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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 ..."
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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.
0Efficient Triangulations of 3Manifolds
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2002
"... 0–efficient triangulations of 3–manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3–manifold M can be modified to a 0–efficient triangulation or M can be shown to be one of the manifolds S3, RP3 or L(3, 1). Similarly, any triangulation of a c ..."
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Cited by 44 (9 self)
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0–efficient triangulations of 3–manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3–manifold M can be modified to a 0–efficient triangulation or M can be shown to be one of the manifolds S3, RP3 or L(3, 1). Similarly, any triangulation of a compact, orientable, irreducible, ∂–irreducible 3–manifold can be modified to a 0–efficient triangulation. The notion of a 0–efficient ideal triangulation is defined. It is shown if M is a compact, orientable, irreducible, ∂–irreducible 3–manifold having no essential annuli and distinct from the 3–cell, then ◦ M admits an ideal triangulation; furthermore, it is shown that any ideal triangulation of such a 3–manifold can be modified to a 0–efficient ideal triangulation. A 0–efficient triangulation of a closed manifold has only one vertex or the manifold is S3 and the triangulation has precisely two vertices. 0–efficient triangulations of 3–manifolds with boundary, and distinct from the 3–cell, have all their vertices in the boundary and then just one vertex in each boundary
Normal Surface Qtheory
, 1998
"... We describe an approach to normal surface theory for triangulated 3manifolds which uses only the quadrilateral disk types (Qdisks) to represent a nontrivial normal surface. Just as with regular normal surface theory, interesting surfaces are among those associated with the vertices of the projecti ..."
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Cited by 10 (0 self)
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We describe an approach to normal surface theory for triangulated 3manifolds which uses only the quadrilateral disk types (Qdisks) to represent a nontrivial normal surface. Just as with regular normal surface theory, interesting surfaces are among those associated with the vertices of the projective solution space of this new Qtheory.
Almost Normal Heegaard Splittings
, 2001
"... The study of threemanifolds via their Heegaard splittings was initiated by Poul Heegaard in 1898 in his thesis. Our approach to the subject is through almost normal surfaces, as introduced by Hyam Rubinstein [28] and distance, as introduced by John Hempel [12]. Among the results presented... ..."
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Cited by 9 (4 self)
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The study of threemanifolds via their Heegaard splittings was initiated by Poul Heegaard in 1898 in his thesis. Our approach to the subject is through almost normal surfaces, as introduced by Hyam Rubinstein [28] and distance, as introduced by John Hempel [12]. Among the results presented...
Knots with infinitely many incompressible Seifert surfaces
 Department of Mathematics 1 Shields Avenue University of California, Davis Davis, CA 95616 USA
"... Abstract. We show that a knot in S 3 with an infinite number of incompressible Seifert surfaces contains a closed incompressible surface in its complement. 1. ..."
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Cited by 7 (1 self)
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Abstract. We show that a knot in S 3 with an infinite number of incompressible Seifert surfaces contains a closed incompressible surface in its complement. 1.
FINDING PLANAR SURFACES IN KNOT AND LINKMANIFOLDS
, 2008
"... Abstract. It is shown that given any linkmanifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. The method uses normal surface theory but does not follow the classical approach. Here the proof uses a re ..."
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Abstract. It is shown that given any linkmanifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. The method uses normal surface theory but does not follow the classical approach. Here the proof uses a rewriting method for normal surfaces in a fixed triangulation and may not find the desired solution among the fundamental surfaces. Two major results are obtained under certain boundary conditions. Given a linkmanifold M, a component B of ∂M, and a slope γ on B, it is shown that there is an algorithm to decide if there is an embedded punctureddisk in M with boundary γ and punctures in ∂M \ B; if there is one, the algorithm will construct one. Again, while normal surfaces are used, we may not find a solution among the fundamental surfaces. In this case we use induction on the number of boundary components of the linkmanifold. It also is shown that given a linkmanifold M, a component B of ∂M, and a meridian slope µ on B, there is an algorithm to decide if there is an embedded punctureddisk with boundary a longitude on B and punctures