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
• Optional inputs:
  • useFaceClass => ..., default value false, Option to return faces in the class Face
• Outputs:
  • a matrix, with one row, whose entries are squarefree monomials representing the faces of dimension i of D

Description

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

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