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faces -- the i-faces of a simplicial complex

Synopsis

Usage:
faces(i,D)
Inputs:
- i, an integer, the dimension of the faces
- D, a simplicial complex
Outputs:
- a matrix, with one row, whose entries are squarefree monomials representing the faces of dimension i of D

Description

In Macaulay2, every simplicial complex is equipped with a polynomial ring, and the matrix of i-faces is defined over this ring.

i1 : loadPackage "SimplicialComplexes";

This triangulation of the real projective plane has 6 vertices, 15 edges and 10 triangles.

i2 : R = ZZ[a..f]

o2 = R

o2 : PolynomialRing

i3 : D = simplicialComplex monomialIdeal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f,
                                                   b*c*d,b*d*e,b*e*f,c*d*f,c*e*f)

o3 = | def aef bdf bcf acf cde bce abe acd abd |

o3 : SimplicialComplex

i4 : faces(-1,D)

o4 = | 1 |

             1       1
o4 : Matrix R  <--- R

i5 : faces(0,D)

o5 = | a b c d e f |

             1       6
o5 : Matrix R  <--- R

i6 : faces(1,D)

o6 = | ab ac ad ae af bc bd be bf cd ce cf de df ef |

             1       15
o6 : Matrix R  <--- R

i7 : faces(2,D)

o7 = | abd abe acd acf aef bce bcf bdf cde def |

             1       10
o7 : Matrix R  <--- R

i8 : fVector D

o8 = HashTable{-1 => 1}
               0 => 6
               1 => 15
               2 => 10

o8 : HashTable

To avoid repeated computation, the matrix of i-faces is cached at D.cache.faces#i. This function will use this value if it has already been computed.

See also

SimplicialComplexes -- simplicial complexes
facets -- the facets of a simplicial complex
boundary -- boundary operator
fVector -- the f-vector of a simplicial complex

Ways to use `faces` :

faces(ZZ,SimplicialComplex)