Before describing the topaz specific aspects of data representation in polymake, we would like to stress that the general concept of data representation in polymake holds for the topaz application as well. You can find more about that here.
A simplicial complex is represented as a list of its inclusion
maximal faces, the facets, encoded as their ordered sets of vertices.
The topaz standard requires the list of facets to be
free of redundancies and the
vertices to be numbered consecutively starting with 0.
The empty complex is represented as a single empty set.
The standard section for
the facets of a simplicial complex is
FACETS and the standard section for the vertices is
called VERTICES.
Therefore nearly all topaz clients expect a file to have
a non-empty section FACETS containing inclusion maximal sets
of integer ranging from 0 to n-1.
In case you are not sure if your FACETS section
conforms to the topaz standard, try the
facets_consitency client to check and maybe correct
your file. If you want your input auto-corrected, choose any not
standard name for the facets of your complex and use the
-auto option.
Since the vertex numbering is restricted,
you are welcome to define VERTEX_LABELS. This
section is an array, containing a string of
characters (letters, digits,
underscore, punctuation characters, etc. but no blanks) for each
vertex. You may choose any distinct name
for each vertex. As the default topaz uses the
vertex numbering as labels.
Any operation identifying vertices (e.g. the non-disjoint union of two complexes) relies on these labels. The labels also help you to keep track of special vertices throughout a series of constructions. Clients producing new complexes keep the old labels whenever appropriate, or construct the new labels from the old ones basing on a well-defined scheme.
All clients constructing a new simplicial complexes from scratch
produce a new file containing a
FACETS section according to the topaz
standard. If a client constructs a new simplicial complex from
others, it still produces a new file containing a
FACETS section, yet it labels the vertices
depending on the original labeling.
Most of the topaz clients consider a subcomplex as a
complex in its own right and therefore produce a new file containing a
FACETS and a VERTEX_LABELS section
(e.g. k_skeleton).
However sometimes a subcomplex is viewed as a property of the original complex
(e.g. boundary_of_pseudo_manifold). In this case, the
client produces a new section within the original file and
does not adjust the numbering. Therefore the vertices of the
subcomplex have the same numbering as in
FACETS, rather than being renumbered from 0 to n-1. If
you would like to view the subcomplex as a separate complex, use
the extract_subcomplex client.