Results 1  10
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27
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.
Towards the Poincaré Conjecture and the Classification of 3Manifolds
, 2003
"... The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying a ..."
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Cited by 27 (0 self)
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The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3dimensional manifolds. The final paragraph provides a brief description of the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work will be provided in a subsequent note by Michael Anderson.
Subgroups of word hyperbolic groups in dimension 2
 Jour. London Math. Soc
, 1996
"... If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are denned for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbol ..."
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Cited by 22 (10 self)
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If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are denned for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbolicity. A sufficient condition for hyperbolicity of a general graph is given along with an application to 'relative hyperbolicity'. Finitely presented subgroups of Lyndon's small cancellation groups of hyperbolic type are word hyperbolic. Finitely presented subgroups of hyperbolic 1relator groups are hyperbolic. Finitely presented subgroups of free Burnside groups are finite in the stable range. 1.
FINITE GROUP EXTENSIONS AND THE ATIYAH CONJECTURE
"... In 1976, Atiyah [2] constructed the L2Betti numbers of a compact Riemannian manifold. They are defined in terms of the spectrum of the Laplace operator on the universal covering of M. ByAtiyah’sL2index theorem [2], they can be used e.g. to compute the Euler characteristic of M. ..."
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Cited by 11 (5 self)
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In 1976, Atiyah [2] constructed the L2Betti numbers of a compact Riemannian manifold. They are defined in terms of the spectrum of the Laplace operator on the universal covering of M. ByAtiyah’sL2index theorem [2], they can be used e.g. to compute the Euler characteristic of M.
Singular surfaces, mod 2 homology, and hyperbolic volume, I. math.GT/0506396
"... Abstract. The main theorem of this paper states that if M is a closed orientable hyperbolic 3manifold of volume at most 3.08, then the dimension of H1(M; Z2) is at most 7, and that it is at most 6 unless M is “strange. ” To say that a closed, orientable 3manifold M, for which H1(M; Z2) has dimensi ..."
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Cited by 11 (8 self)
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Abstract. The main theorem of this paper states that if M is a closed orientable hyperbolic 3manifold of volume at most 3.08, then the dimension of H1(M; Z2) is at most 7, and that it is at most 6 unless M is “strange. ” To say that a closed, orientable 3manifold M, for which H1(M; Z2) has dimension 7, is strange means that the Z2vector space H1(M; Z2) has a 2dimensional subspace X such that for every homomorphism ψ: H1(M; Z2) → Z2 with X ⊂ kerψ, the twosheeted covering space ˜ M of M associated to ψ has the property that H1 ( ˜ M; Z4) is a free Z4module of rank 6. We state a grouptheoretical conjecture which implies that strange 3manifolds do not exist. One consequence of the main theorem is that if M is a closed, orientable, hyperbolic 3
HOMOLOGICAL CHARACTERIZATION OF THE UNKNOT
, 2003
"... Given a knot K in the 3sphere, let QK be its fundamental quandle as introduced by D. Joyce. Its first homology group is easily seen to be H1(QK) ∼ = Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) ∼ = Z whenever K is nontrivial. An analogous result holds for links, thus c ..."
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Cited by 8 (2 self)
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Given a knot K in the 3sphere, let QK be its fundamental quandle as introduced by D. Joyce. Its first homology group is easily seen to be H1(QK) ∼ = Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) ∼ = Z whenever K is nontrivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: let Qπ K be the conjugacy class of a meridian in the knot group π1(S3�K). We show that H2(Qπ K) ∼ = Zp, where p is the number of prime summands in a connected sum decomposition of K.
Jr.: The Homotopy Groups of Knots I. How to Compute the Algebraic 2Type
 Pacific J. Math
"... Let K be a CW complex with an aspherical splitting, i.e., with subcomplexes ϋΓ _ and K such that (a) K—KUK+ and (b) iΓ_, K =KΠK K are connected and aspherical. 0 +1 + The main theorem of this paper gives a practical procedure for computing the homology H*K of the universal cover K of K. It also pr ..."
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Cited by 5 (2 self)
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Let K be a CW complex with an aspherical splitting, i.e., with subcomplexes ϋΓ _ and K such that (a) K—KUK+ and (b) iΓ_, K =KΠK K are connected and aspherical. 0 +1 + The main theorem of this paper gives a practical procedure for computing the homology H*K of the universal cover K of K. It also provides a practical method for computing the algebraic 2type of K i.e., the triple consisting of the 9 fundamental group π K, the second homotopy group π K x 2 as a TΓilΓmodule, and the first /^invariant kK. The effectiveness of this procedure is demonstrated by letting K denote the complement of a smooth 2knot (S 4, JcS 2). Then the above mentioned methods provide a way for computing the algebraic 2type of 2knots, thus solving problem 36 of R. H. Fox in his 1962 paper, "Some problems in knot theory. " These methods can also be used to compute the algebraic 2type of 3manifolds from their Heegaard splittings. This approach can be applied to many other well known classes of spaces. Various examples of the computation are given.
SINGULAR SURFACES, MOD 2 HOMOLOGY, AND HYPERBOLIC VOLUME, I
, 2008
"... Abstract. If M is a simple, closed, orientable 3manifold such that π1(M) contains a genusg surface group, and if H1(M; Z2) has rank at least 4g − 1, we show that M contains an embedded closed incompressible surface of genus at most g. As an application we show that if M is a closed orientable hype ..."
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Cited by 5 (4 self)
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Abstract. If M is a simple, closed, orientable 3manifold such that π1(M) contains a genusg surface group, and if H1(M; Z2) has rank at least 4g − 1, we show that M contains an embedded closed incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3manifold of volume at most 3.08, then the rank of H1(M; Z2) is at most 6. 1. Introduction and
Quandle Coverings and their Galois Correspondence
, 2008
"... This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: ( ˜Q, ˜q) → (Q,q). This in turn leads us to define the algebraic fundamental group π1(Q,q): = Aut(p) = {g ∈ Adj(Q) ′  q g = ..."
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Cited by 3 (0 self)
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This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: ( ˜Q, ˜q) → (Q,q). This in turn leads us to define the algebraic fundamental group π1(Q,q): = Aut(p) = {g ∈ Adj(Q) ′  q g = q}, where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π1(Q,q). Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire’s algebraic covering theory of perfect groups. A detailed investigation also reveals some crucial differences, which we illustrate by numerous examples. As an application we obtain a simple formula for the second (co)homology group of a quandle Q. It has long been known that H1(Q) ∼ = H 1 (Q) ∼ = Z[π0(Q)], and we construct natural isomorphisms H2(Q) ∼ = π1(Q,q)ab and H 2 (Q,A) ∼ = Ext(Q,A) ∼ = Hom(π1(Q,q),A), reminiscent of the classical Hurewicz isomorphisms in degree 1. This means that whenever π1(Q,q) is known, (co)homology calculations in degree 2 become very easy.
THE FUNDAMENTAL CROSSED MODULE OF THE COMPLEMENT Of A Knotted Surface
, 2009
"... We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, the ..."
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We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π2(M,M 1) →Gcan be rescaled to a homotopy invariant IG(M), depending only on the algebraic 2type of M. We describe an algorithm for calculating π2(M,M (1) ) as a crossed module over π1(M (1)), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0 and 1handles. Here, Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant IG yields a nontrivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.