  It is important to have effective methods for simplifying 3-manifold
triangulations without losing any topological information. In theory this is
difficult: we might need to make a triangulation super-exponentially more
complex before we can make it smaller than its original size. Here we present
experimental work that suggests the reality is far different: for an exhaustive
census of 81,800,394 one-vertex triangulations that span 1,901 distinct closed
orientable 3-manifolds, we never need to add more than two extra tetrahedra, we
never need more than a handful of Pachner moves (or bistellar flips), and the
average number of Pachner moves decreases as the number of tetrahedra grows. If
they generalise, these extremely surprising results would have significant
implications for decision algorithms and the study of triangulations in
3-manifold topology.
  Key techniques include polynomial-time computable signatures that identify
triangulations up to isomorphism, the isomorph-free generation of non-minimal
triangulations, theoretical operations to reduce sequences of Pachner moves,
and parallel algorithms for studying finite level sets in the infinite Pachner
graph.
