In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the central role that braid

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 c 1 n 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 triangulated orientable 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 c 2 t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M . We obtain explicit values for c1 and c2 . Keywords: knot theory, knot diagram, Reidemeister move, normal surfaces, computational complexity This paper grew out of work begun while the authors were visiting the Mathematical Sciences Research Institute in Berkeley in 1996/7. Research at MSRI is supported in part by NSF grant DMS-9022140. The first au...

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The study of three-manifolds 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...

We address the question: how common is it for a 3–manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3–manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3–manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3–manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown’s algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a “magic splitting sequence ” and work of Mirzakhani proves the main theorem. The 3–manifold situation contrasts markedly with random 2–generator 1–relator groups; in particular, we show that such groups “fiber ” with probability strictly

We describe an approach to normal surface theory for triangulated 3-manifolds which uses only the quadrilateral disk types (Q-disks) 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 Q-theory.

We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of non-trivial tangling in biomolecules, modeled as alpha complexes.

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.