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One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that ...
A simplicial 3-complex. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). ). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy th
Let be an abstract simplicial complex on a vertex set of size . The Alexander dual X ∗ {\displaystyle X^{*}} of X {\displaystyle X} is defined as the simplicial complex on V {\displaystyle V} whose faces are complements of non-faces of X {\displaystyle X} .
Given an abstract simplicial complex Δ on the vertex set {x 1,...,x n} and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k[Δ], is obtained from the polynomial ring k[x 1,...,x n] by quotienting out the ideal I Δ generated by the square-free monomials corresponding to the non-faces of Δ:
Constructing the Čech complex of a set of points sampled from a circle. In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distribution it is drawn from.
An abstract simplicial complex (ASC) is family of sets that is closed under taking subsets (the subset of a set in the family is also a set in the family). Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each set with k elements in the ASC is mapped to a (k-1)-dimensional simplex in the GSC.
In algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors of abstract simplicial complexes.It includes as a special case the Erdős–Ko–Rado theorem and can be restated in terms of uniform hypergraphs.
Simplex; Simplicial complex. Polytope; Triangulation; Barycentric subdivision; Simplicial approximation theorem; Abstract simplicial complex; Simplicial set