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Simplicial moves on complexes and manifolds
 GEOMETRY & TOPOLOGY MONOGRAPHS VOLUME 2: PROCEEDINGS OF THE KIRBYFEST PAGES 299–320
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Decompositions of Simplicial Balls and Spheres With Knots Consisting of Few Edges
, 1999
"... Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that nonconstructible triangulations of the ddimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] a ..."
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Cited by 16 (6 self)
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Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that nonconstructible triangulations of the ddimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] about nonshellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a nontrivial knot with "few edges" in a 3sphere or 3ball, and a similar hierarchy for 3balls with a knotted spanning arc that consists of "few edges."
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.
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
How to make a triangulation of S 3 polytopal
 Trans. Am. Math. Soc
, 2004
"... We introduce a method of studying triangulations T of S 3 that connects topological and geometrical approaches. In its center is a new numerical invariant p(T), called polytopality. Although its definition is purely topological, it is sensitive for geometric properties of T, as follows. We consider ..."
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Cited by 4 (1 self)
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We introduce a method of studying triangulations T of S 3 that connects topological and geometrical approaches. In its center is a new numerical invariant p(T), called polytopality. Although its definition is purely topological, it is sensitive for geometric properties of T, as follows. We consider local moves called expansions, that generalize stellar subdivisions of simplicial complexes. Let d(T) be the length of a shortest sequence of expansions relating T with the boundary complex of a convex 4–polytope. In this paper we obtain both lower and upper bounds for d(T) in terms of p(T). Using previous results [9] based on the Rubinstein–Thompson algorithm, we obtain an upper bound for d(T) in terms of the number n of tetrahedra of T. The bound is exponential in n 2, and we prove here that in general one can not replace it by a subexponential bound. Our results yield another recognition algorithm for S 3 that is conceptionally much simpler, though slower, as the Rubinstein–Thompson algorithm.
Combinatorial Structure of Constructible Complexes
 Master’s thesis
, 1997
"... This thesis is a study of the combinatorial structure of a certain kind of complexes: constructible complexes and strongly constructible complexes. The notion of constructible complexes is known as a weaker notion than that of shellable complexes, and this thesis is aimed to be a foundation of the s ..."
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Cited by 2 (0 self)
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This thesis is a study of the combinatorial structure of a certain kind of complexes: constructible complexes and strongly constructible complexes. The notion of constructible complexes is known as a weaker notion than that of shellable complexes, and this thesis is aimed to be a foundation of the study of shellable complexes. The theme of this thesis is a characterization of constructible complexes in terms of their face posets, and the relation between constructibility and partitionability. After reviewing some preliminaries on posets and complexes in Chapter 2, the property of constructible complexes is studied in Chapter 3. In this chapter, a new notion named recursively dividable posets is defined and it is shown that a complex is constructible if and only if its face poset is recursively dividable. Usually constructibility is defined only for simplicial complexes, but corresponding to the notion of recursively dividable posets, constructibility is generalized for nonsimplicial c...
CROSSING NUMBER OF LINKS FORMED BY EDGES OF A TRIANGULATION
, 2001
"... Abstract. We study the crossing number of links that are formed by edges of a triangulation T of S 3 with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n 2. In general, this bound can not be replaced by a subexponential bound. However, if T is pol ..."
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Cited by 1 (1 self)
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Abstract. We study the crossing number of links that are formed by edges of a triangulation T of S 3 with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n 2. In general, this bound can not be replaced by a subexponential bound. However, if T is polytopal (resp. shellable) then there is a quadratic (resp. biquadratic) upper bound in n for the crossing number. In our proof, we use a numerical invariant p(T), called polytopality, that we have introduced in [5].