Related papers: The augmented external activity complex of a matro…
Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground…
A Euclidean oriented matroid program yields a partial ordering of the cocircuits of its cocircuit graph. We show that every linear extension of that ordering yields a topological sweep and induces a recursive atom-ordering (a shelling of…
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable.…
Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave: 1. The matroid complex of independent subsets of E. 2. The broken circuit complex of…
This paper investigates the shellability of $r$-independence complexes $\mathcal{I}_r(G)$, a generalization of classical independence complexes introduced by Paolini and Salvetti. For a graph $G$, a subset $A \subseteq V(G)$ is…
We provide a combinatorial study of split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. A nice feature of split matroids is that they generalize paving matroids, while being…
We introduce the notion of a quasi-matroidal class of ordered simplicial complexes: an approximation to the idea of a matroid cryptomorphism in the landscape of ordered simplicial complexes. A quasi-matroidal class contains pure shifted…
Imposing a strong condition on the linear order of shellable complexes, we introduce strong shellability. Basic properties, including the existence of dimension-decreasing strong shelling orders, are developed with respect to nonpure…
The splitting operation on a $p$-matroid does not necessarily preserve connectivity. It is observed that there exists a single element extension of the splitting matroid which is connected. In this paper, we define the element splitting…
We consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the…
Shellability is a well-known combinatorial criterion for verifying that a simplicial complex is Cohen-Macaulay. Another notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the…
The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell…
We say that a pure simplicial complex ${\mathbf K}$ of dimension $d$ satisfies the removal-collapsibility condition if ${\mathbf K}$ is either empty or ${\mathbf K}$ becomes collapsible after removing $\tilde \beta_d ({\mathbf K}; {\mathbb…
We propose a mechanism enabling the appearance of border cells -- neurons firing at the boundaries of the navigated enclosures. The approach is based on the recent discovery of discrete complex analysis on a triangular lattice, which allows…
The complement of a hyperplane arrangement in $\mathbb{C}^n$ deformation retracts onto an $n$-dimensional cell complex, but the known procedures only apply to complexifications of real arrangements (Salvetti) or the cell complex produced…
The h-vector of a matroid M is an important invariant related to the independence complex of M and can also be recovered from an evaluation of its Tutte polynomial. A well-known conjecture of Stanley posits that the h-vector of a matroid is…
We show that chiral symmetry can be broken spontaneously in one-component systems with isotropic interactions, i.e. many-particle systems having maximal a priori symmetry. This is achieved by designing isotropic potentials that lead to…
Let K be the face ring of the independence complex of a matroid. We show that if T is a generic linear system of parameters, then K/T satisfies a weak form of the Hard Lefschetz Theorem. As a result, the first half of the h-vector of the…
Let $X=\{x_i\}_{i=1}^m$ be a set of unit vectors in $\RR^n$. The coherence of $X$ is $\coh(X):=\max_{i\not=j}|\langle x_i, x_j\rangle|$. A vector $x\in X$ is said to be isolable if there are no unit vectors $x'$ arbitrarily close to $x$…
We showed in the first paper of this series that the generic $C_2^1$-cofactor matroid is the unique maximal abstract $3$-rigidity matroid. In this paper we obtain a combinatorial characterization of independence in this matroid. This solves…