Figure 1: The points
The new version 1.2.0 now supports checkpointing for enumerations of triangulations, circuits, and cocircuits.
Versions starting at 1.1.0 use vastly improved enumeration methods for
enumeration of all triangulations,
enumeration of circuits from points,
enumeration of cocircuits from points.
Moreover, lock-free multithreading is used on demand for enumeration and/or symmetry processing.
For really large problem instances (co-dimension above 13), the exploration-by-flipping of the regular component of the flip-graph in TOPCOM’s points2(n)triangs takes a lot of memory. The exploration of the (in general smaller) subregular component of the flip graph can be done with less memory consumption by using MPTOPCOM by Jordan, Joswig, and Kastner (https://polymake.org/doku.php/mptopcom). The enumeration of all triangulations, however, can be done with similar memory consumption now by the TOPCOM client points2(n)alltriangs. This client, for the time being, is the fastest method in TOPCOM to enumerate all triangulations of point configurations.
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.
All programs read the input from stdin and write the result to stdout so that you can pipe the results to the next command. Up to four units of data input are read: a point configuration; the symmetries of the configuration; the symmetries required for triangulations; a seed triangulation, where, e.g., the flip-graph exploration starts.
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]]
The symmetries of a configuration with n elements are given by a list of permutations, where a permutation is given as a list of the images of 0, 0,1,…,n-1. Each permutation must represent a combinatorial symmetry of the configuration. That is: it must map signed cocircuits to signed cocircuits. The symmetry generators of the square read as follows (observe that the count starts at 0):
[[3,2,1,0],[2,3,0,1],[0,2,1,3]]
The required symmetries for the resulting triangulations are given in the same way. They are only observed if the command-line option --triangsymmetries is used. To generate only triangulations of the square symmetric w.r.t. the diagonal of the first quadrant, add the following line:
[[3,1,2,0]]
A seed triangulation can be given by a set of sets of numbers, where elements of a set are enclosed in curly brackets and separated by commas. For example, for the square:
{{0,1,2},{1,2,3}}
If you want to prescribe a seed but no (required) symmetries, specify [] for the
symmetries.
Before, inbetween, and after the data units comments may be added: Whenever the hash symbol “#” is detected, the rest of the line is ignored. Note that inside the data units comments are not allowed.
The following commands are provided:
(New.) Displays the point and their symmetry generators in a more readable form.
Computes the chirotope of a point configuration.
Computes the dual of a chirotope.
Computes the circuits of a chirotope.
Dto. for point configurations (using a faster method).
Computes the cocircuits of a chirotope.
Dto. for point configurations (using a faster method).
Computes the facets of a set of cocircuits.
Computes a Gale transform of a point configuration.
Computes the circuits of a point configuration.
Computes the cocircuits of a point configuration.
Computes the facets of a point configuration.
Computes the vertices of a point configuration.
Computes the number of flips of a point configurations and the seed triangulation.
Computes all flips of a point configurations and the seed triangulation.
Computes the placing triangulation of a chirotope given by the numbering of the elements.
Dto. for point configurations.
Computes a full (i.e., using all points) triangulation by placing and pushing (obsolete; use –full instead).
Dto. for point configurations.
Computes all triangulations of a chirotope that are connected by bistellar flips to the seed, which is a regular triangulation if no seed is given in the input file.
Dto. for point configurations.
Computes the number of all triangulations of a chirotope that are connected by bistellar flips to the seed, which is a regular triangulation if no seed is given in the input file.
Dto. for point configurations.
Computes all full triangulations (i.e., using all points) of a chirotope that are connected by bistellar flips to a full seed triangulation (obsolete; use –full instead).
Dto. for point configurations.
Computes the number of all full triangulations (i.e., using all points) of a chirotope that are connected by bistellar flips to a full seed triangulation (obsolete; use –full instead).
Dto. for point configurations.
Computes all triangulations of a chirotope.
Dto. for point configurations.
Computes the number of all triangulations of a chirotope.
Dto. for point configurations.
Computes all full triangulations (i.e., using all points) of a chirotope (obsolete; use –full instead).
Dto. for point configurations.
Computes the number of all full triangulations (i.e., using all points) of a chirotope (obsolete; use –full instead).
Dto. for point configurations.
Computes a triangulation of a chirotope with a minimum number of simplices.
Dto. for point configurations.
