The class of all embedded complexes, not necessarily simplicial or compact or equidimensional. These are complexes with coordinates assigned to their vertices.
The following functions return complexes:
simplex -- Simplex in the variables of a polynomial ring
boundaryCyclicPolytope -- The boundary complex of a cyclic polytope with standard projective space vertices
fullCyclicPolytope -- The full cyclic polytope with moment curve vertices
convHull -- The convex hull
hull -- The positive hull
boundaryOfPolytope -- The boundary of a polytope
newEmptyComplex -- Generates an empty complex.
idealToComplex -- The complex associated to a reduced monomial ideal
dualize -- The dual of a co-complex.
complement -- The complement of a co-complex.
complex -- Make a complex from a list of faces
complexFromFacets -- Make a complex from a list of facets
embeddingComplex -- The complex containing a subcomplex
For examples see the documentation of these functions.
The data stored in a complex C:
C.simplexRing, the polynomial ring of vertices of C.
C.grading, is C.simplexRing.grading, a matrix with the coordinates of the vertices of C in its rows.
C.facets, a list with the facets of C sorted into lists by dimension.
C.dim, the dimension of the complex.
C.isSimp, a Boolean indicating whether C is simplicial.
C.isEquidimensional, a Boolean indicating whether C is equidimensional.
If not just the facets but the faces of C a known (e.g., after computed with fc) then the following data is present:
C.fc, a ScriptedFunctor with the faces of C sorted and indexed by dimension.
C.fvector, a List with the F-vector of C.
The following may be present (if known due to creation of C or due to calling some function):
C.dualComplex, the dual co-complex of C in the sense of dual faces of a polytope. See dualize.
C.isPolytope, a Boolean indicating whether C is a polytope.
C.polytopalFacets, a List with the boundary faces of the polytope C.
C.complementComplex, the complement co-complex of C (if C is a subcomplex of a simplex). See complement.