We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection 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 self-intersection 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.

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 component. As tools, we introduce the concepts of barrier surface and shrinking, as well as the notion of crushing a triangulation along a normal surface. A number of applications are given, including an algorithm to construct an irreducible decomposition of a closed, orientable 3-manifold, an algorithm to construct a maximal collection of pairwise disjoint, normal 2-spheres in a closed 3-manifold, an alternate algorithm for the 3-sphere recognition problem, results on edges of low valence in minimal triangulations of 3-manifolds, and a construction of irreducible knots in closed 3-manifolds.

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 1-skeleton of the interior of a compact, orientable, triangulated PL 3-manifold 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.

Abstract. We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold 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 3-manifold. There are finitely many homotopy classes of 2-plane fields which carry tight contact structures. Theorem 0.2. Every closed, oriented, atoroidal 3-manifold 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 Kronheimer-Mrowka 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].

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 so-called arc-presentations of links and established some of their basic properties. Here we exhibit a further development of that technique and of the theory of arc-presentations, and prove that any arc-presentation of the unknot admits a (non-strictly) 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 arc-presentation 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.

Abstract. We give a proof of the so-called 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

It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model[12, 3, 1]. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems [20]. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class. That is, a one clean qubit computer can approximate these Jones polynomials in time polynomial in both the number of strands and number of crossings, and the problem of simulating a one clean qubit computer is reducible to approximating the Jones polynomial of the trace closure of a braid. 1 One Clean Qubit The one clean qubit model of quantum computation originated as an idealized model of quantum computation on highly mixed initial states, such as appear in NMR implementations[20, 4]. In this model, one is given an initial quantum state consisting of a single qubit in the pure state |0〉, and n qubits in the maximally mixed

Abstract. In this paper, we use normal surface theory to study Dehn filling on a knot-manifold. First, it is shown that there is a finite computable set of slopes on the boundary of a knot-manifold that bound normal and almost normal surfaces in a one-vertex triangulation of that knot-manifold. 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 knot-manifold. If a filled manifold contains an essential surface then the knot-manifold 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.

Abstract. For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R 3 having less than 10n + 9 edges, with the property that any Piecewise-Linear triangulated disk spanning the curve contains at least 2 n−1 triangles. 1. Introduction. Let K be a closed polygonal curve in R3 consisting of n line segments. Assume that K is unknotted, so that it is the boundary of an embedded disk in R3. This paper considers the question: How many triangles are needed to triangulate a Piecewise-Linear (PL) spanning disk of K? The main result, Theorem 1 below,

this paper. REFERENCES