Results 1  10
of
45
Straightening polygonal arcs and convexifying polygonal cycles
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2000
"... Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles ..."
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Cited by 77 (30 self)
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Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewisedifferentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the wellstudied carpenter’s rule conjecture.
The Symplectic Thom Conjecture
"... In this paper, we demonstrate a relation among SeibergWitten invariants which arises from embedded surfaces in fourmanifolds whose selfintersection number is negative. These relations, together with Taubes' basic theorems on the SeibergWitten invariants of symplectic manifolds, are then used to ..."
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Cited by 43 (8 self)
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In this paper, we demonstrate a relation among SeibergWitten invariants which arises from embedded surfaces in fourmanifolds whose selfintersection number is negative. These relations, together with Taubes' basic theorems on the SeibergWitten invariants of symplectic manifolds, are then used to prove the Symplectic Thom Conjecture: a symplectic surface in a symplectic fourmanifold is genusminimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative selfintersection in fourmanifolds. 1.
The Virtual Haken Conjecture: experiments and examples
 Geom. Topol
"... ABSTRACT. A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we desc ..."
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Cited by 32 (2 self)
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ABSTRACT. A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3manifolds. We took the complete HodgsonWeeks census of 10,986 smallvolume closed hyperbolic 3manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every nontrivial Dehn surgery on the figure8 knot is virtually Haken.
Symplectic Fillings and Positive Scalar Curvature
 GEOM. AND TOPOLOGY
, 1998
"... Let X be a 4manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b^+_ 2 (X) > 0 or the bou ..."
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Cited by 26 (9 self)
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Let X be a 4manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b^+_ 2 (X) > 0 or the boundary of X is disconnected. As an application we show that the Poincaré homology 3sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3manifold which is not symplectically semifillable. Using work of Fr yshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3spheres having positive scalar curvature metrics.
Witten’s conjecture and property
"... Let K be a knot in S 3 and let Y1 be the oriented 3manifold obtained by +1surgery on K. The following is one formulation of the “Property P ” conjecture for knots: Conjecture 1. If K is a nontrivial knot, then Y1 is not a homotopy 3sphere. The purpose of this note is to prove the conjecture. The ..."
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Cited by 25 (2 self)
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Let K be a knot in S 3 and let Y1 be the oriented 3manifold obtained by +1surgery on K. The following is one formulation of the “Property P ” conjecture for knots: Conjecture 1. If K is a nontrivial knot, then Y1 is not a homotopy 3sphere. The purpose of this note is to prove the conjecture. The ingredients of the argument
Some Results in Geometric Topology and Geometry”, Thesis submitted for the degree
 of PhD, Warwick Maths Institute
, 1997
"... ..."
Knots with unknotting number one and Heegaard Floer homology
"... Abstract. We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal ..."
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Cited by 18 (2 self)
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Abstract. We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with ten crossings and unknotting number equal to one. 1.
The arithmetic and geometry of Salem numbers
 Bull. Amer. Math. Soc
, 1991
"... Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and ar ..."
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Cited by 14 (2 self)
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Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the ‘minimization problem’. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from 1. Contents
Akbulut's Corks and HCobordisms of Smooth, Simply Connected 4Manifolds
"... F35> 0 = @A 1 , is an involution [9]. Corollary: Any homotopy 4sphere, \Sigma 4 , can be constructed by cutting out a contractible 4manifold, A 0 from S 4 and gluing it back in by an involution of @A 0 . Remark: Since there are many examples of nontrivial hcobordisms (the first ones were di ..."
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Cited by 11 (1 self)
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F35> 0 = @A 1 , is an involution [9]. Corollary: Any homotopy 4sphere, \Sigma 4 , can be constructed by cutting out a contractible 4manifold, A 0 from S 4 and gluing it back in by an involution of @A 0 . Remark: Since there are many examples of nontrivial hcobordisms (the first ones were discovered by Donaldson [4]), there are as many examples of nontrivial, rel boundary, hcobordisms A. However these A are delicate objects; their nontriviality vanishes when a trivial hcobordism is added. That is, if we add A 0 \Theta I to A along @A 0 \Theta I, then it follows from the Addenda that we have an hcobordism between S 4 on the bottom as well as S 4 on the top; thus the hcobordism is the trivial<F54.1
Transversal torus knots
 Geometry and Topology
, 1999
"... Abstract. We classify positive transversal torus knots in tight contact structures up to transversal isotopy. 1. ..."
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Cited by 7 (1 self)
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Abstract. We classify positive transversal torus knots in tight contact structures up to transversal isotopy. 1.