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Heegaard diagrams and surgery descriptions for twisted facepairing 3manifolds
 Algebr. Geom. Topol
"... Abstract. We give a simple algorithmic construction of a Heegaard diagram for an arbitrary twisted facepairing 3manifold. One family of meridian curves in the Heegaard diagram corresponds to the face pairs, and the other family is obtained from the first by a product of powers of Dehn twists. Thes ..."
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Abstract. We give a simple algorithmic construction of a Heegaard diagram for an arbitrary twisted facepairing 3manifold. One family of meridian curves in the Heegaard diagram corresponds to the face pairs, and the other family is obtained from the first by a product of powers of Dehn twists. These Dehn twists are along curves which correspond to the edge cycles and the powers are the multipliers. From the Heegaard diagram, one can easily construct a framed link in the 3sphere such that Dehn surgery on this framed link gives the twisted facepairing manifold. 1.
CONSTRUCTING NONPOSITIVELY CURVED SPACES AND GROUPS
"... Abstract. The theory of nonpositively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test – in real ti ..."
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Abstract. The theory of nonpositively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test – in real time – whether a random finite piecewise Euclidean complex is nonpositively curved. In this article I focus on the question of how to construct examples of nonpositively curved spaces and groups, highlighting in particular the boundary between what is currently doable and what is not yet feasible. Since this is intended primarily as a survey, the key ideas are merely sketched with references pointing the interested reader to the original articles. Over the past decade or so, the consequences of nonpositive curvature for geometric group theorists have been throughly investigated, most prominently in the book by Bridson and Haefliger [26]. See also the recent review article by Kleiner in the Bulletin of the AMS [59] and the related books by Ballmann [4], BallmannGromovSchroeder [5] and the original long article by Gromov [48]. In this article
A SURVEY OF TWISTED FACEPAIRING 3MANIFOLDS
"... Abstract. The twisted facepairing construction gives an efficient way to generate facepairing descriptions for many interesting closed 3manifolds. Our work in this paper is directed toward the goal of determining which closed, connected, orientable 3manifolds can be generated from this construct ..."
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Abstract. The twisted facepairing construction gives an efficient way to generate facepairing descriptions for many interesting closed 3manifolds. Our work in this paper is directed toward the goal of determining which closed, connected, orientable 3manifolds can be generated from this construction. We succeed in proving that all lens spaces, the Heisenberg manifold (Nil geometry), S 2 × S 1, and all connected sums of twisted facepairing manifolds are twisted facepairing manifolds. We show how to obtain most closed, connected, orientable, Seifertfibered manifolds as twisted facepairing manifolds. It still seems unlikely that all closed, connected, orientable 3manifolds can be so obtained. The twisted facepairing construction of our earlier papers [1], [2], [3] gives an efficient way of generating, mechanically and with little effort, myriads of relatively simple facepairing descriptions of interesting closed 3manifolds. Papers [1] and [2] established the basic properties of these manifolds. In [3] we investigated a special subclass of twisted facepairing manifolds, namely the ample manifolds,
BITWIST 3MANIFOLDS
"... Abstract. Our earlier twistedfacepairing construction showed how to modify an arbitrary orientationreversing facepairing ɛ on a faceted 3ball in a mechanical way so that the quotient is automatically a closed, orientable 3manifold. The modifications were, in fact, parametrized by a finite set o ..."
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Abstract. Our earlier twistedfacepairing construction showed how to modify an arbitrary orientationreversing facepairing ɛ on a faceted 3ball in a mechanical way so that the quotient is automatically a closed, orientable 3manifold. The modifications were, in fact, parametrized by a finite set of positive integers, arbitrarily chosen, one integer for each edge class of the original facepairing. This allowed us to find very simple facepairing descriptions of many, though presumably not all, 3manifolds. Here we show how to modify the construction to allow negative parameters, as well as positive parameters, in the twistedfacepairing construction. We call the modified construction the bitwist construction. We prove that all closed connected orientable 3manifolds are bitwist manifolds. As with the twist construction, we analyze and describe the Heegaard splitting naturally associated with a bitwist description of a manifold. 1.
