[[File:Tetrahedron.svg|thumb|The tetrahedron is a 2-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.
== Link of a vertex == Given an abstract simplicial complex {{mvar|X}} and <math display="inline">v</math> a vertex in <math display="inline">V(X)</math>, its '''link''' <math display="inline">\operatorname{Lk}(v,X)</math> is a set containing every face <math display="inline">\tau \in X</math> such that <math display="inline">v\not\in \tau</math> and <math display="inline"> \tau\cup \{v\}</math> is a face of {{mvar|X}}.
* In the special case in which {{mvar|X}} is a 1-dimensional complex (that is: a graph), <math display="inline">\operatorname{Lk}(v,X)</math> contains all vertices <math display="inline">u\neq v</math> such that <math display="inline">\{u,v\}</math> is an edge in the graph; that is, <math display="inline">\operatorname{Lk}(v, X)=\mathcal{N}(v)=</math>the neighborhood system of <math display="inline">v</math> in the graph.
Given a geometric simplicial complex {{mvar|X}} and <math display="inline">v\in V(X)</math>, its '''link''' <math display="inline">\operatorname{Lk}(v,X)</math> is a set containing every face <math display="inline">\tau \in X</math> such that <math display="inline">v\not\in \tau</math> and there is a simplex in <math display="inline"> X</math> that has <math display="inline">v</math> as a vertex and <math display="inline"> \tau</math> as a face.<ref name=":0">{{Citation |last=Bryant |first=John L. |title=Chapter 5 - Piecewise Linear Topology |date=2001-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780444824325500068 |work=Handbook of Geometric Topology |pages=219–259 |editor-last=Daverman |editor-first=R. J. |place=Amsterdam |publisher=North-Holland |language=en |isbn=978-0-444-82432-5 |access-date=2022-11-15 |editor2-last=Sher |editor2-first=R. B.}}</ref>{{Rp|page=3}} Equivalently, the join <math display="inline">v \star \tau</math> is a face in <math display="inline"> X</math>.<ref name=":1">{{Cite book |first1=Colin P.|last1= Rourke |author1-link=Colin P. Rourke|first2=Brian J.|last2= Sanderson |url=https://link.springer.com/book/10.1007/978-3-642-81735-9 |title=Introduction to Piecewise-Linear Topology |year=1972 |language=en |doi=10.1007/978-3-642-81735-9|isbn=978-3-540-11102-3 }}</ref>{{Rp|page=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.thumb|The link of a vertex of a tetrahedron is the triangle. An alternative definition is: the '''link''' of a vertex <math display="inline">v\in V(X)</math> is the graph {{math|Lk(''v'', ''X'')}} constructed as follows. The vertices of {{math|Lk(''v'', ''X'')}} are the edges of {{mvar|X}} incident to {{mvar|v}}. Two such edges are adjacent in {{math|Lk(''v'', ''X'')}} iff they are incident to a common 2-cell at {{mvar|v}}.
* The graph {{math|Lk(''v'', ''X'')}} is often given the topology of a ball of small radius centred at {{mvar|v}}; it is an analog to a sphere centered at a point.<ref>{{Citation |last1=Bridson |first1=Martin |author1-link=Martin Bridson|title=Metric spaces of non-positive curvature |year=1999 |publisher=Springer |isbn=3-540-64324-9 |last2=Haefliger |first2=André |author2-link=André Haefliger}}</ref>
== Link of a face == The definition of a link can be extended from a single vertex to any face.
Given an abstract simplicial complex {{mvar|X}} and any face <math display="inline">\sigma</math> of {{mvar|X}}, its '''link''' <math display="inline">\operatorname{Lk}(\sigma,X)</math> is a set containing every face <math display="inline">\tau \in X</math> such that <math display="inline">\sigma, \tau</math> are disjoint and <math display="inline"> \tau\cup \sigma</math> is a face of {{mvar|X}}: <math display="inline">\operatorname{Lk}(\sigma,X) := \{\tau\in X: ~\tau\cap \sigma = \emptyset,~ \tau\cup \sigma \in X\}</math>.
