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SimplicialComplexes :: SimplicialComplexes

SimplicialComplexes -- simplicial complexes

Description

SimplicialComplexes is a package for manipulating 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:

Authors

Version

This documentation describes version 1.1 of SimplicialComplexes.

Source code

The source code from which this documentation is derived is in the file SimplicialComplexes.m2.

Exports

  • Functions and commands
    • boundary -- boundary operator
    • buchbergerComplex -- Buchberger complex of a monomial ideal
    • faces -- the i-faces of a simplicial complex
    • facets -- the facets of a simplicial complex
    • fVector -- the f-vector of a simplicial complex
    • isPure -- whether the facets are equidimensional
    • label -- labels with monomials the faces of simplicial complex
    • link -- link of a face in a simplicial complex
    • lyubeznikComplex -- Simplicial complex supporting the Lyubeznik resolution of a monomial ideal
    • simplicialChainComplex (missing documentation)
    • simplicialComplex -- create a simplicial complex
    • superficialComplex -- Simplicial complex supporting a superficial resolution of a monomial ideal
  • Methods
    • boundary(SimplicialComplex) -- the boundary simplicial complex of D
    • boundary(ZZ,SimplicialComplex) -- the boundary map from i-faces to (i-1)-faces
    • chainComplex(SimplicialComplex) (missing documentation)
    • coefficientRing(SimplicialComplex) -- get the coefficient ring
    • dim(SimplicialComplex) -- dimension of a simplicial complex
    • dual(SimplicialComplex) -- the Alexander dual of a simplicial complex
    • faces(ZZ,SimplicialComplex), see faces -- the i-faces of a simplicial complex
    • facets(SimplicialComplex), see facets -- the facets of a simplicial complex
    • fVector(SimplicialComplex), see fVector -- the f-vector of a simplicial complex
    • HH SimplicialComplex (missing documentation)
    • HH_ZZ SimplicialComplex (missing documentation)
    • HH_ZZ(SimplicialComplex,Ring) (missing documentation)
    • homology(Nothing,SimplicialComplex) (missing documentation)
    • homology(Nothing,SimplicialComplex,Ring) (missing documentation)
    • homology(SimplicialComplex,Ring) (missing documentation)
    • ideal(SimplicialComplex) -- the ideal of minimal nonfaces (the Stanley-Reisner ideal)
    • isPure(SimplicialComplex), see isPure -- whether the facets are equidimensional
    • label(SimplicialComplex,List), see label -- labels with monomials the faces of simplicial complex
    • link(SimplicialComplex,RingElement), see link -- link of a face in a simplicial complex
    • monomialIdeal(SimplicialComplex) -- the monomial ideal of minimal nonfaces (the Stanley-Reisner ideal)
    • net(SimplicialComplex) (missing documentation)
    • ring(SimplicialComplex) -- get the associated ring of an object
    • simplicialChainComplex(List,SimplicialComplex) (missing documentation)
    • SimplicialComplex == SimplicialComplex (missing documentation)
  • Symbols
    • faceIdeal (missing documentation)