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In some conventions, [7] the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes.
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation. Thus every simplex has exactly ...
In general, an n-simplex is an object made up from a list of n + 1 vertices (which are 0-simplices) and n + 1 faces (which are (n − 1)-simplices). The vertices of the i-th face are the vertices of the n-simplex minus the i-th vertex. The vertices of a simplex need not be distinct and a simplex is not determined by its vertices and faces: two ...
Consider is a given simplex, is a given node corresponding to the last vertex of , is the label associate to that node, is the depth of that node, is the dimension of the simplicial complex, is the maximal number of operations to access in a dictionary (if the dictionary is a red-black tree, = ((())) is the complexity) .
For each k ≤ n, this has a subcomplex , the k-th horn inside , corresponding to the boundary of the n-simplex, with the k-th face removed. This may be formally defined in various ways, as for instance the union of the images of the n maps Δ n − 1 → Δ n {\displaystyle \Delta ^{n-1}\rightarrow \Delta ^{n}} corresponding to all the other ...
Pure simplicial complexes can be thought of as triangulations and provide a definition of polytopes. A facet is a maximal simplex, i.e., any simplex in a complex that is not a face of any larger simplex. [2] (Note the difference from a "face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same ...
Example of singular 1-chains: The violet and orange 1-chains cannot be realized as a boundary of a 2-chain. The usual construction of singular homology proceeds by defining formal sums of simplices, which may be understood to be elements of a free abelian group, and then showing that we can define a certain group, the homology group of the topological space, involving the boundary operator.
Geometric realization of a 3-dimensional abstract simplicial complex. 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 subsets, i.e., every subset of a set in the family is also in the family.