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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 20 (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."
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 14 (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.
Random Discrete Morse Theory and a New Library of Triangulations. arXiv:1303.6422 [cs.CG
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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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Cited by 9 (4 self)
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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.
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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