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
Generating All Vertices of a Polyhedron Is Hard
Discrete & Computational Geometry
The complexity of the normal surface solution space
Proceedings of the twenty-sixth annual symposium on Computational geometry
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
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The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding ''admissible faces''. Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.