Join, Star and Link are important operations on simplectical complexes. This page discusses a FriCAS implementation of these.
Join
The code includes a join operation like this:
simplicialJoin : (a : %, b : %) > % 
The simplectial join of A and B is a simplectical complex with the vertices and simplexes defined as follows:
Vertices
Are disjoint union of vertices of A and vertices of B
vertexSet(simplicialJoin(A,B)) = vertexSet(A) vertexSet(B)
If A and B have common vertices then the vertices of B will be renamed to make then different from A.
simplexes
A subset of A B is a simplex of simplicialJoin(A,B) if and only if it is:
 a simplex of A or
 a simplex of B or
 the union of a simplex of A and a simplex of B
Properties of simplicialJoin
May be geometrically thought of as a cone over A with tip of shape B.
simplicialJoin is commutative and associative.
dim(simplicialJoin(A,B)) = dim(A) + dim(B) + 1
Example 1
Here is an example of the join of 2 lines: [1, 2] and [3, 4] This gives 4 points and the following simplexes:

but each simplex contains all its sub simplexes so the join is just:
(1, 2, 3, 4)
We can calculate this using FriCAS like this: Note: the implementation of simplicialJoin refactors the indices of the second operand, if necessary, to make sure the points being joined are disjoint. 

The above example is just one simplex so here is an example where the operands have multiple simplexes. 
Each operand has two simplexes. The result has all 4 combinations of these simplexes. 

Star
We can calculate this using FriCAS like this: 

Link
We can calculate this using FriCAS like this: 
