How good are convex hull algorithms?
Computational Geometry: Theory and Applications
The computational complexity of knot and link problems
Journal of the ACM (JACM)
A census of cusped hyperbolic 3-manifolds
Mathematics of Computation
3-manifold knot genus is NP-complete
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Enumeration of Non-Orientable 3-Manifolds Using Face-Pairing Graphs and Union-Find
Discrete & Computational Geometry
Generating All Vertices of a Polyhedron Is Hard
Discrete & Computational Geometry
Maximal admissible faces and asymptotic bounds for the normal surface solution space
Journal of Combinatorial Theory Series A
Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations
Proceedings of the 36th international symposium on Symbolic and algebraic computation
The least spanning area of a knot and the optimal bounding chain problem
Proceedings of the twenty-seventh annual symposium on Computational geometry
A tree traversal algorithm for decision problems in knot theory and 3-manifold topology
Proceedings of the twenty-seventh annual symposium on Computational geometry
The pachner graph and the simplification of 3-sphere triangulations
Proceedings of the twenty-seventh annual symposium on Computational geometry
Tracing compressed curves in triangulated surfaces
Proceedings of the twenty-eighth annual symposium on Computational geometry
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Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number of such surfaces grows in relation to the size of the underlying triangulation. Here we address this problem in both theory and practice. In theory, we tighten the exponential upper bound substantially; furthermore, we construct pathological triangulations that prove an exponential bound to be unavoidable. In practice, we undertake a comprehensive analysis of millions of triangulations and find that in general the number of vertex normal surfaces is remarkably small, with strong evidence that our pathological triangulations may in fact be the worst case scenarios. This analysis is the first of its kind, and the striking behaviour that we observe has important implications for the feasibility of topological algorithms in three dimensions.