Published/Accepted Papers

Heegaard Splitting with Boundary and Almost Normal Surfaces , Topology and its Applications 116 (2001) 153-184. (or here, with 2004 Erratum)

This paper generalizes the definition of a Heegaard splitting to unify the concepts of thin position for 3-manifolds, thin position for knots, and normal and 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 algorithmic and finiteness results about Dehn fillings with small Heegaard genus.


Critical Heegaard Surfaces and Index 2 Minimal Surfaces , in Proceedings of the Conference on Heegaard splittings and Dehn surgeries of 3-manifolds, Kyoto (Japan), July 2001.

This paper contains the motivation for the study of critical surfaces in " Critical Heegaard Surfaces. " In that paper, 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.

Critical Heegaard Surfaces , Transactions of the American Mathematical Society 354 (2002), 4015-4042. .

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.


Thin Position for Tangles (joint with Saul Schleimer), Journal of Knot Theory and its Ramifications Vol. 12, No. 1 (2003) 117-122 .

If a tangle, K, in B3 has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.


A note on Kneser-Haken finiteness , Proceedings of the American Mathematical Society 132 (2004) 899-902.
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). 

Large Embedded balls and Heegaard Genus in Negative curvature (joint with Daryl Cooper and Matthew E. White), Algebraic & Geometric Topology 4 (2004) 31-47.
We show if $M$ is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus $g$ then $g \ge \frac{1}{2}\cosh (r)$ where $r$ denotes the radius of any isometrically embedded ball in $M$. Assuming an unpublished result of Pitts and Rubinstein improves this to $g \ge \frac{1}{2}\cosh (r) + \frac{1}{2}.$ We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperimetric surfaces in hyperbolic balls. (Update 5/20/03)


Distance and Bridge Position (joint with Saul Schleimer), Pacific Journal of Mathematics 219, No. 2 (2005) 221-235.

J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudo-Anosov maps have minimal bridge presentations which are thin.

Surface Bundles versus Heegaard Splittings (joint with Saul Schleimer), Communications in Analysis and Geometry 13, No. 5 (2005) 1-26.
This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre.  The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex.  We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard splitting.  As a consequence, if a splitting surface has small genus compared to the translation distance of the monodromy, the splitting is standard. (Updated 12/6/02)

Sweepouts of amalgamated 3-manifolds (joint with S. Schleimer and E. Sedgwick), Algebraic & Geometric Topology 6 (2006) 171-194.

We show that if two 3-manifolds with toroidal boundary are glued via a ``sufficiently complicated" map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, suppose $X \cup _F Y$ is a manifold obtained by gluing $X$ and $Y$, two connected small manifolds with incompressible boundary, along a closed surface $F$. Then the following inequality on genera is obtained:
g(X \cup _F Y)\ge \frac{1}{2}\left(g(X)+g(Y)-2g(F)\right).
Both results follow from a new technique to simplify the intersection between an incompressible surface and a strongly irreducible Heegaard splitting.


Non-isotopic Heegaard splittings of Seifert fibered spaces (joint with Ryan Derby-Talbot), Algebraic & Geometric Topology 6 (2006) 351-374.

We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3-manifolds. Using this invariant, we show that a closed, totally orientable Seifert fibered space $M$ has infinitely many isotopy classes of Heegaard splittings of the same genus if and only if $M$ has an irreducible, horizontal Heegaard splitting, has a base orbifold of positive genus, and is not a circle bundle. This characterizes precisely which Seifert fibered spaces satisfy the converse of Waldhausen's conjecture.