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58
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
The number of Reidemeister Moves Needed for Unknotting
, 2008
"... There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embe ..."
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Cited by 35 (11 self)
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There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1skeleton of the interior of a compact, orientable, triangulated PL 3manifold M. There is a positive constant c2 such that for each t ≥ 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c2t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M. We obtain explicit values for c1 and c2.
Scalar curvature and geometrization conjectures for 3manifolds
 in Comparison Geometry (Berkeley 1993–94), MSRI Publications
, 1997
"... Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization ..."
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Cited by 31 (9 self)
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Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof.
Towards the Poincaré Conjecture and the Classification of 3Manifolds
, 2003
"... The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying a ..."
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Cited by 27 (0 self)
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The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3dimensional manifolds. The final paragraph provides a brief description of the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work will be provided in a subsequent note by Michael Anderson.
On the coarse classification of tight contact structures
 the proceedings of the 2002 Georgia International Topology Conference
"... Abstract. We present a sketch of the proof of the following theorems: (1) Every 3manifold has only finitely many homotopy classes of 2plane fields which carry tight contact structures. (2) Every closed atoroidal 3manifold carries finitely many isotopy classes of tight contact structures. In this ..."
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Cited by 19 (2 self)
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Abstract. We present a sketch of the proof of the following theorems: (1) Every 3manifold has only finitely many homotopy classes of 2plane fields which carry tight contact structures. (2) Every closed atoroidal 3manifold carries finitely many isotopy classes of tight contact structures. In this article we explain how to normalize tight contact structures with respect to a fixed triangulation. Using this technique, we obtain the following results: Theorem 0.1. Let M be a closed, oriented 3manifold. There are finitely many homotopy classes of 2plane fields which carry tight contact structures. Theorem 0.2. Every closed, oriented, atoroidal 3manifold carries a finite number of tight contact structures up to isotopy. P. Kronheimer and T. Mrowka [KM] had previously shown Theorem 0.1 for (weakly) symplectically (semi)fillable contact structures. Our theorem is a genuine improvement of the KronheimerMrowka theorem because there exist tight structures which are not fillable [EH]. Now, since every Reebless foliation is a limit of tight contact structures [Co4, ET], we obtain a new proof of a recent result of D. Gabai [Ga].
Solvable Fundamental Groups of compact 3manifolds
 Trans. Amer. Math. Soc
, 1972
"... Abstract. A classification is given for groups which can occur as the fundamental group of some compact 3manifold. In most cases we are able to determine the topological structure of a compact 3manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups ..."
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Cited by 12 (0 self)
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Abstract. A classification is given for groups which can occur as the fundamental group of some compact 3manifold. In most cases we are able to determine the topological structure of a compact 3manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed 3manifolds to the more general class of compact 3manifolds. In the final section it is shown that each nonfinitely generated abelian group which occurs as a subgroup of the fundamental group of a 3manifold is a subgroup of the additive group of rationals. (1) Introduction. This paper is primarily concerned with the classification of those solvable groups which can occur as the fundamental group of a compact 3manifold. We also consider the problem of determining the structure of a
Scalar Curvature And The Existence Of Geometric Structures On 3Manifolds, I.
, 1999
"... This paper analyses the convergence and degeneration of sequences of metrics on a 3manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar curvaturetype functional and their degeneration is related to the sphe ..."
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Cited by 10 (5 self)
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This paper analyses the convergence and degeneration of sequences of metrics on a 3manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar curvaturetype functional and their degeneration is related to the sphere and torus decompositions of the 3manifold under certain conditions.
Heegaard surfaces and measured laminations, I: the Waldhausen conjecture
 Invent. Math
"... Abstract. We give a proof of the socalled generalized Waldhausen conjecture, which says that an orientable irreducible atoroidal 3–manifold has only finitely many Heegaard splittings in each genus, up to isotopy. Jaco and Rubinstein have announced a proof of this conjecture using different methods. ..."
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Cited by 10 (3 self)
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Abstract. We give a proof of the socalled generalized Waldhausen conjecture, which says that an orientable irreducible atoroidal 3–manifold has only finitely many Heegaard splittings in each genus, up to isotopy. Jaco and Rubinstein have announced a proof of this conjecture using different methods. Contents
The size of triangulations supporting a given link
, 2000
"... Abstract. Let T be a triangulation of S 3 containing a link L in its 1skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces. 1. ..."
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Cited by 8 (6 self)
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Abstract. Let T be a triangulation of S 3 containing a link L in its 1skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces. 1.