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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.
The standard simplex or probability simplex [2] is the (k − 1)-dimensional simplex whose vertices are the k standard unit vectors in , or in other words {: + + =, =, …,}. In topology and combinatorics , it is common to "glue together" simplices to form a simplicial complex .
Let K be a geometric simplicial complex (GSC). A subdivision of K is a GSC L such that: [1]: 15 [2]: 3 |K| = |L|, that is, the union of simplices in K equals the union of simplices in L (they cover the same region in space). each simplex of L is contained in some simplex of K.
For a single simplex s, the star of s is the set of simplices in K that have s as a face. The star of S is generally not a simplicial complex itself, so some authors define the closed star of S (denoted S t S {\displaystyle \mathrm {St} \ S} ) as C l s t S {\displaystyle \mathrm {Cl} \ \mathrm {st} \ S} the closure of the star of S.
Let K and L be two geometric simplicial complexes (GSC). A simplicial map of K into L is a function : such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex , ((())).
Given an abstract simplicial complex X and a vertex in (), its link (,) is a set containing every face such that and {} is a face of X.. In the special case in which X is a 1-dimensional complex (that is: a graph), (,) contains all vertices such that {,} is an edge in the graph; that is, (,) = = the neighborhood of in the graph.
In this popular version of solitaire almost every hand is a winner.
Assuming the segments all are oriented left-to-right (in increasing order from A k to A k+1), the boundary is A 6 − A 1. A closed polygonal curve, assuming consistent orientation, has null boundary. The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a k-chain is a (k−1)-chain. Note ...