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.

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is

This paper shows that non--normal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic. Normal monomodal logics can simulate all others 1 This paper is dedicated to our teacher, Wolfgang Rautenberg x1. Introduction. A simulation of a logic by a logic \Theta is a translation of the expressions of the language for into the language of \Theta such that the consequence relation defined by is reflected under the translation by the consequence relation of \Theta. A well--known case is provided by the Godel translation, which simulates intuitionistic logic by Grzegorczyk's logic (cf. [11] and [5]). Such simulations not only yield technical results but may also ...

We apply a construction of Rips to show that a number of algorithmic problems concerning certain small cancellation groups and, in particular, word hyperbolic groups, are recursively unsolvable. Given any integer k> 2, there is no algorithm to determine whether or not any small cancellation group can be generated by either two elements or more than k elements. There is a small cancellation group E such that there is no algorithm to determine whether or not any finitely generated subgroup of E is all of E, or is finitely presented, or has a finitely generated second integral homology group.

The Andrews-Curtis conjecture states that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of the elementary Nielsen transformations and conjugations. In this paper we describe all balanced presentations of the trivial group on two generators and with the total length of relators 12. We show that all these presentations satisfy the Andrews-Curtis conjecture.

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Abstract. We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most g is NP-complete. 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 NP-hard. 1.

this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25

Following the course set by A. Markov (1951), S. Adjan (1958), and M. Rabin (1958), C. ' O'D'unlaing (1983) has shown that certain properties of finitely generated Thue congruences are undecidable in general. Here we prove that many of these undecidability results remain valid even when only finitely generated Thue congruences on a fixed twoletter alphabet \Sigma 2 are considered. In contrast to a construction given by P. Schupp (1976) which applies to groups only, we use a modified version of a technical lemma from A. Markov's original paper. Based on this technical result we can carry the result of A. Sattler-Klein (1996), which says that certain Markov properties remain undecidable even when they are restricted to finitely generated Thue congruences that are decidable, over to the alphabet \Sigma 2 . 1 Introduction A string-rewriting system R on some alphabet \Sigma is a set of pairs of strings over \Sigma. It induces a congruence $ R on \Sigma , the Thue congruence generat...

this paper. Our special interest is in groups which are presented to us as being generated by a small set of elements, be these permutations of vertices of a graph, matrices describing automorphisms of linear codes, or classes of homotopies of a knot described only by the relations that they satisfy (see [Hac 87]). Although the axioms that define the notion of a group are rather simple, and in spite of the abundance of knowledge about large classes of groups, one is frequently frustrated by the paucity of methods for dealing with groups described by a small set of generators and relations that hold between them. What is lacking in the standard texts of classical algebra and group theory is a counterpart of numerical methods in differential equations. Yet, such computational methods in group theory have been developed along the years under the influence of external problems as well as from within, especially by the needs of the classification of finite simple groups and the Burnside Groups problem. It should be emphasized that the present computational methods build on careful analysis of algorithmic aspects of known theories. Also, that the practical use of these algorithms became possible in a meaningful way only with the advent of computer technology. We will present in this article some of the known computational methods for investigating groups given by generators and relations, comment