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The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
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
Arc presentations of links: Monotonic simplification
 ArXiv:math.GT/0208153 v2 8
, 2003
"... In the beginning of 90’s J.Birman and W. Menasco worked out a nice technique for studying links presented in the form of a closed braid. The technique is based on certain foliated surfaces and uses tricks that were introduced earlier by D. Bennequin for a slightly different foliation. A few years la ..."
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Cited by 14 (0 self)
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In the beginning of 90’s J.Birman and W. Menasco worked out a nice technique for studying links presented in the form of a closed braid. The technique is based on certain foliated surfaces and uses tricks that were introduced earlier by D. Bennequin for a slightly different foliation. A few years later P.Cromwell adapted Birman–Menasco’s method for studying socalled arcpresentations of links and established some of their basic properties. Here we exhibit a further development of that technique and of the theory of arcpresentations, and prove that any arcpresentation of the unknot admits a (nonstrictly) monotonic simplification by elementary moves; this yields a simple algorithm for recognizing the unknot. We show also that the problem of recognizing split links and that of factorizing a composite link can be solved in a similar manner. We also define two easily checked sufficient conditions for knottedness. Our principal contribution to the technique is this. We describe how to handle a disk with a given arcpresentation of the unknot as boundary in order to make simplification possible. We fill a gap in Cromwell’s arguments, a gap which was “borrowed ” from Birman–Menasco’s proof of the claim that any braid representing a composite link can be made composite by applying finitely many conjugations and exchange moves.
Spaces Which Are Not Negatively Curved
 Comm. in Anal. and Geom
, 1997
"... this paper will be 2dimensional. Definition of a lamination ..."
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this paper will be 2dimensional. Definition of a lamination
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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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.
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 8 (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...
The computational complexity of knot genus and spanning area
 electronic), arXiv: math.GT/0205057. MR MR2219001
"... Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NPha ..."
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Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NPhard. 1.
Algorithms for recognizing knots and 3manifolds
 Chaos, Solitons and Fractals
, 1998
"... Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution. ..."
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Cited by 6 (3 self)
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Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution.
ON THE EXISTENCE OF INFINITELY MANY ESSENTIAL SURFACES OF BOUNDED GENUS
"... if M is an irreducible, ∂irreducible 3manifold with boundary a single torus, and if M contains no genus one essential (incompressible and ∂incompressible) surfaces, then M cannot contain infinitely many distinct isotopy classes of essential surfaces of uniformly bounded genus. The main result in ..."
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Cited by 5 (0 self)
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if M is an irreducible, ∂irreducible 3manifold with boundary a single torus, and if M contains no genus one essential (incompressible and ∂incompressible) surfaces, then M cannot contain infinitely many distinct isotopy classes of essential surfaces of uniformly bounded genus. The main result in this paper is a generalization: If M is an irreducible ∂irreducible 3manifold with boundary, and M contains no genus one or genus zero essential surfaces, then M cannot contain infinitely many isotopy classes of essential surfaces of uniformly bounded genus. 1. Introduction. In this paper, a Haken manifold is an orientable, irreducible, ∂irreducible 3manifold containing a (2sided) incompressible surface. Even if a 3manifold is not Haken, we shall always, for simplicity, assume that it is orientable. An irreducible, ∂irreducible manifold M is simple if it contains no incompressible,