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 $n$ 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

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

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

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

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 $g$ Heegaard splittings that must be stabilized roughly $g$ 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

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