Given a geometric simplicial complex {{mvar|X}} and any face <math display="inline">\sigma \in X</math>, its '''link''' <math display="inline">\operatorname{Lk}(\sigma,X)</math> is a set containing every face <math display="inline">\tau \in X</math> such that <math display="inline">\sigma, \tau</math> are disjoint and there is a simplex in <math display="inline"> X</math> that has both <math display="inline">\sigma</math> and <math display="inline"> \tau</math> as faces.<ref name=":0" />{{Rp|page=3}}
== 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.<gallery class="center" widths="350" heights="112"> File:Simplicial complex link.svg|alt=A vertex and its link.|A {{color|#fc3|vertex}} and its {{color|#093|'''link'''}}. </gallery>
== Properties ==
* For any simplicial complex {{mvar|X}}, every link <math display="inline">\operatorname{Lk}(\sigma,X)</math> is downward-closed, and therefore it is a simplicial complex too; it is a sub-complex of {{mvar|X}}. * Because {{mvar|X}} is simplicial, there is a set isomorphism between <math display="inline">\operatorname{Lk}(\sigma,X)</math> and the set <math>X_{\sigma} := \{\rho \in X\text{ such that }\sigma \subseteq \rho\}</math>: every <math display="inline">\tau\in \operatorname{Lk}(\sigma,X)</math> corresponds to <math display="inline">\tau \cup \sigma</math>, which is in <math>X_{\sigma}</math>.
== Link and star{{Anchor|star}} == A concept closely related to the link is the '''star'''.
Given an abstract simplicial complex {{mvar|X}} and any face <math display="inline">\sigma \in X</math>,<math display="inline">V(X)</math>, its '''star''' <math display="inline">\operatorname{St}(\sigma,X)</math> is a set containing every face <math display="inline">\tau \in X</math> such that <math display="inline"> \tau\cup \sigma</math> is a face of {{mvar|X}}. In the special case in which {{mvar|X}} is a 1-dimensional complex (that is: a graph), <math display="inline">\operatorname{St}(v,X)</math> contains all edges <math display="inline">\{u,v\}</math> for all vertices <math display="inline">u</math> that are neighbors of <math display="inline">v</math>. That is, it is a graph-theoretic star centered at ''<math display="inline">u</math>''.
Given a geometric simplicial complex {{mvar|X}} and any face <math display="inline">\sigma \in X</math>, its '''star''' <math display="inline">\operatorname{St}(\sigma,X)</math> is a set containing every face <math display="inline">\tau \in X</math> such that there is a simplex in <math display="inline"> X</math> having both <math display="inline">\sigma </math> and <math display="inline"> \tau</math> as faces: <math display="inline">\operatorname{St}(\sigma,X) := \{\tau\in X: \exists \rho\in X: \tau, \sigma \text{ are faces of }\rho \}</math>. In other words, it is the closure of the set <math display="inline">\{\rho\in X: \sigma \text{ is a face of }\rho \}</math> -- the set of simplices having <math display="inline">\sigma </math> as a face.
So the link is a subset of the star. The star and link are related as follows:
* For any <math display="inline">\sigma\in X</math>, <math display="inline">\operatorname{Lk}(\sigma,X) = \{\tau\in \operatorname{St}(\sigma,X): \tau\cap \sigma=\emptyset \}</math>. <ref name=":0" />{{Rp|page=3}} * For any <math display="inline">v\in V(X)</math>, <math display="inline">\operatorname{St}(v,X) = v \star \operatorname{Lk}(v,X) </math>, that is, the star of <math display="inline">v</math> is the cone of its link at <math display="inline">v</math>.<ref name=":1" />{{Rp|page=20}}
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.<gallery class="center" widths="350" heights="112"> File:Simplicial complex star.svg|A {{color|#fc3|vertex}} and its {{color|#093|'''star'''}}. </gallery>
== See also ==
* Vertex figure - a geometric concept similar to the simplicial link.
==References== {{Reflist}} Category:Geometry