Results 1  10
of
47
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)
 Add to MetaCart
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 ..."
Abstract

Cited by 50 (8 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 44 (9 self)
 Add to MetaCart
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 Virtual Haken Conjecture: experiments and examples
 GEOM. TOPOL
, 2002
"... A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe compu ..."
Abstract

Cited by 31 (2 self)
 Add to MetaCart
A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3manifolds. We took the complete HodgsonWeeks census of 10,986 smallvolume closed hyperbolic 3manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every nontrivial Dehn surgery on the figure8 knot is virtually Haken.
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 ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
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.
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... ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
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.