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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Tetrahedron.svg/220px-Tetrahedron.svg.png)
Link of a vertex
editGiven 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.
Given a geometric simplicial complex X and , its link
is a set containing every face
such that
and there is a simplex in
that has
as a vertex and
as a face.[1]: 3 Equivalently, the join
is a face in
.[2]: 20
- 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, the join of v with the edge is a triangle (one of the three triangles at the sides of the tetrahedron); and the join of v with the triangle itself is the entire tetrahedron.
The link of a vertex of a tetrahedron is the triangle.
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.
Link of a face
editThe definition of a link can be extended from a single vertex to any face.
Given an abstract simplicial complex X and any face of X, its link
is a set containing every face
such that
are disjoint and
is a face of X:
.
Given a geometric simplicial complex X and any face , its link
is a set containing every face
such that
are disjoint and there is a simplex in
that has both
and
as faces.[1]: 3
Examples
editThe 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.
Another example is illustrated below. 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 vertex and its link.
Properties
edit- For any simplicial complex X, every link
is downward-closed, and therefore it is a simplicial complex too; it is a sub-complex of X.
- Because X is simplicial, there is a set isomorphism between
and the set
: every
corresponds to
, which is in
.
Link and star
editA concept closely related to the link is the star.
Given an abstract simplicial complex X and any face ,
, its star
is a set containing every face
such that
is a face of X. In the special case in which X is a 1-dimensional complex (that is: a graph),
contains all edges
for all vertices
that are neighbors of
. That is, it is a graph-theoretic star centered at
.
Given a geometric simplicial complex X and any face , its star
is a set containing every face
such that there is a simplex in
having both
and
as faces:
. In other words, it is the closure of the set
-- the set of simplices having
as a face.
So the link is a subset of the star. The star and link are related as follows:
An example is illustrated below. 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.
- A vertex and its star.
See also
edit- Vertex figure - a geometric concept similar to the simplicial link.
References
edit- ^ a b c Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15
- ^ a b Rourke, Colin P.; Sanderson, Brian J. (1972). Introduction to Piecewise-Linear Topology. doi:10.1007/978-3-642-81735-9. ISBN 978-3-540-11102-3.
- ^ Bridson, Martin; Haefliger, André (1999), Metric spaces of non-positive curvature, Springer, ISBN 3-540-64324-9