Maximal Simplex Research Articles

Shellable tilings on relative simplicial complexes and their h-vectors

Abstract An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.

Root polytopes and Jaeger‐type dissections for directed graphs

We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid to what we call the semi-balanced case, that is, when each cycle has the same number of edges pointing in the two directions. Given a ribbon structure, we identify a natural class of spanning trees and show that, in the semi-balanced case, they induce a shellable dissection of the root polytope into maximal simplices. This allows for a computation of the h ∗ $h^*$ -vector of the polytope and for showing some properties of this new graph invariant, such as a product formula and that in the planar case, the h ∗ $h^*$ -vector is equivalent to the greedoid polynomial of the dual graph. We obtain a general recursion relation as well. We also work out the case of layer-complete directed graphs, where our method recovers a previously known triangulation. Indeed our dissection is often but not always a triangulation; we address this with a series of examples.

Automorphisms and opposition in spherical buildings of exceptional type, I

AbstractTo each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf {E}_6$ , $\mathsf {F}_4$ , and $\mathsf {G}_2$ . Moreover, for all split spherical buildings of exceptional type, we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

Strong collapse and persistent homology

In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. We also focus on the problem of computing persistent homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice. Finally, we can compromise between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.

Strong collapse and persistent homology

In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. We also focus on the problem of computing persistent homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice. Finally, we can compromise between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.

Shellability and Sphericity of Finite Quasi-arc Complexes

Quasi-triangulations of a non-orientable surface were introduced by Dupont and Palesi (J Algebr Comb 42(2):429–472, 2015). The quasi-arc complex provides an intricate description of the combinatorics of these quasi-triangulations. This is the simplicial complex where vertices correspond to quasi-arcs and maximal simplices to quasi-triangulations. We prove that when the quasi-arc complex is finite then it is shellable and, as a consequence, it is homeomorphic to a sphere.

Building Efficient and Compact Data Structures for Simplicial Complexes

The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the ST while retaining its functionalities. In addition, we propose two new data structures called the Maximal Simplex Tree and the Simplex Array List. We analyze the compressed ST, the Maximal Simplex Tree, and the Simplex Array List under various settings.

On the complexity of the representation of simplicial complexes by trees

In this paper, we investigate the problem of the representation of simplicial complexes by trees. We introduce and analyze local and global tree representations. We prove that the global tree representation is more efficient in terms of time complexity for searching a given simplex and we show that the local tree representation is more efficient in terms of size of the structure. The simplicial complexes are modeled by hypergraphs. We then prove that the associated combinatorial optimization problems are very difficult to solve and to approximate even if the set of maximal simplices induces a planar graph of maximum degree at most three or a bounded degree hypergraph. However, we prove polynomial time algorithms that compute constant factor approximations and optimal solutions for some classes of instances.

Dyck path triangulations and extendability (extended abstract)

We introduce the Dyck path triangulation of the cartesian product of two simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$ using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever$m\geq k>n$, any triangulations of $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique triangulation of $\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory. Nous introduisons la triangulation par chemins de Dyck du produit cartésien de deux simplexes $\Delta_{n-1}\times\Delta_{n-1}$. Les simplexes maximaux de cette triangulation sont donnés par des chemins de Dyck, et cette construction se généralise de façon naturelle pour produire des triangulations $\Delta_{r\ n-1}\times\Delta_{n-1}$ qui utilisent des chemins de Dyck rationnels. Notre étude de la triangulation par chemins de Dyck est motivée par des problèmes de prolongement de triangulations partielles de produits de deux simplexes. On montre que $m\geq k>n$ alors toute triangulation de $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ se prolonge en une unique triangulation de $\Delta_{m-1}\times\Delta_{n-1}$. De plus, avec une construction explicite, nous montrons que la borne $k>n$ est optimale. Nous présentons aussi des interprétations de nos résultats dans le langage des matroïdes orientés tropicaux, qui sont analogues aux résultats classiques de la théorie des matroïdes orientés.

Dyck path triangulations and extendability

We introduce the Dyck path triangulation of the cartesian product of two simplices Δn−1×Δn−1. The maximal simplices of this triangulation are given by Dyck paths, and the construction naturally generalizes to certain rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever m≥k>n, any triangulation of the product of the k-skeleton of Δm−1 with Δn−1 extends to a unique triangulation of Δm−1×Δn−1. Moreover, using the Dyck path triangulation, we prove that the bound k>n is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids that are analogous to classical results in oriented matroid theory.

Does a simplex formed by endmembers really yield maximal volume?

One of commonly used criteria for finding an endmember set is based on the fact that for a given number of endmembers, p, a p-vertex simplex with its all p vertices specified by endmembers always yields the maximal volume. Since there are also other criteria that have been used for endmember extraction, an interesting issue arises: 'Does a p-vertex simplex with its all vertices specified by p endmembers really produce maximal volume?' In other words, 'Is the maximal simplex volume a better and more effective measure than other criteria currently being used by endmember extraction such as orthogonal projection, least squares error, etc.?' This paper explores this issue by investigating a number of popular endmember extraction algorithms designed by various criteria via a comparative study and analysis through real image experiments.

Unique Minimal Liftings for Simplicial Polytopes

For a minimal inequality derived from a maximal lattice-free simplicial polytope in ℝn, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers ℝn. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in ℝn with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points.

