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.

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with nonempty boundary, then the classifying space BDiff (M rel ∂M) of the space of diffeomorphisms of M which restrict to the identity map on ∂M has the homotopy type of a finite aspherical CW-complex. This answers, for this class of manifolds, a question posed by M. Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group H(M rel ∂M) is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.

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.

Abstract. We prove the existence of pure braids with arbitrarily many strands which are small, i.e. they contain no closed incompressible surface in the complement which is not boundary parallel. This implies the existence of irreducible non-Haken 3-manifolds of arbitrarily high Heegaard genus. 1.

Abstract. We describe a procedure for creating infinite families of knots, each having the maximum degree of their HOMFLY polynomial strictly less than twice their canonical genus. §0

ISSN numbers are printed here 1 Topology of knot spaces in dimension 3

ISSN numbers are printed here 1 JSJ-decompositions of knot and link complements in S 3

ISSN numbers are printed here 1 Topology of knot spaces in dimension 3

3-manifolds with genus 2 one-sided Heegaard

ISSN numbers are printed here 1 JSJ-decompositions of knot and link complements in S 3