SimplicialComplexes -- simplicial complexes

A simplicial complex on a set of vertices is a collection of subsets D of these vertices, such that if F is in D, then every subset of F is also in D. In Macaulay2, the vertices are variables in a polynomial ring, and each subset is represented as a product of the corresponding variables.

There is a bijection between simplicial complexes and squarefree monomial ideals. This package exploits this correspondence by using commutative algebra routines to perform most of the necessary computations.

This package includes the following functions:

• boundary -- boundary operator
• buchbergerComplex -- Buchberger complex of a monomial ideal
• chainComplex(SimplicialComplex) -- The chain complex of boundary maps.
• coefficientRing(SimplicialComplex) -- get the coefficient ring
• dim(SimplicialComplex) -- dimension of a simplicial complex
• dual(SimplicialComplex) -- the Alexander dual of a simplicial complex
• faces -- the i-faces of a simplicial complex
• facets -- the facets of a simplicial complex
• fVector -- the f-vector of a simplicial complex
• HH SimplicialComplex -- Compute the homology of a simplicial complex.
• ideal(SimplicialComplex) -- the ideal of minimal nonfaces (the Stanley-Reisner ideal)
• isPure -- whether the facets are equidimensional
• label -- labels with monomials the faces of simplicial complex
• lyubeznikComplex -- Simplicial complex supporting the Lyubeznik resolution of a monomial ideal
• monomialIdeal(SimplicialComplex) -- the monomial ideal of minimal nonfaces (the Stanley-Reisner ideal)
• ring(SimplicialComplex) -- get the associated ring of an object
• simplicialComplex -- create a simplicial complex
• simplicialChainComplex -- The chain complex of boundary maps.
• superficialComplex -- Simplicial complex supporting a superficial resolution of a monomial ideal