Reduced simplicial homology groups H0,...,Hd (integer coefficients), listed in increasing dimension order.
Each group G is encoded as a sequence ({ (t1 m1) ... (tn mn) } f) of non-negative integers,
with t1 > t2 > ... > tn > 1, plus an extra non-negative integer f.
The group G is isomorphic to (Z/t1)m_1 × ... × (Z/tn)m_n × Zf,
where Z0 is the trivial group.
Representatives of cycle groups, listed in increasing dimension order.
The first component in each dimension is a matrix of integer coefficients,
the second component is a vector of faces. To obtain the chains, one must multiply (symbolically)
both components.
An orientation of the facets of an ORIENTED_PSEUDO_MANIFOLD, such that the induced orientations
of a common ridge of two neighboring facets cancel each other out. Each facet is marked with true,
iff the orientation is given by the increasing vertex ordering and is marked with false, if the
orientation is obtained from the increasing vertex ordering by a transposition.
A finite representation of the fundamental group.
The fundamental group is represented as a pair of an integer,
the number of generators, and a list of relations. The generators are numbered
consecutively starting with zero. A relation is encoded as a list of pairs,
each pair consisting of a generator and its exponent.
You may use the fundamental2gap method to produce a GAP file.
Labels of the generators of the FUNDAMENTAL_GROUP.
The labels can be chosen freely. If the FUNDAMENTAL_GROUP is computed
by the polymake system, the generators correspond to the edges of the
complex. Hence they are labeled g followed by the vertices of the edge, e.g.
g3_6 corresponds to the edge {36}.
A Morse matching is a reorientation of the arcs in the Hasse diagram of a simplicial complex
such that at most one arc incident to each face is reoriented (matching condition) and the
resulting orientation is acyclic (acyclicity condition). Morse matchings capture the main
structure of discrete Morse functions, see
Robin Forman: Morse Theory for Cell-Complexes,
Advances in Math., 134 (1998), pp. 90-145.
This property is computed by one of two heuristics. The default heuristic is
a simple greedy algorithm (greedy). The alternative is to use a canceling algorithm
due to Forman (cancel). Note that the computation of a Morse matching of largest
size is NP-hard. See
Michael Joswig, Marc E. Pfetsch: Computing Optimal Morse Matchings
Coordinates for the vertices of the simplicial complex, such that the complex is embedded without crossings in some Re.
Vector (x1 .. xe) represents a point in Euclidean e-space.
The nodes of the mixed graph are the nodes of the primal GRAPH and
the DUAL_GRAPH. Additional to the primal and dual edges, there is
an edge between a primal and a dual node iff the primal node represents
a vertex of the corresponding facet of the dual node.
Signature of a geometric simplicial complex embedded in the integer lattice.
Like DUAL_GRAPH_SIGNATURE,
but only simplices with odd normalized volume are counted.
organized as a directed graph. Each node corresponds to some face
of the simplical complex. It is represented as the list of vertices
comprising the face. The outgoing arcs point to the containing faces
of the next dimension. An artificial top node is added to represent
the entire complex.