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.

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 Topologically Minimal Surfaces III: Bounded Combinatorics

We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a particular normal form with respect to any triangulation.

Heegaard structure respects complicated JSJ-decompositions (joint with R. Derby-Talbot and E. Sedgwick)

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $\phi:T_1 \to T_2$ be a homeomorphism, $M_\phi$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $\phi$, and $T$ the image of $T_1$ in $M_\phi$. We show that if $\phi$ is “sufficiently complicated” then any incompressible or strongly irreducible surface in $M_\phi$ can be isotoped to be disjoint from $T$. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.

Surfaces that become isotopic after Dehn filling (joint with R. Derby-Talbot and E. Sedgwick)

We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a constrained way.