• Corpus ID: 10401375

3-manifold knot genus is NP-complet

@inproceedings{Agol20023manifoldKG,
  title={3-manifold knot genus is NP-complet},
  author={Ian Agol and Joel Hass and William P. Thurston},
  booktitle={Symposium on the Theory of Computing},
  year={2002},
  url={https://api.semanticscholar.org/CorpusID:10401375}
}
View Paper
25 Citations
Highly Influential Citations
1
Background Citations
14
Methods Citations
6

25 Citations

Linkless and flat embeddings in 3-space and the unknot problem

This is the first polynomial-time algorithm for constructing flat embeddings in 3-space and thereby settle a problem proposed by Lovasz [29], which is a fundamental problem in knot theory and low dimensional topology.

The complexity of the normal surface solution space

A comprehensive analysis of millions of triangulations is undertaken and finds that in general the number of vertex normal surfaces is remarkably small, with strong evidence that the authors' pathological triangulation may in fact be the worst case scenarios.

Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations

This paper shows how to preempt bad triangulations by detecting genus in partially-constructed vertex links, allowing us to prune the enumeration tree substantially, and describes new data sets that have been obtained using high-performance computing facilities.

Fundamental normal surfaces and the enumeration of Hilbert bases

The most successful of these algorithms combines a maximal admissible face decomposition with the primal Hilbert basis algorithm of Bruns, Ichim and Koch, and can solve 168-dimensional enumeration problems (based on 24-tetrahedron knot complements) in a matter of hours.

Closed, oriented, connected 3-manifolds are subtle equivalence classes of plane graphs

A blink is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, it is shown that the homeomorphisms classes of closed oriented 3-manifolds are in…

Uniformly polynomial-time classification of surface homeomorphisms

We describe an algorithm which, given two essential curves on a surface S, computes their distance in the curve graph of S, up to multiplicative and additive errors. As an application, we present…

Closed oriented 3-manifolds are subtle equivalence classes of plane graphs

A {\em blink} is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, I show that the homeomorphisms classes of closed oriented 3-manifolds are…

Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time

This work shows that the following two basic tasks of computational topology, namely performing a Dehn-twist of a curve along another curve, and computing the geometric intersection number of two curves can be solved in polynomial time even in the succinct normal coordinate representation.

Quadrilateral–Octagon Coordinates for Almost Normal Surfaces

Joint coordinates is introduced, a system with only 3n dimensions for octagonal almost normal surfaces that has appealing geometric properties and can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein, and Thompson, reducing experimental running times by factors of thousands.

Algorithms for Normal Curves and Surfaces

This paper presents efficient algorithms for counting the number of connected components of curves and surfaces, deciding whether two curves are isotopic, and computing the algebraic intersection numbers of two curves.