An algorithm to construct one-vertex triangulations of Heegaard splittings

Abstract

Following work of Jaco and Rubinstein (2006), which (non-constructively) proved that any 3-manifold admits a one-vertex layered triangulation, we present an algorithm, with implementation using Regina, that uses a combinatorial presentation of a Heegaard diagram to construct a generalised notion of a layered triangulation. We show that work of Husz\'ar and Spreer (2019) extends to our construction: given a genus-$g$ Heegaard splitting, our algorithm generates a triangulation with cutwidth bounded above by $4g-2$. Beyond Heegaard splittings, our construction actually extends to a natural generalisation of Dehn fillings: given a one-vertex triangulation with a genus-$g$ boundary component $B$, we can construct a one-vertex triangulation of any 3-manifold obtained by filling $B$ with a handlebody. To demonstrate the usefulness of our algorithm, we present findings from preliminary computer searches using this algorithm.