Most of my research is focussed on different aspects of Heegaard surfaces and bridge surfaces in 3-manifolds and knot/link complements. The list below is divided into narrower categories within this area.
Any two Heegaard surfaces for the same 3-manifold are related by repeating a construction called stabilization, which increases the genus of the surface. The problem is to determine how many time the construction needs to be repeated, i.e. how high the genus of the common stabilization must be.
An upper bound on common stabilizations of Heegaard splittings.
This paper improves the upper bound on the genus of the smallest common stabilization of two Heegaard splittings. Given Heegaard splittings of genus p and q, we find a common stabilization of genus at most 3/2 p + 2q - 1. Previously, the best known upper bound was 5p + 8q - 9 (for non-Haken manifolds).
Bounding the stable genera of Heegaard splittings from below
Journal of Topology, 3(2010) no. 3, 668--690.
This paper shows that for any g, there is a 3-manifold with Heegaard splittings of genera g and g + 1 whose smallest common stabilization has genus g + g/2 - 1. The proof uses a combinatorial/topological argument inspired by a geometric proof by Hass, Thompson and Thurston, proving a similar result for transversely oriented Heegaard surfaces.
Flipping and stabilizing Heegaard splittings.
This paper contains a combinatorial/topological proof of a result proved by Hass, Thompson and Thurston using hyperbolic geometry and harmonic surfaces. Namely, it shows that for every g, there are genus g Heegaard surfaces such that the smallest stabilization that can be flipped (isotoped onto itself with the reverse orientation) has genus 2g. This paper was combined with the above paper when it was published in JoT.
Flipping bridge surfaces and bounds on the stable bridge number
(with Maggy Tomova), to appear in Algebraic & Geometric Topology.
There is a similar notion of stabilization for bridge surfaces of knots and links. Maggy Tomova and I adapted my examples of Heegaard surfaces requiring many stabilizations to bridge surfaces, with a slightly lower bound on the number of stabilizations needed.
Stable functions and common stabilizations of Heegaard splittings,
Transactions of the AMS, (2009)361: 3747-3765.
The Rubinstein-Scharlemann graphic is the image of the discriminant set of a certain stable function defined by a pair of Heegaard splittings. This paper describes an upper bound on the genus of the smallest common stabilization that can be read off from the graphic.
The mapping class group of a Heegaard surface is its group of symmetries, i.e. the group of automorphisms of the 3-manifold that take the surface onto itself. These groups can be very large or very small, depending on the Heegaard splitting. The classic result in this area is that the group for a genus two Heegaard splitting of the 3-sphere is finitely generated. However, this is still open for higher genus splittings of the 3-sphere.
Heegaard splittings and open books.
An open book decomposition for a 3-manifold is a link and a surface bundle structure on the complement of the link. Every open book decomposition determines a Heegaard surface consisting of the union of two pages of the open book. The mapping class group of this Heegaard surface has an infinite cyclic subgroup defined by "spinning" the surface around the monodromy of the open book. In this paper, I show that if the monodromy map of the open book decomposition is sufficiently complicated then the induced Heegaard surface is the unique minimal genus Heegaard surface for the 3-manifold and its mapping class group is exactly the cyclic group induced by the monodromy of the open book. This implies that the open book decomposition is the unique minimal open book for the ambient 3-manifold.
Mapping class groups of once-stabilized Heegaard splittings.
Scharlemann recently found a finite generating set for the mapping class group of a (standard) genus g+1 Heegaard splitting of a genus g handlebody. If you stabilize any Heegaard splitting once, then you can think of the new Heegaard surface as a genus g+1 Heegaard splitting of each of the handlebodies in the original splitting. Thus the group classified by Scharlemann appears as two different subgroups of the new mapping class group. We show that if the original Heegaard splitting has sufficiently high distance then the mapping class group of the stabilized splitting is generated by these two subgroups, and is thus finitely generated.
Mapping class groups of medium distance Heegaard splittings
to appear in Proceedings of the American Mathematical Society.
This paper shows that for a generic Heegaard splitting, the mapping class group is finite, and in fact isomorphic to the mapping class group of the ambient 3-manifold. Generic here means that the Hempel distance (a measure of the complexity of the gluing map that defines the Heegaard splitting) is at most three.
The space of Heegaard Splittings
(with D. McCullough), to appear in Crelle.
