Making an abstract simplicial complex -- information about the basic constructors

An abstract simplicial complex on a finite set is a family of subsets that is closed under taking subsets. The elements in the abstract simplicial complex are called its faces. The faces having cardinality 1 are its vertices and the maximal faces (order by inclusion) are its facets. Following the combinatorial conventions, every nonempty abstract simplicial complex has the empty set as a face.

In this package, a simplicial complex is represented by its Stanley–Reisner ideal. The vertices are identified with a subset of the variables in a polynomial ring and each face is identified with the product of the corresponding variables. A nonface is any subset of the variables that does not belong to the simplicial complex and each nonface is again identified with product of variables. The Stanley-Reisner ideal of a simplicial complex is generated by monomials corresponding to its nonfaces.

Basic constructors for abstract simplicial complexes

• simplicialComplex(List) -- make an abstract simplicial complex from a list of faces
• simplicialComplex(MonomialIdeal) -- make a simplicial complex from its Stanley–Reisner ideal
• SimplicialComplex -- the class of all abstract simplicial complexes
• isWellDefined(SimplicialComplex) -- whether underlying data is uncontradictory

Classic examples of abstract simplicial complexes

• simplexComplex(ZZ,PolynomialRing) -- make the simplex as an abstract simplicial complex
• bartnetteSphereComplex(PolynomialRing) -- make a non-polytopal 3-sphere with 8 vertices
• bjornerComplex(PolynomialRing) -- make a shellable 2-polyhedron with 6 vertices
• dunceHatComplex(PolynomialRing) -- make a realization of the dunce hat with 8 vertices
• poincareSphereComplex(PolynomialRing) -- make a homology 3-sphere with 16 vertices
• nonPiecewiseLinearSphereComplex(PolynomialRing) -- make a non-piecewise-linear 5-sphere with 18 vertices
• rudinBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 41 facets
• grunbaumBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 29 facets
• zieglerBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 10 vertices and 21 facets
• smallManifold -- get a small manifold from the Frank Lutz database

Other operations producing abstract simplicial complexes

• prune(SimplicialComplex) -- make a minimal presentation of an abstract simplicial complex
• inducedSubcomplex(SimplicialComplex,List) -- make the induced simplicial complex on a subset of vertices
• dual(SimplicialComplex) -- make the Alexander dual of an abstract simplicial complex
• link(SimplicialComplex,RingElement) -- make the link of a face in an abstract simplicial complex
• skeleton(ZZ,SimplicialComplex) -- make a new simplicial complex generated by all faces of a bounded dimension
• star(SimplicialComplex,RingElement) -- make the star of a face
• SimplicialComplex * SimplicialComplex -- make the join for two abstract simplicial complexes
• wedge(SimplicialComplex,SimplicialComplex,RingElement,RingElement) -- make the wedge sum of two abstract simplicial complexes