On the complexity of computing the homology type of a triangulation
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics)
Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics)
Computing closed essential surfaces in knot complements
Proceedings of the twenty-ninth annual symposium on Computational geometry
Computing closed essential surfaces in knot complements
Proceedings of the twenty-ninth annual symposium on Computational geometry
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The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3-manifold topology: it enabled the first practical implementations of 3-sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in state-of-the-art algorithms for unknot recognition and testing for essential surfaces. Although the crushing operation will always reduce the size of a triangulation, it might alter its topology, and so it requires a careful theoretical analysis for the settings in which it is used. The aim of this paper is to make the crushing operation more accessible to practitioners, and easier to generalise to new settings. When the crushing operation was first introduced, the analysis was powerful but extremely complex. Here we give a new treatment that reduces the crushing process to a sequential combination of three "atomic" operations on a cell decomposition, all of which are simple to analyse. As an application, we generalise the crushing operation to the setting of non-orientable 3-manifolds, where we obtain a new practical and robust algorithm for non-orientable prime decomposition.