Mathematics – Geometric Topology
Scientific paper
2011-10-27
Mathematics
Geometric Topology
39 pages, 15 figures, 10 tables. This is the journal version of arXiv:1011.4169 and contains significant new material; see the
Scientific paper
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.
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