Constructing Bitwisted Face Pairing 3Manifolds
, 2008
"... (ABSTRACT) The bitwist construction, originally discovered by Cannon, Floyd, and Parry, gives us a new method for finding face pairing descriptions of 3manifolds. In this paper, I will describe the construction in a way suitable for a more general audience than the original research papers, which i ..."
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(ABSTRACT) The bitwist construction, originally discovered by Cannon, Floyd, and Parry, gives us a new method for finding face pairing descriptions of 3manifolds. In this paper, I will describe the construction in a way suitable for a more general audience than the original research papers, which include [2], [3], [4], and [5]. Along the way, I will describe Dehn Surgery and a set of moves which allows us to change the framings of a link without changing the topology of the manifold obtained by Dehn Surgery. Once the theory has been developed, I will apply it to find several bitwist representations of the Poincare ́ Sphere and 3Torus. Finally, I discuss how one might attempt to find a set of moves that can take one bitwist representation of a manifold to any other bitwist representation of the same manifold. Acknowledgments First and foremost, I would like to thank Bill Floyd for all the time he spent helping me to understand the bitwist construction and more general concepts in low dimensional topology. I’d also like to thank Jim Thomson and Bud Brown for being on my committee, and being some of the best teachers I’ve had at Virginia Tech. I also thank Peter Linnell and Leslie
TWISTED FACEPAIRING 3MANIFOLDS WHICH ARE HYPERBOLIC
"... Abstract. We construct a family of 3balls using cones which represent closed orientable 3manifolds and study twisted facepairing construction due to Cannon, Floyd and Parry to understand the structure of such manifolds. Moreover, we prove that those manifolds are hyperbolic. 1. ..."
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Abstract. We construct a family of 3balls using cones which represent closed orientable 3manifolds and study twisted facepairing construction due to Cannon, Floyd and Parry to understand the structure of such manifolds. Moreover, we prove that those manifolds are hyperbolic. 1.
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, 806
"... Abstract. Our earlier twistedfacepairing construction showed how to modify an arbitrary orientationreversing facepairing ǫ on a faceted 3ball in a mechanical way so that the quotient is automatically a closed, orientable 3manifold. The modifications were, in fact, parametrized by a finite set ..."
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Abstract. Our earlier twistedfacepairing construction showed how to modify an arbitrary orientationreversing facepairing ǫ on a faceted 3ball in a mechanical way so that the quotient is automatically a closed, orientable 3manifold. The modifications were, in fact, parametrized by a finite set of positive integers, arbitrarily chosen, one integer for each edge class of the original facepairing. This allowed us to find very simple facepairing descriptions of many, though presumably not all, 3manifolds. Here we show how to modify the construction to allow negative parameters, as well as positive parameters, in the twistedfacepairing construction. We call the modified construction the bitwist construction. We prove that all closed connected orientable 3manifolds are bitwist manifolds. As with the twist construction, we analyze and describe the Heegaard splitting naturally associated with a bitwist description of a manifold. 1.
SPINES OF 3MANIFOLDS AS POLYHEDRA WITH IDENTIFIED FACES
"... Abstract. In this article we establish the relation between the spines of 3manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3manifolds can be presented through polyhedra with identified faces in a very natural way. We als ..."
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Abstract. In this article we establish the relation between the spines of 3manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3manifolds can be presented through polyhedra with identified faces in a very natural way. We also prove the equivalence between the special spines and a certain type of polyhedra, and other related results. In this article we will consider the polyhedra with identified faces and the spines of 3manifolds. Spines have been studied broadly by Matveev, among other mathematicians, and their definition, as well as the main theorems concerning them, are available in [8]. Further results on spine theory can be found in [7], [9], [10] and [11]. On the other hand,