P.L. homeomorphic manifolds are equivalent by elementary shellings
European Journal of Combinatorics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics)
Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics)
Enumeration of Non-Orientable 3-Manifolds Using Face-Pairing Graphs and Union-Find
Discrete & Computational Geometry
The complexity of the normal surface solution space
Proceedings of the twenty-sixth annual symposium on Computational geometry
Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Computing the Crosscap Number of a Knot Using Integer Programming and Normal Surfaces
ACM Transactions on Mathematical Software (TOMS)
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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 super-exponentially more complex before we can make it smaller than its original size. Here we present experimental work suggesting that for 3-sphere triangulations the reality is far different: 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 interesting in their own right. Key techniques include the isomorph-free generation of all 3-manifold triangulations of a given size, polynomial-time computable signatures that identify triangulations uniquely up to isomorphism, and parallel algorithms for studying finite level sets in the infinite Pachner graph.