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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. 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.

  4. 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.

  5. Subdivision (simplicial complex) - Wikipedia

    en.wikipedia.org/wiki/Subdivision_(simplicial...

    Let K be an abstract simplicial complex (ASC). The face poset of K is a poset made of all nonempty simplices of K , ordered by inclusion (which is a partial order). For example, the face-poset of the closure of {A,B,C} is the poset with the following chains:

  6. Vietoris–Rips complex - Wikipedia

    en.wikipedia.org/wiki/Vietoris–Rips_complex

    In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ.

  7. 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.

  8. h-vector - Wikipedia

    en.wikipedia.org/wiki/H-vector

    Let Δ be an abstract simplicial complex of dimension d − 1 with f i i-dimensional faces and f −1 = 1. These numbers are arranged into the f-vector of Δ, = (,, …,).An important special case occurs when Δ is the boundary of a d-dimensional convex polytope.

  9. Link (simplicial complex) - Wikipedia

    en.wikipedia.org/wiki/Link_(simplicial_complex)

    In this example, the link can be visualized by cutting off the vertex with a plane; formally, intersecting the tetrahedron with a plane near the vertex – the resulting cross-section is the link. Another example is illustrated below. There is a two-dimensional simplicial complex. At the left, a vertex is marked in yellow.