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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 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.

  4. 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

  5. Stanley–Reisner ring - Wikipedia

    en.wikipedia.org/wiki/Stanley–Reisner_ring

    Consequently, its homology can be studied by combinatorial and geometric methods. An abstract simplicial complex Δ is called Cohen–Macaulay over k if its face ring is a Cohen–Macaulay ring. [3] In his 1974 thesis, Gerald Reisner gave a complete characterization of such complexes.

  6. 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 ErdÅ‘s–Ko–Rado theorem and can be restated in terms of uniform hypergraphs.

  7. Triangulation (topology) - Wikipedia

    en.wikipedia.org/wiki/Triangulation_(topology)

    The dimension of an abstract simplicial complex is defined as () = {():}. [ 1 ] Abstract simplicial complexes can be realized as geometrical objects by associating each abstract simplex with a geometric simplex, defined below.

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  9. Simplicial homology - Wikipedia

    en.wikipedia.org/wiki/Simplicial_homology

    Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. [2]: sec.8.6 As a result, it gives a computable way to distinguish one space from another.