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11
Lectures on Polytopes
, 1994
"... uptodate electronically. Thus, this is an electronic preprint, the newest, latest and hottest version of which you should always be able to get via our WWWserver, at ..."
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Cited by 359 (5 self)
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uptodate electronically. Thus, this is an electronic preprint, the newest, latest and hottest version of which you should always be able to get via our WWWserver, at
Nonconstructible complexes and the bridge index
, 1999
"... We show that if a 3dimensional polytopal complex has a knot in its 1skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shella ..."
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Cited by 11 (2 self)
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We show that if a 3dimensional polytopal complex has a knot in its 1skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3spheres. We also obtain similar bounds concluding that a 3sphere or 3ball is nonshellable or not vertex decomposable. These two last bounds are sharp.
Small examples of nonconstructible simplicial balls and spheres
 SIAM J. Discrete Math
, 2004
"... We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1 ..."
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Cited by 10 (4 self)
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We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1
The size of triangulations supporting a given link
, 2000
"... Abstract. Let T be a triangulation of S 3 containing a link L in its 1skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces. 1. ..."
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Cited by 8 (6 self)
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Abstract. Let T be a triangulation of S 3 containing a link L in its 1skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces. 1.
Onepoint suspensions and wreath products of polytopes and spheres
"... Abstract. It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many ..."
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Cited by 7 (2 self)
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Abstract. It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many nonPL spheres as well as contractible simplicial complexes with a vertextransitive group of automorphisms can be obtained in this way. 1.
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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Cited by 6 (0 self)
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
Higher minors and Van Kampen’s obstruction
 Math. Scand
"... We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes H and K and every nonnegative integer m, we prove that if H is a minor of K then the non vanishing of Van Kampen’s obstruction in dimension m (a characteristic class indicating non embed ..."
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Cited by 6 (2 self)
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We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes H and K and every nonnegative integer m, we prove that if H is a minor of K then the non vanishing of Van Kampen’s obstruction in dimension m (a characteristic class indicating non embeddability in the (m − 1)sphere) for H implies its non vanishing for K. As a corollary, based on results by Van Kampen [20] and Flores [5], if K has the dskeleton of the (2d+2)simplex as a minor, then K is not embeddable in the 2dsphere. We answer affirmatively a problem asked by Dey et. al. [3] concerning topologypreserving edge contractions, and conclude from it the validity of the generalized lower bound inequalities for a special class of triangulated spheres. 1
Combinatorial 3manifolds with 10 vertices
, 2007
"... We give a complete enumeration of all combinatorial 3manifolds with 10 vertices: There are precisely 247882 triangulated 3spheres with 10 vertices as well as 518 vertexminimal triangulations of the sphere product S 2 ×S 1 and 615 triangulations of the twisted sphere product S 2 × S 1. All the 3s ..."
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Cited by 4 (2 self)
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We give a complete enumeration of all combinatorial 3manifolds with 10 vertices: There are precisely 247882 triangulated 3spheres with 10 vertices as well as 518 vertexminimal triangulations of the sphere product S 2 ×S 1 and 615 triangulations of the twisted sphere product S 2 × S 1. All the 3spheres with up to 10 vertices are shellable, but there are 29 vertexminimal nonshellable 3balls with 9 vertices.
On locally constructible spheres and balls
 Acta Math
"... The unsolved question whether there are only exponentiallymany combinatorial types of simplicial 3spheres is crucial for the convergence of models for 3D quantum gravity. Working towards this question, Durhuus and Jonsson (1995) introduced the restriction to “locally constructible ” (LC) 3spheres ..."
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Cited by 4 (1 self)
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The unsolved question whether there are only exponentiallymany combinatorial types of simplicial 3spheres is crucial for the convergence of models for 3D quantum gravity. Working towards this question, Durhuus and Jonsson (1995) introduced the restriction to “locally constructible ” (LC) 3spheres, and showed that there are only exponentiallymany LC 3spheres. We characterize the LC property for dspheres (“the sphere minus a facet collapses to a (d − 2)complex”) and for dballs. Thus we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from this study are: – Not all simplicial 3spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.) – There are only exponentially many shellable simplicial 3spheres with given number of facets. (This answers a question by Kalai.) – All simplicial constructible 3balls are collapsible. (This answers a question by Hachimori.) 1