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In a simplicial complex, the link of a vertex v is the collection of subsets of vertices such that, if they are added to v, the result is a simplex. As an example, suppose v is the top vertex of the tetrahedron at the left. Then the link of v is the triangle at the base of the tetrahedron. This is because, for each edge of that triangle, if the edge is added to v, the result is a triangle.
The link of a vertex encodes information about the local structure of the complex at the vertex. It is a graph-theoretic analog to a sphere centered at a point.
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