Related papers: Link complexes of subspace arrangements
This article surveys recent developments on Hessenberg varieties, emphasizing some of the rich connections of their cohomology and combinatorics. In particular, we will see how hyperplane arrangements, representations of symmetric groups,…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain…
The colourful simplicial depth conjecture states that any point in the convex hull of each of d+1 sets, or colours, of d+1 points in general position in R^d is contained in at least d^2+1 simplices with one vertex from each set. We verify…
A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring such that no two vertices in the same edge have the same color, and each $k$-element subset of colors appears on at most one edge. The harmonious number $h(H)$ is…
A bi-arrangement of hyperplanes in a complex affine space is the data of two sets of hyperplanes along with a coloring information on the strata. To such a bi-arrangement, one naturally associates a relative cohomology group, that we call…
We provide first a categorical exploration of, and then completion of the mapping of the relationships among, three fundamental perspectives on binary relations: as the incidence matrices of hypergraphs, as the formal contexts of concept…
A simplicial complex $\Delta$ is called flag if all minimal nonfaces of $\Delta$ have at most two elements. The following are proved: First, if $\Delta$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $\Delta$ (i)…
In this paper, we give a new algebraic criterion for the {\em shellability} of (non-pure) simplicial complex $\Delta$ over $[n]$, shellable in the sense of Bj\"orner and Wachs \cite{BW}. We show that the spanning simplicial complex of…
Fixing two concordant links in $3$--space, we study the set of all embedded concordances between them, as knotted annuli in $4$--space. When regarded up to surface-concordance or link-homotopy, the set $\mathcal{C}(L)$ of concordances from…
We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module $M$ over the Laurent polynomial ring $\Lambda_{\mu}=\mathbb{Z}[t_1^{\pm1},\dots,t_{\mu}^{\pm1}]$. If $D$ is a diagram of a…
For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $\psi:V(G)\rightarrow V(H)$ such that $ij\in E(G)\Rightarrow\psi(i)\psi(j)\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\hom(G,H)$. We prove the following: for all…
A plane graph $H$ is a {\em plane minor} of a plane graph $G$ if there is a sequence of vertex and edge deletions, and edge contractions performed on the plane, that takes $G$ to $H$. Motivated by knot theory problems, it has been asked if…
Let X be a projective scheme carrying a circle action S with isolated fixed points. We associate a simplicial complex Delta(X,S) of "closure chains" using a refinement of its Morse/Bialynicki-Birula decomposition. If this decomposition is a…
The matching complex of a simple graph $G$ is a simplicial complex consisting of the matchings on $G$. Jeli\'c Milutinovi\'c et al. studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the…
We show that the order dimension of the weak order on a Coxeter group of type A, B or D is equal to the rank of the Coxeter group, and give bounds on the order dimensions for the other finite types. This result arises from a unified…
This paper verifies a conjecture of Edelman and Reiner regarding the homology of the $h$-complex of a Boolean algebra. A discrete Morse function with no low-dimensional critical cells is constructed, implying a lower bound on connectivity.…
This paper presents a construction of fibered links $(K,\Sigma)$ out of chord diagrams $\sL$. Let $\Gamma$ be the incidence graph of $\sL$. Under certain conditions on $\sL$ the symmetrized Seifert matrix of $(K,\Sigma)$ equals the bilinear…
For a simplicial complex $X$, the $d$-clique complex $\Delta_d(X)$ is the simplicial complex having all subsets of vertices whose $(d + 1)$-subsets are contained by $X$ as its faces. We prove that if $p = n^{\alpha}$, with $\alpha <…
M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for…
We generalize the construction of connected branched polymers and the notion of the volume of the space of connected branched polymers studied by Brydges and Imbrie, and Kenyon and Winkler to any hyperplane arrangement A. The volume of the…