TOPCOM User Manual
Jörg Rambau
Universtät Bayreuth, Germany
May 7, 2008
1 What is TOPCOM?
TOPCOM is a collection of clients to compute Triangulations Of Point Configurations
and Oriented Matroids, resp.
The algorithms use only combinatorial data of the point configuration as is given by
its oriented matroid. Some basic commands for computing and manipulating oriented
matroids can also be accessed by the user.
2 How do I use TOPCOM?
All programs read the input from stdin and write the result to stdout so that you can
pipe the results to the next command.
A point configuration is given by a matrix (enclosed in square brackets) whose
columns (enclosed in square brackets) are the homogeneous coordinates (seperated by
commas) of the points in the configuration. A square could be specified as
follows.
[[0,0,1],[0,1,1],[1,0,1],[1,1,1]]

You may specify generators of the combinatorial symmetry of a point configuration
as permutations of the vertex numbers. The symmetry of the square reads as follows
(observe that the count starts at 0!):
[[3,2,1,0],[2,3,0,1],[0,2,1,3]]

3 Commands
The following commands are provided:

points2chiro
 Computes the chirotope of a point configuration.

chiro2dual
 Computes the dual of a chirotope.

chiro2circuits
 Computes the circuits of a chirotope.

chiro2cocircuits
 Computes the circuits of a chirotope.

cocircuits2facets
 Computes the facets of a set of cocircuits.

points2facets
 Computes the facets of a point configuration.

points2nflips
 Computes the number of flips of a point configurations and the
seed triangulation.

points2flips
 Computes all flips of a point configurations and the seed
triangulation.

chiro2placingtriang
 Computes the placing triangulation of a chirotope given
by the numbering of the elements.

points2placingtriang
 dto. for point configurations.

chiro2finetriang
 Computes a fine (i.e., using all vertices) triangulation by
placing and pushing.

points2finetriang
 dto. for point configurations.

chiro2triangs
 Computes all triangulations of a chirotope that are connected by
bistellar flips to the regular triangulations.

points2triangs
 dto. for point configurations.

chiro2ntriangs
 Computes the number of all triangulations of a chirotope that
are connected by bistellar flips to the regular triangulations.

points2ntriangs
 dto. for point configurations.

chiro2finetriangs
 Computes all fine triangulations of a chirotope that are
connected by bistellar flips to a fine seed triangulation.

points2finetriangs
 dto. for point configurations.

chiro2nfinetriangs
 Computes the number of all fine triangulations of a
chirotope that are connected by bistellar flips to a fine seed triangulation.

points2nfinetriangs
 dto. for point configurations.

chiro2alltriangs
 Computes all triangulations of a chirotope.

points2alltriangs
 dto. for point configurations.

chiro2nalltriangs
 Computes the number of all triangulations of a chirotope.

points2nalltriangs
 dto. for point configurations.

chiro2allfinetriangs
 Computes all fine triangulations of a chirotope.

points2allfinetriangs
 dto. for point configurations.

chiro2nallfinetriangs
 Computes the number of all fine triangulations of a
chirotope.

points2nallfinetriangs
 dto. for point configurations.

cube d
 Computes the vertices and symmetry generators of a dcube.

cyclic n d
 Computes the vertices and symmetry generators of the cyclic
dpolytope with n vertices.

cross d
 Computes the vertices of the ddimensional crosspolytope.

hypersimplex k d
 Computes the vertices and symmetry generators of the kth
hypersimplex in dimension d.

santos_triang
 Computes the point configuration, the symmetry, and the Santos
triangulation (without flips).
4 Command Line Options
The following command line options are supported:
Options controlling the overall behaviour of clients

d
 Debug.

h
 Print a usage message.

v
 Verbose.
Options controlling what is computed

cardinality [k]
 Count only triangulations with exactly k simplices.

checktriang
 Check seed triangulation.

flipdeficiency
 Check triangulations for flip deficiency.

frequency [k]
 Check every kth triangulation for regularity and stop if one is
found.

heights
 Output a height vector for every regular triangulation (implies
regular).

noinsertion
 Never add a point that is unused in the seed triangulation.

reducepoints
 Try to greedily minimize the number of vertices used; keep a
global upper bound on the current minimal number of vertices and do not
accept triangulations with more vertices.

regular
 Search for regular triangulations only (checked liftings are w.r.t. the
last homogeneous coordinate, e.g., last coordinates all ones is fine); note that
this may reduce the effort of exploration, since regular triangulations are
connected by themselves.

nonregular
 Output nonregular triangulations only; note that this does not
reduce the effort of exploration, since nonregular triangulations are in
general not connected by themselves.
Options controlling the internals of the clients

chirocache [n]
 Set the chirotope cache to n elements.

localcache [n]
 Set the cache for local operations.

memopt
 Save memory by using caching techniques.

soplex
 Use soplex instead of cdd for regularity checks (unstable).
4.1 Options for warm starts from previous calculations

dump
 Write intermediate results into a file.

dumpfile [dumpfilename]
 Write intermediate results into file dumpfilename
(default: TOPCOM.dump).

dumpfrequency [k]
 Dump the results of each kth BFS round

dumprotations [k]
 Dump into k different rotating files.

read
 Read intermediate results from a file.

readfile [readfilename]
 Read intermediate results from file dumpfilename
(default: TOPCOM.dump).
5 Examples
In the subdirectory examples you find some example inputs for TOPCOM routines. For
example,
points2chiro < lattice_3_3.dat

outputs the sign string of the chirotope of the sublattice of integer points (i,j) with
i,j = 0, 1, 2.
points2chiro < lattice_3_3.dat  chiro2ntriangs

or
points2ntriangs < lattice_3_3.dat

yields the number of triangulations that are connected to the regular ones by
bistellar flips.
points2ntriangs r affine < moae_testfile

counts all regular triangulations of the “mother of all examples”, two nested triangles
in the plane.
points2chiro < lattice_3_3.dat  chiro2nalltriangs

yields the number of all triangulations via a branch & bound algorithm. For large
examples this routine may take a lot of time but since it branches in a DFS manner it
does not take a lot of memory.
The example r12.chiro is the chirotope of the oriented matroid R12 with
disconnected realization space, constructed by Jürgen RichterGebert. If you want to
compute, e.g., a placing triangulation of R12 then type
chiro2placingtriang < r12.chiro

The facets of a 4cube can be computed by
cube 4  points2chiro  chiro2dual  \
chiro2circuits  cocircuits2facets

but be aware of the fact that this is not an efficient way of computing facets of a
point configuration. It is, however, numerically stable because rational arithmetics is
used.
Finally, you can check the Santos triangulation by
santos_triang  points2nflips v memopt checktriang

Recall that the options mean:

v
 verbose;

memopt
 save memory;

checktriang
 check seed triangulation.
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