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The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
Abstract

Cited by 55 (6 self)
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We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without selfintersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
Linking of Random pSpheres in Z^d
"... We consider the number of embeddings of k pspheres in Z , with p+2 d 2p+1, stratified by the pdimensional volumes of the spheres. We show for p + 2 ! d that the number of embeddings of a fixed link type of k equivolume pspheres grows with increasing pdimensional volume at an exponential ..."
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We consider the number of embeddings of k pspheres in Z , with p+2 d 2p+1, stratified by the pdimensional volumes of the spheres. We show for p + 2 ! d that the number of embeddings of a fixed link type of k equivolume pspheres grows with increasing pdimensional volume at an exponential rate which is independent of the link type. For d = p+2 we derive similar results both for links of unknotted pspheres and for "augmented" links where each component psphere can have any knot type, and similar but weaker results when the spheres are of specified knot type.