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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 78 (9 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.
Hyperbolic Groups With Low Dimensional Boundary
 Ann. Sci. École Norm. Sup
, 2000
"... If a torsionfree hyperbolic group G has 1dimensional boundary @1G, then @1G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When @1G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3dimensional hyperbolic Poincar'e duality group ..."
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Cited by 53 (8 self)
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If a torsionfree hyperbolic group G has 1dimensional boundary @1G, then @1G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When @1G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3dimensional hyperbolic Poincar'e duality group. We also construct a "topologically rigid" hyperbolic group G: any homeomorphism of @1G is induced by an element of G.
The Thurston norm, fibered manifolds and twisted Alexander polynomials
, 2005
"... Every element in the first cohomology group of a 3–manifold is dual to embedded surfaces. The Thurston norm measures the minimal ‘complexity ’ of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3–sphere. We show that the degrees of twisted A ..."
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Cited by 48 (25 self)
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Every element in the first cohomology group of a 3–manifold is dual to embedded surfaces. The Thurston norm measures the minimal ‘complexity ’ of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3–sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm, generalizing work of McMullen and Turaev. Our bounds attain their most elegant form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. The bounds are very powerful and can be easily implemented with a computer. We show that these lower bounds determine the genus of all knots with 12 crossings or less, including the Conway knot and the Kinoshita–Terasaka knot which have trivial Alexander polynomial. We also give many examples of closed manifolds and link complements where twisted Alexander polynomials detect the correct Thurston norm. We also give obstructions to fibering 3–manifolds using twisted Alexander polynomials and detect all knots with 12 crossings or less that are not fibered. For some
Twisted Alexander polynomials detect fibered 3manifolds
 Monopoles and ThreeManifolds, New Mathematical Monographs (No. 10), Cambridge University Press. , Knots, sutures and excision
"... Abstract. A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3–manifolds. In this paper we show that ..."
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Cited by 41 (11 self)
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Abstract. A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3–manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3–manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S 1 × N 3 admits a symplectic structure, then N fibers over S 1. In fact we will completely determine the symplectic cone of S 1 × N in terms of the fibered faces of the Thurston norm ball of N. 1.
A W Reid, Essential closed surfaces in bounded 3manifolds
 553–563 MR1431827 Geometry & Topology, Volume 12 (2008
, 1997
"... A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a ..."
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Cited by 38 (11 self)
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A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a map i: S � M of a closed,
Convex decomposition theory
, 2001
"... We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3manifold with nonempty boundary, and (2) prove that every toroidal 3manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures. ..."
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Cited by 32 (11 self)
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We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3manifold with nonempty boundary, and (2) prove that every toroidal 3manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures.
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 20 (7 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.
Graph manifolds with boundary are virtually special
 Journal of Topology
"... Abstract. Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in pi1M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a “sufficient ” collection of surfaces in M, then pi1M is virtually t ..."
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Cited by 19 (4 self)
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Abstract. Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in pi1M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a “sufficient ” collection of surfaces in M, then pi1M is virtually the fundamental group of a special nonpositively curved cube complex. We provide a sufficient collection for graph manifolds with boundary thus proving that their fundamental groups are virtually special, and hence linear. 1.
Twisted Alexander polynomials and symplectic structures
, 2006
"... Abstract. Let N be a closed, oriented 3–manifold. A folklore conjecture states that S 1 ×N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showin ..."
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Cited by 19 (15 self)
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Abstract. Let N be a closed, oriented 3–manifold. A folklore conjecture states that S 1 ×N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showing that their behavior is the same as of those of fibered 3–manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S¹ × N. As an application of these results we will show that S¹ × N(P) does not admit a symplectic structure, where N(P) is the 0–surgery along the pretzel knot P = (5, −3, 5), answering a question of Peter Kronheimer.
Covering spaces of arithmetic 3orbifolds
"... This paper investigates properties of finite sheeted covering spaces of arithmetic hyperbolic 3orbifolds (see §2). The main motivation is a central unresolved question in the theory of closed hyperbolic 3manifolds; namely whether a closed ..."
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Cited by 15 (10 self)
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This paper investigates properties of finite sheeted covering spaces of arithmetic hyperbolic 3orbifolds (see §2). The main motivation is a central unresolved question in the theory of closed hyperbolic 3manifolds; namely whether a closed