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39
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.
Word hyperbolic dehn surgery
 Invent. Math
"... In the late 1970’s, Thurston dramatically changed the nature of 3manifold theory with the introduction of his Geometrisation Conjecture, and by proving it in the case of Haken 3manifolds [23]. The conjecture for general closed orientable 3manifolds remains perhaps the most important unsolved prob ..."
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Cited by 50 (8 self)
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In the late 1970’s, Thurston dramatically changed the nature of 3manifold theory with the introduction of his Geometrisation Conjecture, and by proving it in the case of Haken 3manifolds [23]. The conjecture for general closed orientable 3manifolds remains perhaps the most important unsolved problem in the subject.
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...
The disjoint curve property
, 2004
"... A Heegaard splitting of a closed, orientable threemanifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the d ..."
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Cited by 8 (0 self)
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A Heegaard splitting of a closed, orientable threemanifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve property. This paper shows that in a closed, orientable threemanifold all splittings of sufficiently large genus have the disjoint curve property. From this and a solution to the generalized Waldhausen conjecture it would follow that any closed, orientable three manifold contains only finitely many full splittings.
Decision problems in the space of Dehn fillings
 Topology
, 2003
"... Abstract. In this paper, we use normal surface theory to study Dehn filling on a knotmanifold. First, it is shown that there is a finite computable set of slopes on the boundary of a knotmanifold that bound normal and almost normal surfaces in a onevertex triangulation of that knotmanifold. This ..."
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Cited by 8 (2 self)
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Abstract. In this paper, we use normal surface theory to study Dehn filling on a knotmanifold. First, it is shown that there is a finite computable set of slopes on the boundary of a knotmanifold that bound normal and almost normal surfaces in a onevertex triangulation of that knotmanifold. This is combined with existence theorems for normal and almost normal surfaces to construct algorithms to determine precisely which manifolds obtained by Dehn filling: 1) are reducible, 2) contain two–sided incompressible surfaces, 3) are Haken, 4) fiber over S 1, 5) are the 3–sphere, and 6) are a lens space. Each of these algorithms is a finite computation. Moreover, in the case of essential surfaces, we show that the topology of the filled manifolds is strongly reflected in the triangulation of the knotmanifold. If a filled manifold contains an essential surface then the knotmanifold contains an essential vertex solution that caps off to an essential surface of the same type in the filled manifold. (Vertex solutions are the premier class of normal surface and are computable.) 1.
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.
Incompressible surfaces and Dehn Surgery on 1bridge Knots in handlebodies
, 1996
"... Given a knot K in a 3manifold M, we use N(K) to denote a regular neighborhood of K. Suppose γ is a slope (i.e an isotopy class of essential simple closed curves) on ∂N(K). The surgered manifold along γ is denoted by (H, K; γ), which by definition is the manifold obtained by gluing a solid torus to ..."
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Cited by 7 (1 self)
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Given a knot K in a 3manifold M, we use N(K) to denote a regular neighborhood of K. Suppose γ is a slope (i.e an isotopy class of essential simple closed curves) on ∂N(K). The surgered manifold along γ is denoted by (H, K; γ), which by definition is the manifold obtained by gluing a solid torus to H − IntN(K) so that γ bounds a meridianal disk. We say that M is ∂reducible if ∂M is compressible in M, and we call γ a ∂reducing slope of K if (H, K; γ) is ∂reducible. Since incompressible surfaces play an important rule in 3manifold theory, it is interesting to know what slopes of a given knot are ∂reducing. In generic case there are at most three ∂reducing slopes for a given knot [12], but there is no known algorithm to find these slopes. An exceptional case is when M is a solid torus, which has been well studied by Berge, Gabai and Scharlemann [1, 4, 5, 10]. It is now known that a knot in a solid torus has ∂reducing slopes only if it is a 1bridge braid. Moreover, all such knots and its corresponding ∂reducing slopes are classified in [1]. For 1bridge braids with small bridge width, a geometric method of detecting ∂reducing slopes has also been given in [5]. It was conjectured that a similar result holds for handlebodies, i.e, if K is a knot in a handlebody with H − K ∂irreducible, then K has ∂reducing slopes only if K is a 1bridge knot (see below for definitions). One is referred to [13] for some discussion of this conjecture and related problems. The main result of the present paper is to give an algorithm which will determine all ∂reducing slopes for a given 1bridge knot in a handlebody. Given a 1bridge presentation of a knot K in a handlebody H, the Main Algorithm in Section 7 will do the following. (1). Determine if K is disjoint from some compressing disk of ∂H. If it is, then ∂H is compressible after all surgeries, so all slopes are ∂reducing. (2). If K intersects all compressing disk of ∂H, determine if K is isotopic to a simple closed curve on ∂H. If it is, then ∂H is compressible in (H, K; γ) if and only if ∆(γ, γ 0) ≤ 1, where γ 0 is the the boundary of an annulus in E(K) whose other boundary component is on ∂H.
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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Cited by 6 (0 self)
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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.