Link (simplicial complex)

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 structure of the complex at the vertex.

Given an abstract simplicial complex X and

Given a geometric simplicial complex X and

[2]: 20 An alternative definition is: the link of a vertex

is the graph Lk(v, X) constructed as follows.

The vertices of Lk(v, X) are the edges of X incident to v. Two such edges are adjacent in Lk(v, X) iff they are incident to a common 2-cell at v. The definition of a link can be extended from a single vertex to any face.

Given an abstract simplicial complex X and any face

σ , τ

τ ∪ σ

τ ∩ σ = ∅ ,

τ ∪ σ ∈

Given a geometric simplicial complex X and any face

σ , τ

[1]: 3 The link of a vertex of a tetrahedron is a triangle – the three vertices of the link corresponds to the three edges incident to the vertex, and the three edges of the link correspond to the faces incident to the vertex.

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.

There is a two-dimensional simplicial complex.

At the left, a vertex is marked in yellow.

At the right, the link of that vertex is marked in green.A concept closely related to the link is the star.

Given an abstract simplicial complex X and any face

τ ∪ σ

In the special case in which X is a 1-dimensional complex (that is: a graph),

That is, it is a graph-theoretic star centered at

Given a geometric simplicial complex X and any face

In other words, it is the closure of the set

So the link is a subset of the star.

The star and link are related as follows:

There is a two-dimensional simplicial complex.

At the left, a vertex is marked in yellow.

At the right, the star of that vertex is marked in green.