Link (geometry)
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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.


Link of a vertex

Given an
abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
and v a vertex in V(X), its link \operatorname(v,X) is a set containing every face \tau \in X such that v\not\in \tau and \tau\cup \ is a face of . * In the special case in which is a 1-dimensional complex (that is: a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
), \operatorname(v,X) contains all vertices u\neq v such that \ is an edge in the graph; that is, \operatorname(v, X)=\mathcal(v)=the neighborhood of v in the graph. Given a
geometric simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial se ...
and v\in V(X), its link \operatorname(v,X) is a set containing every face \tau \in X such that v\not\in \tau and there is a simplex in X that has v as a vertex and \tau as a face. Equivalently, the
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two top ...
v \star \tau is a face in X. * 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. An alternative definition is: the link of a vertex v\in V(X) is the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
constructed as follows. The vertices of are the edges of incident to . Two such edges are adjacent in iff they are incident to a common 2-cell at . * The graph is often given the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of a ball of small radius centred at ; it is an analog to a sphere centered at a point.


Link of a face

The definition of a link can be extended from a single vertex to any face. Given an
abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
and any face \sigma of , its link \operatorname(\sigma,X) is a set containing every face \tau \in X such that \sigma, \tau are disjoint and \tau\cup \sigma is a face of : \operatorname(\sigma,X) := \. Given a
geometric simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial se ...
and any face \sigma \in X, its link \operatorname(\sigma,X) is a set containing every face \tau \in X such that \sigma, \tau are disjoint and there is a simplex in X that has both v and \tau as faces.


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. 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. File:Simplicial complex link.svg, alt=A vertex and its link., A and its .


Properties

* For any simplicial complex , every link \operatorname(\sigma,X) is downward-closed, and therefore it is a simplicial complex too; it is a sub-complex of . * Because is simplicial, there is a set isomorphism between \operatorname(\sigma,X) and the set X_ := \: every \tau\in \operatorname(\sigma,X) corresponds to \tau \cup \sigma, which is in X_.


Link and star

A concept closely-related to the link is the star. Given an
abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
and any face \sigma \in X,V(X), its star \operatorname(\sigma,X) is a set containing every face \tau \in X such that \tau\cup \sigma is a face of . In the special case in which is a 1-dimensional complex (that is: a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
), \operatorname(v,X) contains all edges \ for all vertices u that are neighbors of v. That is, it is a graph-theoretic star centered at ''u''. Given a
geometric simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial se ...
and any face \sigma \in X, its star \operatorname(\sigma,X) is a set containing every face \tau \in X such that there is a simplex in X having both \sigma and \tau as faces: \operatorname(\sigma,X) := \. In other words, it is the closure of the set \ -- the set of simplices having \sigma as a face. So the link is a subset of the star. The star and link are related as follows: * For any \sigma\in X, \operatorname(\sigma,X) = \. * For any v\in V(X), \operatorname(v,X) = v \star \operatorname(v,X) , that is, the star of v is the
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
of its link at v. 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. File:Simplicial complex star.svg, A and its .


See also

*
Vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
- a geometric concept similar to the simplicial link.


References

{{Reflist Geometry