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  2. Abstract simplicial complex - Wikipedia

    en.wikipedia.org/wiki/Abstract_simplicial_complex

    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 ...

  3. Simplicial complex - Wikipedia

    en.wikipedia.org/wiki/Simplicial_complex

    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

  4. Alexander duality - Wikipedia

    en.wikipedia.org/wiki/Alexander_duality

    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} .

  5. Stanley–Reisner ring - Wikipedia

    en.wikipedia.org/wiki/Stanley–Reisner_ring

    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:

  6. Čech complex - Wikipedia

    en.wikipedia.org/wiki/Čech_complex

    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.

  7. Simplicial complex recognition problem - Wikipedia

    en.wikipedia.org/wiki/Simplicial_complex...

    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.

  8. Kruskal–Katona theorem - Wikipedia

    en.wikipedia.org/wiki/Kruskal–Katona_theorem

    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.

  9. List of algebraic topology topics - Wikipedia

    en.wikipedia.org/wiki/List_of_algebraic_topology...

    Simplex; Simplicial complex. Polytope; Triangulation; Barycentric subdivision; Simplicial approximation theorem; Abstract simplicial complex; Simplicial set