The mapping class group of a Heegaard surface contains a canonical subgroup called the isotopy subgroup. We show that space of all embedded surfaces isotopic to a given Heegaard surface is a classifying space for the isotopy subgroup of the mapping class group of the Heegaard surface.
Extending pseudo-Anosov maps to compression bodies
(with I. Biringer and Y. Minsky.)
The boundaries of essential disks in a handlebody define a subset of the projective measured lamination space of its boundary. If an automorphism of the handlebody defines a pseudo-Anosov map on the boundary then its stable lamination is in the closure of this set. We show a converse of this - That if a pseudo-Anosov map has a stable lamination in the closure of the disk set for a handlebody then some power of this map is the restriction to the boundary of an automorphism of a non-trivial compression body embedded in the handlebody. We also give examples that show that this is the strongest converse one could prove.
Automorphisms of the three-torus preserving a genus three Heegaard splitting
to appear in Pacific Journal of Mathematics.
We present a finite set of generators for the mapping class group of the (unique) genus three Heegaard splitting of the 3-torus. This was the first example of a genus three Heegaard splitting whose mapping class group was classified.
Mapping class groups of Heegaard splittings
(with Hyam Rubinstein.)
We present a number of families of examples of Heegaard splittings with infinite mapping class groups. We also extend the Nielsen realization theorem to Heegaard splittings, showing that every finite subgroup of the mapping class group of a Heegaard surface comes form the deck transformations of a regular covering.
Mapping class groups of Heegaard splittings of surface bundles.
Every surface bundle has a canonical Heegaard splitting formed by attaching tubes between two copies of the leaf of the bundle. If the displacement distance of the monodromy map is sufficiently high, it is known that this canonical Heegaard splitting is the unique minimal splitting for the manifold. We classify the mapping class groups of such Heegaard splittings.
The classic problem when studying 3-manifolds is to classify the Heegaard splittings for a given 3-manifold. This consists of two parts: determining a class of surfaces that represent all the isotopy classes of Heegaard surfaces, then determining when two representatives are in the same isotopy class.
One-sided and two-sided Heegaard splittings.
A one-sided surface whose complement in the ambient 3-manifold is an open handlebody determines a one-sided Heegaard splitting. We prove an analogue of Scharlemann-Tomova's Theorem for one-sided splittings, which has implications for the two-sided splitting that is naturally induced by a one-sided splitting. We also show that for a "high distance" one-sided splitting, the mapping class group of the induced two-sided Heegaard splitting is isomorphic to the fundamental group of the one-sided surface.
Calculating isotopy classes of Heegaard splittings.
We show that for hyperbolic 3-manifolds with boundary, there is an algorithm to classify the isotopy classes of Heegaard surfaces with genus below any given bound. The algorithm uses normal surface theory and the partially flat, angled ideal triangulations constructed by Lackenby. The proof that the algorithm works is based on an axiomatic approach to thin position
Heegaard splittings with large subsurface distances
(with Yair Minsky and Yoav Moriah) Algebraic & Geometric Topology, 10 (2010) 2251–-2275.
The Hempel distance of a Heegaard splitting measures how complicated the topology around the surface is. Scharlemann and Tomova have shown that if a Heegaard surface has large Hempel distance then every other Heegaard surface for that manifold with bounded genus is a stabilization of the high distance surface. Subsurface distance generalizes Hempel distance by measuring how complicated the Heegaard splitting is on a given subsurface of the Heegaard surface and a Heegaard surface with low Hempel distance may still have high distance on certain subsurfaces. We generalize Scharlemann-Tomova's result to show that if a Heegaard surface has large subsurface distance then every other Heegaard surface with low genus is parallel to the high distance subsurface of the original Heegaard surface.
Horizontal Heegaard splittings of Seifert fibered spaces
Pacific Journal of Mathematics, 245(2010), no. 1, 151--165.
Heegaard splittings of Seifert fibered spaces have been divided into two classes, called horizontal and vertical. The isotopy classes of vertical Heegaard splittings were almost entirely classified by Lustig and Moriah. Bachman and Derby-Talbot showed that a toroidal Seifert fibered space admitting a horizontal Heegaard surface admits infinitely many, though they were unable to give a complete classification. This paper completes their classification, showing that the isotopy class of a horizontal Heegaard surface is uniquely determined by its intersection slope with a complete collection of incompressible tori.