The Lefschetz-Hopf theorem and axioms for the Lefschetz number

The reduced Lefschetz number, that is, where denotes the Lefschetz number, is proved to be the unique integer-valued function on self-maps of compact polyhedra which is constant on homotopy classes such that (1) for and ; (2) if is a map of a cofiber sequence into itself, then ; (3) , where is a self-map of a wedge of circles, is the inclusion of a circle into the th summand, and is the projection onto the th summand. If is a self-map of a polyhedron and is the fixed point index of on all of , then we show that satisfies the above axioms. This gives a new proof of the normalization theorem: if is a self-map of a polyhedron, then equals the Lefschetz number of . This result is equivalent to the Lefschetz-Hopf theorem: if is a self-map of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of is the sum of the indices of all the fixed points of .

Classification of Solid Transitive Simplex Tilings in Simply Connected 3-Spaces II. Metric realizations of the maximal simplex tilings

Classification of Solid Transitive Simplex Tilings in Simply Connected 3-Spaces II. Metric realizations of the maximal simplex tilings

A decomposition of 2-weak vertex-packing polytopes

The 2-weak vertex-packing polytope of a loopless graphG withd vertices is the subset of the unitd-cube satisfyingxi+xj≤1 for every edge (i,j) ofG. The dilation by 2 of this polytope is a polytope[Figure not available: see fulltext.] with integral vertices. We triangulate[Figure not available: see fulltext.] with lattice simplices of minimal volume and label the maximal simplices with elements of the hyperoctahedral groupBd. This labeling gives rise to a shelling of the triangulation[Figure not available: see fulltext.] of[Figure not available: see fulltext.], where theh-vector of[Figure not available: see fulltext.] (and the Ehrharth*-vector of[Figure not available: see fulltext.] can be computed as a descent statistic on a subset ofBd defined in terms ofG. A recursive way of computing theh-vector of[Figure not available: see fulltext.] is also given, and a recursive formula for the volume of[Figure not available: see fulltext.].

Generating combinatorial complexes of polyhedral type

The paper describes a method for generating combinatorial complexes of polyhedral type. Building blocks B {\mathbf {B}} are implanted into the maximal simplices of a simplicial complex C {\mathbf {C}} , on which a group operates as a combinatorial reflection group. Of particular interest is the case where B {\mathbf {B}} is a polyhedral block and C {\mathbf {C}} the barycentric subdivision of a regular incidence-polytope K {\mathbf {K}} together with the action of the automorphism group of K {\mathbf {K}} .

On the Least Number of Fixed Points

THEOREM ([8], see also [1], [3], and [6]). If X has no local separating points and dim X 2 3, then for any self-map f of X we have N(f) = M(f). Here by a local separating point we mean a point x E X at which H1(X, X x) ? 0, or, equivalently, at which there is a connected neighborhood U such that U x is not connected. For any triangulation K of X, X has no local separating points iff every maximal simplex of K is of dimension >2 and for every vertex v of K the boundary of stK v is connected. The purpose of this paper is to improve the above theorem.

The Broken-Circuit Complex

The broken-circuit complex introduced by H. Wilf (Which polynomials are chromatic?, Proc. Colloq. Combinational Theory (Rome, 1973)) of a matroid G is shown to be a cone over a related complex, the reduced broken-circuit complex $\mathcal {C}’(G)$. The topological structure of $\mathcal {C}’(G)$ is studied, its Euler characteristic is computed, and joins and skeletons are shown to exist in the class of all such complexes. These computations and constructions are compared with analogous results in the theory of the independent set complex of a matroid. Reduced broken-circuit complexes are partially characterized by conditions concerning which subcomplexes are pure (i.e., have equicardinal maximal simplices). In particular, every matroid complex is a reduced broken-circuit complex. Three proofs are given that the simplex numbers of the cone of $\mathcal {C}’(G)$ are the coefficients which appear in the characteristic polynomial $\chi (G)$ of G. This relates the work of W. Tutte on externally inactive bases to that of H. Whitney on sets which do not contain broken circuits. One of these proofs gives a combinatorial correspondence between these sets. Properties of the characteristic polynomial are then given topological proofs.

The broken-circuit complex

The broken-circuit complex introduced by H. Wilf (Which polynomials are chromatic?, Proc. Colloq. Combinational Theory (Rome, 1973)) of a matroid G is shown to be a cone over a related complex, the reduced broken-circuit complex C ′ ( G ) \mathcal {C}’(G) . The topological structure of C ′ ( G ) \mathcal {C}’(G) is studied, its Euler characteristic is computed, and joins and skeletons are shown to exist in the class of all such complexes. These computations and constructions are compared with analogous results in the theory of the independent set complex of a matroid. Reduced broken-circuit complexes are partially characterized by conditions concerning which subcomplexes are pure (i.e., have equicardinal maximal simplices). In particular, every matroid complex is a reduced broken-circuit complex. Three proofs are given that the simplex numbers of the cone of C ′ ( G ) \mathcal {C}’(G) are the coefficients which appear in the characteristic polynomial χ ( G ) \chi (G) of G. This relates the work of W. Tutte on externally inactive bases to that of H. Whitney on sets which do not contain broken circuits. One of these proofs gives a combinatorial correspondence between these sets. Properties of the characteristic polynomial are then given topological proofs.

On maximal simplices inscribed in a central convex set

On maximal simplices inscribed in a central convex set