A Greedy Algorithm to Compute Arrangements of Lines in the Projective Plane

We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.

Cosheaf representations of relations and Dowker complexes

The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes.

Matching Complexes of $3 \times n$ Grid Graphs

The matching complex of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. In the last few years the matching complexes of grid graphs have gained much attention among the topological combinatorists. In 2017, Braun and Hough obtained homological results related to the matching complexes of $2 \times n$ grid graphs. Further in 2019, Matsushita showed that the matching complexes of $2 \times n$ grid graphs are homotopy equivalent to a wedge of spheres. In this article we prove that the matching complexes of $3\times n$ grid graphs are homotopy equivalent to a wedge of spheres. We also give the comprehensive list of the dimensions of spheres appearing in the wedge.

The Gordian complex of theta-curves

In this paper, we study the Gordian metric on the set of all theta-curves and give a lower bound of it. We define the Gordian complex of theta-curves, which is a simplicial complex whose vertices consist of all theta-curves in the 3-dimensional Euclidean space [Formula: see text]. We show that for any given theta-curve [Formula: see text], there exists an infinite family of theta-curves containing [Formula: see text] such that the Gordian distance between any two distinct members of this family is equal to one.

LS-category of moment-angle manifolds and higher order Massey products

Using the combinatorics of the underlying simplicial complex K, we give various upper and lower bounds for the Lusternik–Schnirelmann (LS) category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} . We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS-category. In particular, we characterize the LS-category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} over triangulated d-manifolds K for d ≤ 2 {d\leq 2} , as well as higher-dimensional spheres built up via connected sum, join, and vertex doubling operations. We show that the LS-category closely relates to vanishing of Massey products in H * ⁢ ( 𝒵 K ) {H^{*}(\mathcal{Z}_{K})} , and through this connection we describe first structural properties of Massey products in moment-angle manifolds. Some of the further applications include calculations of the LS-category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and k-neighborly complexes, which double as important examples of hyperbolic manifolds.

Local-to-global Urysohn width estimates

The notion of the Urysohn d-width measures to what extent a metric space can be approximated by a d-dimensional simplicial complex. We investigate how local Urysohn width bounds on a Riemannian manifold affect its global width. We bound the 1-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of n-manifolds of considerable ( n - 1 ) {(n-1)} -width in which all unit balls have arbitrarily small 1-width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.

A Construction of Sequentially Cohen–Macaulay Graphs

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.