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KustinMiller :: faces

faces -- Returns the faces of a complex

Synopsis

Description

Compute the faces of a simplicial Complex C. The result is a List L. The j-th entry L#j of L is a list of the faces of C of dimension j.

If the second argument d is specificed then a list of the faces of dimension d is returned.

i1 : K=QQ;
i2 : R=K[x_0..x_4]

o2 = R

o2 : PolynomialRing
i3 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0)

o3 = ideal (x x , x x , x x , x x , x x )
             0 1   1 2   2 3   3 4   0 4

o3 : Ideal of R
i4 : C=idealToComplex I

o4 = {x  x  , x  x  , x  x  , x  x  , x  x  }
       2  4    0  3    0  2    1  3    1  4

o4 : complex with 5 facets on the vertices x  x  x  x  x  
                                            0  1  2  3  4
i5 : faces(C)

o5 = {{{}}, {x  , x  , x  , x  , x  }, {x  x  , x  x  , x  x  , x  x  , x  x 
              1    0    3    2    4      1  4    1  3    2  4    0  2    0  3
     ------------------------------------------------------------------------
     }, {}, {}, {}}

o5 : List

i6 : K=QQ;
i7 : R=K[x_0..x_4]

o7 = R

o7 : PolynomialRing
i8 : I=ideal(x_0*x_1,x_2*x_3*x_4)

o8 = ideal (x x , x x x )
             0 1   2 3 4

o8 : Ideal of R
i9 : C=idealToComplex I

o9 = {x  x  x  , x  x  x  , x  x  x  , x  x  x  , x  x  x  , x  x  x  }
       0  2  3    1  2  3    1  3  4    1  2  4    0  3  4    0  2  4

o9 : complex with 6 facets on the vertices x  x  x  x  x  
                                            0  1  2  3  4
i10 : faces(C)

o10 = {{{}}, {x  , x  , x  , x  , x  }, {x  x  , x  x  , x  x  , x  x  , x 
               4    1    2    0    3      0  4    2  4    2  3    3  4    1
      -----------------------------------------------------------------------
      x  , x  x  , x  x  , x  x  , x  x  }, {x  x  x  , x  x  x  , x  x  x  ,
       2    1  3    0  2    0  3    1  4      0  2  4    0  3  4    1  2  4  
      -----------------------------------------------------------------------
      x  x  x  , x  x  x  , x  x  x  }, {}, {}}
       1  2  3    1  3  4    0  2  3

o10 : List
i11 : faces(C,1)

o11 = {x  x  , x  x  , x  x  , x  x  , x  x  , x  x  , x  x  , x  x  , x  x 
        0  4    2  4    2  3    3  4    1  2    1  3    0  2    0  3    1  4
      -----------------------------------------------------------------------
      }

o11 : List

See also

  • Complex -- The class of all simplicial complexes.
  • Face -- The class of faces of simplicial complexes.
  • facets -- The facets of a simplicial complex.
  • idealToComplex -- Compute the Stanley-Reisner complex.

Ways to use faces :

  • faces(Complex)
  • faces(Complex,ZZ)