2023/12/29 by He, Alexander, Morgan, James, Thompson, Em K. · 2 citations
#57N10 #57Q15 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2312.17556
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 Huszár 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.