Link (geometry)

In geometry, the link of a vertex of a 2-dimensional simplicial complex is a graph that 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.

Definition

The tetrahedron is a 2-complex.

The link of a vertex of a tetrahedron is the triangle.

Let $X$ be a simplicial complex. The link of a vertex $v$ is the graph $Lk(v, X)$ constructed as follows. The vertices of $Lk(v, X)$ are precisely 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 graph $Lk(v, X)$ is often given the topology of a ball of small radius centred at $v$ .

Similarly, for an abstract simplicial complex and a face $F$ of $X$ , there is also a notion of the link of a face $F$ , denoted $Lk(F, X)$ . $Lk(F, X)$ is the set of faces $G$ such that

G\cap F=\emptyset {\text{ and }}G\cup F\in X

.

Because $X$ is simplicial, there is a set isomorphism between $Lk(F, X)$ and

X_{F}=\{G\in X{\text{ such that }}F\subset G\}

.

Examples

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.

References

Bridson, Martin; Haefliger, André (1999), Metric spaces of non-positive curvature, Springer, ISBN 3-540-64324-9

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