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]]
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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]]
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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 d-cube.
-
cyclic n d
- Computes the vertices and symmetry generators of the cyclic
d-polytope with n vertices.
-
cross d
- Computes the vertices of the d-dimensional cross-polytope.
-
hypersimplex k d
- Computes the vertices and symmetry generators of the k-th
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 k-th 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 non-regular triangulations only; note that this does not
reduce the effort of exploration, since non-regular 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
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outputs the sign string of the chirotope of the sub-lattice of integer points (i,j) with
i,j = 0, 1, 2.
points2chiro < lattice_3_3.dat | chiro2ntriangs
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or
points2ntriangs < lattice_3_3.dat
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yields the number of triangulations that are connected to the regular ones by
bistellar flips.
points2ntriangs -r --affine < moae_testfile
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counts all regular triangulations of the “mother of all examples”, two nested triangles
in the plane.
points2chiro < lattice_3_3.dat | chiro2nalltriangs
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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 Richter-Gebert. If you want to
compute, e.g., a placing triangulation of R12 then type
chiro2placingtriang < r12.chiro
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The facets of a 4-cube can be computed by
cube 4 | points2chiro | chiro2dual | \
chiro2circuits | cocircuits2facets
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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
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Recall that the options mean:
-
-v
- verbose;
-
--memopt
- save memory;
-
--checktriang
- check seed triangulation.