologically Minimal Surfaces
In earlier work we introduced topologically minimal surfaces as the analogue
of geometrically minimal surfaces. Here we strengthen the analogy by showing
that complicated amalgamations act as barriers to low genus, topologically
minimal surfaces.Comment: 13 pages, 1 figur
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.Comment: 22 pages, 6 figure
Normalizing Topologically Minimal Surfaces I: Global to Local Index
We show that in any triangulated 3-manifold, every index n topologically
minimal surface can be transformed to a surface which has local indices (as
computed in each tetrahedron) that sum to at most n. This generalizes classical
theorems of Kneser and Haken, and more recent theorems of Rubinstein and
Stocking, and is the first step in a program to show that every topologically
minimal surface has a normal form with respect to any triangulation.Comment: 36 pages, 17 figures. First in a series of three papers. arXiv admin
note: text overlap with arXiv:0901.020
Critical Heegaard Surfaces and Index 2 Minimal Surfaces
This paper contains the motivation for the study of critical surfaces. In
previous work the only justification given for the definition of this new class
of surfaces is the strength of the results. However, when viewed as the
topological analogue to index 2 minimal surfaces, critical surfaces become
quite natural.Comment: 9 pages; For proceedings of the conference "Heegaard Splittings and
Dehn surgeries of 3-manifolds", Kyoto University Research Institute for
Mathematical Sciences, June 200
Critical Heegaard Surfaces
In this paper we introduce "critical surfaces", which are described via a
1-complex whose definition is reminiscent of the curve complex. Our main result
is that if the minimal genus common stabilization of a pair of strongly
irreducible Heegaard splittings of a 3-manifold is not critical, then the
manifold contains an incompressible surface. Conversely, we also show that if a
non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then
it does not contain a critical Heegaard surface. In the final section we
discuss how this work leads to a natural metric on the space of strongly
irreducible Heegaard splittings, as well as many new and interesting open
questions.Comment: 28 pages, 8 figures, to appear in Transactions of the AM
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 earlier
work 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.Comment: 49 pages, 16 figure
A note on Kneser-Haken finiteness
Kneser-Haken Finiteness asserts that for each compact 3-manifold M there is
an integer c(M) such that any collection of k>c(M) closed, essential, 2-sided
surfaces in M must contain parallel elements. We show here that if M is closed
then twice the number of tetrahedra in a (pseudo)-triangulation of M suffices
for c(M).Comment: 4 pages, 1 figure; to appear in Proceedings of the AM
Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)
We construct families of manifolds that have pairs of genus Heegaard
splittings that must be stabilized roughly times to become equivalent. We
also show that when two unstabilized, boundary-unstabilized Heegaard splittings
are amalgamated by a "sufficiently complicated" map, the resulting splitting is
unstabilized. As a corollary, we produce a manifold that has distance one
Heegaard splittings of arbitrarily high genus. Finally, we show that in a
3-manifold formed by a sufficiently complicated gluing, a low genus,
unstabilized Heegaard splitting can be expressed in a unique way as an
amalgamation over the gluing surface.Comment: 10 pages, 2 figure
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