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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 [1 ..."
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Cited by 17 (5 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."
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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We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1
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 10 (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.
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
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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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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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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...
Glicci simplicial complexes
"... ABSTRACT. One of the main open questions in liaison theory is whether every homogeneous CohenMacaulay ideal in a polynomial ring is glicci, i.e. if it is in the Gliaison class of a complete intersection. We give an affirmative answer to this question for StanleyReisner ideals defined by simplicia ..."
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ABSTRACT. One of the main open questions in liaison theory is whether every homogeneous CohenMacaulay ideal in a polynomial ring is glicci, i.e. if it is in the Gliaison class of a complete intersection. We give an affirmative answer to this question for StanleyReisner ideals defined by simplicial complexes that are weakly vertexdecomposable. This class of complexes includes matroid, shifted and Gorenstein complexes respectively. Moreover, we construct a simplicial complex which shows that the property of being glicci depends on the characteristic of the base field. As an application of our methods we establish new evidence for two conjectures of Stanley on partitionable complexes and Stanley decompositions. 1.