In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their n-dimensional counterparts, called simplices...

In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking...

The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local...

triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that admits...

mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional...

In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of...

different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially...

The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether...

simplices to form a simplicial complex. The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex...

Steinitz is related to the notion of an abstract simplicial complex and it differs from a simplicial complex by the property that its elements are no simplices:...

n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the...

interactions among multiple entities and complex hierarchies. This approach leverages structures like simplicial complexes and hypergraphs to capture global...

the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices...

algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which...

for each vertex i of X ', a simplicial complex Li' endowed with a rigid simplicial action of a finite group Γi. a simplicial map φi of Li' onto the link...

represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested...

illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...

Whitehead torsion vanishes. simplicial approximation See simplicial approximation theorem. simplicial complex See simplicial complex; the basic example is a...

and B {\displaystyle B} are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows:: 74, Def.4.2.1  The...

tree is a type of trie used to represent efficiently any general simplicial complex. Through its nodes, this data structure notably explicitly represents...

a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations. If X is a connected simplicial complex, an...

illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...

the task easier. The simplicial homology groups Hn(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with Cn(X) the free...

Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the...

subgraphs) of an undirected graph. The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed...

collapse reduces a simplicial complex (or more generally, a CW complex) to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were...

combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as...

category of sets or the category of simplicial sets to the category of manifolds. A simplicial manifold is a simplicial complex for which the geometric realization...

geometry studies maps between the subrings of the function field. Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring...

continuous maps from the standard n-simplex into X; if K is a simplicial complex then the simplicial chains Cn(K) are formal linear combinations of the n-simplices...