Tunnel number one, genus one fibered knots
(with Kenneth Baker and Elizabeth Kloginski) Communications in Analysis and Geometry, 17(2009) no. 1, 1--16.
A fibered knot is a knot whose complement is a surface bundle. Genus one fibered knots all have Heegaard genus two or three, i.e. tunnel number one or two. We generalize a proof of Cooper-Scharlemann to classify the genus one, once-punctured surface bundles with genus two Heegaard splittings, and use this to classify all the genus-one fibered knots in lens spaces that have tunnel number one.
Surface bundles with genus two Heegaard splittings
Journal of Topology, 1(2008), no. 3, 671--692.
Surface bundles over surfaces of arbitrarily high genus can admit genus two Heegaard splittings. We show that a surface bundle will admit a genus two Heegaard splitting if and only if its monodromy map has a relatively simple form consisting of a rotation composed with some collection of Dehn twists along a certain pair-of-pants decomposition for the fiber.
On the existence of high index topologically minimal surfaces
(with Dave Bachman) Math Research Letters, 17(2010), no. 3, 389--394.
Dave Bachman defined the topological index of a surface as a generalization of the notion of strongly irreducible. This idea has proved to be very useful, at least for low index. We construct Heegaard surfaces of arbitrarily high index. These surface are finite covers of a strongly irreducible (i.e. index-one) Heegaard surface and their index is equal to the degree of the cover.
On tunnel number one knots that are not (1,n)
(with Abigail Thompson) to appear in Journal of Knot Theory and its Ramifications.
The tunnel number of a knot in the 3-sphere and its bridge number with respect to different surfaces two different ways of measuring the complexity of a knot. The bridge number of a knot with respect to any unknotted surface gives an upper bound on its tunnel number. Conversely, we show that there are knots in the 3-sphere with tunnel number one whose bridge number is arbitrarily high with respect to an unknotted torus.
Bridge Number and the Curve Complex. This paper contains a technical result needed in the paper with Abby Thompson. A tunnel-number-one knot is defined by an unknotting tunnel, which determines a Heegaard splitting of the knot complement. I show that there are tunnel-number-one knots in the 3-sphere in which the Hempel distance of the induced Heegaard splitting is arbitrarily high.
The coarse geometry of the Kakimizu complex
(with Roberto Pelayo and Robin Wilson).
The Kakimizu complex for a knot is the simplicial complex whose vertices are minimal genus Seifert surfaces for the knot and whose simplices span sets of pairwise disjoint surfaces. Robin Wilson showed that this complex is finite for atoroidal knots. In this paper, we show that for toroidal knots, the complex is coarsely euclidean, or in other words quasi-isometric to a finitely generated abelian group.
Layered models for closed 3-manifolds.
Yair Minsky defined a combinatorial structure called a model in his proof of the ending lamination conjecture. This construction is oddly reminiscent of the layered triangulations defined recently by Jaco, Rubinstein and Tillman. This paper combines the two ideas to construct cell-like decompositions for closed manifolds using the pieces from Minsky's models for ends of hyperbolic manifolds.
Generalized handlebody sets and non-Haken 3-manifolds
(with Terk Patel) xPacific Journal of Mathematics, 235(2008), no. 1, 35--41.
Many essential loops in the boundary of a handlebody will bound essential, properly embedded surfaces in the handlebody. We show that these loops are dense in the curve complex, i.e. that every loop is distance at most two from such a loop. This is in contrast to the set of loops bounding disks in the handlebody, for with there are loops arbitrarily far away.
Heegaard Splittings and the Pants Complex
Algebraic and Geometric Topology, 6(2006), 853--874.
This paper explores a measure of complexity for Heegaard splittings coming from the complex of pair-of-pants decompositions for the Heegaard surface. Unlike the Hempel distance, which goes to zero when the Heegaard splitting is stabilized, the pants distance is relative stable. We use this fact to define an invariant measure of complexity for the 3-manifold by taking the pants distance after repeatedly stabilizing any given Heegaard splitting.
Locally Unknotted Spines of Heegaard Splittings
Algebraic and Geometric Topology, 5(2005), 1573--1584.
This paper proves a spine for a strongly irreducible Heegaard splitting will intersect any closed ball in the ambient manifold, under fairly general conditions, in an unknotted graph. The conclusions are similar to Scharlemann's work on how a strongly irreducible Heegaard surface may intersect a ball.