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28
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 ..."
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Cited by 55 (6 self)
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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.
The Isomorphism Problem for Toral Relatively Hyperbolic Groups
"... We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually ..."
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Cited by 21 (7 self)
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We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]).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 nmanifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsionfree relatively hyperbolic group with abelian parabolics is
Unsolvable problems about small cancellation and word hyperbolic groups
 BULL. LONDON MATH. SOC
, 1994
"... 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 ca ..."
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Cited by 15 (2 self)
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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.
Normal Monomodal Logics Can Simulate All Others
 Journal of Symbolic Logic
, 1999
"... This paper shows that nonnormal 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 ..."
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Cited by 12 (0 self)
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This paper shows that nonnormal 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 wellknown 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 ...
Balanced Presentations of the Trivial Group on Two Generators and the AndrewsCurtis Conjecture
, 2001
"... The AndrewsCurtis 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 ..."
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Cited by 12 (5 self)
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The AndrewsCurtis 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 AndrewsCurtis conjecture.
The computational complexity of knot genus and spanning area
 electronic), arXiv: math.GT/0205057. MR MR2219001
"... Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. 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 NPha ..."
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Cited by 6 (0 self)
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Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. 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 NPhard. 1.
Questions in Computable Algebra and Combinatorics
, 1999
"... 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 ..."
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Cited by 5 (0 self)
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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
Groups Presented by Finite TwoMonadic ChurchRosser Thue Systems
 Transactions of the American Mathematical Society
, 1986
"... Abstract. It is shown that a group G can be defined by a monoidpresentation of the form (2; 7"), where T is a finite twomonadic ChurchRosser Thue system over 2, if and only if G is isomorphic to the free product of a finitely generated free group with a finite number of finite groups. Introductio ..."
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Cited by 5 (3 self)
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Abstract. It is shown that a group G can be defined by a monoidpresentation of the form (2; 7"), where T is a finite twomonadic ChurchRosser Thue system over 2, if and only if G is isomorphic to the free product of a finitely generated free group with a finite number of finite groups. Introduction. In 1911 M. Dehn formulated three fundamental problems for groups given by presentations of the form (2; L), where 2 is some set of generators, 2 is a disjoint copy of 2, and L ç (2 U 2) * is a set of defining relators [12]. One of these problems is the word problem, which can be stated as follows: Let (2; L) be a group presentation. Given a word w e (2 U 2) * decide in a finite number of steps
Naurois. The complexity of semilinear problems in succinct representation
 Computational Complexity
"... Abstract. We prove completeness results for twentythree problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the BlumShubSmale additive model of com ..."
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Cited by 5 (3 self)
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Abstract. We prove completeness results for twentythree problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the BlumShubSmale additive model of computation. If, in contrast, the circuit is constantfree, then the completeness results are for the Turing model of computation. One such result, the P NP[log]completeness of deciding Zariski irreducibility, exhibits for the first time a problem with a geometric nature complete in this class.