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Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that non-constructible triangulations of the d-dimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] about non-shellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a non-trivial knot with "few edges" in a 3-sphere or 3-ball, and a similar hierarchy for 3-balls with a knotted spanning arc that consists of "few edges."

We construct non-constructible simplicial d-spheres with d + 10 vertices and non-constructible, non-realizable simplicial d-balls with d + 9 vertices for d≥3. 1

We show that if a 3-dimensional polytopal complex has a knot in its 1-skeleton, 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 3-spheres. We also obtain similar bounds concluding that a 3-sphere or 3-ball is non-shellable or not vertex decomposable. These two last bounds are sharp.

INTRODUCTION The k-dimensional skeleton of a d-polytope P is the set of all faces of the polytope of dimension at most k. The 1-skeleton 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

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 sub-exponential bound. Our results yield another recognition algorithm for S 3 that is conceptionally much simpler, though slower, as the Rubinstein–Thompson algorithm.

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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 non-simplicial c...

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].

. The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism group of codimension one immersions in the manifold. The connected sums of manifolds and immersions induce multiplicative structures which are respected by this surjection. We prove that those cubulations which map combinatorially into the standard decomposition of R n for large enough n (called mappable), are equivalent. Finally we classify the cubulations of the 2-sphere. Keywords and phrases: Cubulation, bubble/np-bubble move, f-vector, immersion, bordism, mappable, embeddable, simple, standard, connected sum. 1. Introduction and statement of results 1.1. Outline. Stellar moves were first considered by Alexander ([1]) who proved that they can relate any two triangul...