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SimplicialComplexes :: faces

faces -- the i-faces of a simplicial complex

Synopsis

Description

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

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

i1 : R = ZZ[a..f]

o1 = R

o1 : PolynomialRing
i2 : 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)

o2 = | def aef bdf bcf acf cde bce abe acd abd |

o2 : SimplicialComplex
i3 : faces(-1,D)

o3 = | 1 |

             1       1
o3 : Matrix R  <--- R
i4 : faces(0,D)

o4 = | a b c d e f |

             1       6
o4 : Matrix R  <--- R
i5 : faces(1,D)

o5 = | ab ac ad ae af bc bd be bf cd ce cf de df ef |

             1       15
o5 : Matrix R  <--- R
i6 : faces(2,D)

o6 = | abd abe acd acf aef bce bcf bdf cde def |

             1       10
o6 : Matrix R  <--- R
i7 : fVector D

o7 = HashTable{-1 => 1}
               0 => 6
               1 => 15
               2 => 10

o7 : 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

Ways to use faces :

For the programmer

The object faces is a method function with options.