Title: The Pachner graph and the simplification of 3-sphere triangulations
Speaker: Ben Burton (UQ)
Abstract: It is important to have fast and effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation exponentially more complex before we can make it smaller than its original size. Here we present experimental work suggesting that for 3-sphere triangulations this never happens: instead, we never need to add more than two tetrahedra, and we never need more than a handful of local modifications. If true in general, these extremely surprising results would have significant implications for decision algorithms and the study of triangulations in 3-manifold topology.

The algorithms behind these experiments are also interesting in their own right. Key techniques include the isomorph-free generation of all 3-manifold triangulations of a given size, polynomial-time signatures for triangulations that are unique up to isomorphism, and parallel algorithms for studying finite level sets in the infinite Pachner graph.