Computes the points and symmetry generators of the permutation polytope of the symmetric group of degree n, also known as the Birkhoff polytope.
Computes the points and symmetry generators of the permutation polytope of the alternating group of degree n, also known as the even Birkhoff polytope.
Computes the points and symmetry generators of the permutation polytope of the dihedral group of degree n.
Computes B_S n with an additional center point.
Computes B_A n with an additional center point.
Computes B_D n with an additional center point.
Computes the points and symmetry generators of a d-cube.
Computes the points and symmetry generators of the cyclic d-polytope with n vertices.
Computes the points and symmetry generators of the d-dimensional cross-polytope.
Computes the points and symmetry generators of the k-times dilated n-simplex with all interior lattice points.
Computes the points and symmetry generators of k copies of the n-simplex.
Computes the nm two-dimensional lattice points with non-negative coordinates at most (n-1,m-1) and their symmetry generators.
Computes the points and symmetry generators of the k-th hypersimplex in dimension d. A third parameter makes it the S-hypersimplex with coordinate sums equal to k or ℓ.
Computes the point configuration, the symmetry, and the Santos triangulation (without flips).
The following command line options are supported. Note that not all options are sensible for all clients.
read input from [filename] instead of stdin.
Ignore the symmetries in the input.
(New.) Observe the prescribed symmetries in the input for triangulations in the result.
(New.) Observe the prescribed seed in the input for flip-graph exploration.
Check seed triangulation.
(New.) Check given symmetries by testing invariance on cocircuits.
Print a usage message.
Debug.
Verbose.
In enumerations, report intermediate results after (at least) [n]th discovered nodes.
Output a height vector for every regular triangulation (only in connection with --[find]regular).
(New.) Output the GKZ-vector for every triangulation..
Output all flips in terms of IDs of adjacent triangulations. (Can be used to generate the flip graph.)
When checking for flip-graph connectivity, output flip-paths to regular triangulations in terms of sequences of triangulations and flips.
Write asymptote graphics commands into file (in rank-3 triangulations, points are drawn as well). The graphics contains a view of the point configuration (only in rank 3), the enumeration tree with a classification of enumeration nodes into solutions, non-canonical nodes, deadends, and early detected deadends, as well a statistics file showing a histogram of enumeration node types. The output file has to be processed by the computer graphics compiler asy (https://asymptote.sourceforge.io) using the asy-library Combinatorial_Geometry.asy and the LaTeX-macroes in triangbook_macroes.sty inside share/asy/.
Check triangulations for flip deficiency during flip-graph exploration.
(New form.) Check for regularity and stop if a regular one is found during flip-graph exploration.
Only count symmetry classes, not the total number.
Count/output only triangulations with exactly k simplices.
Count/oputput only triangulations with at most k simplices.
(New.) Output unimodular triangulations only; while this does not reduce the effort of flip graph exploration, since unimodular triangulations are in general not connected by themselves, it does reduce the effort of extension graph exploration linke in points2nalltriangs.
Count/oputput only triangulations with simplices containing point k (currently only supported by points2(n)alltriangs). Note that in this case only symmetries stabilizing that point are considered.
Output non-regular triangulations only; note that this does not reduce the effort of flip-graph exploration, since non-regular triangulations are in general not connected by themselves.
Output non-subregular triangulations only, i.e., those that cannot be gkz-increasingly flipped to a regular triangulation.
Output non-superregular triangulations only, i.e., those that cannot be gkz-decreasingly flipped to a regular triangulation.
(New.) Output non-regular triangulations only if they are not connected to the flip-graph component of the regular triangulations. (Each such triangulation constitutes a significant mathematical result.)
(New.) Output non-regular triangulations only if they are possibly not connected to the flip-graph component of the regular triangulations. This is checked by lex-GKZ-monotone flipping w.r.t. a random permutation.
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 flip-graph exploration, since regular triangulations are connected by themselves. In particular, if the optional given seed is non-regular, the exploration will stop immediately with no triangulations counted, since non-regular triangulations are not followed.
Search for regular partial triangulations only; note that this may reduce the effort of extension-graph exploration, since no non-regular partial triangulation can be completed to a regular triangulation.
Search for full triangulations (i.e., using all points) only.
Never flip-in a point that is unused in the seed triangulation.
Try to greedily minimize the number of points used while flipping; keep a global upper bound on the current minimal number of points and do not accept triangulations with more points.
Never change the cardinality of triangulations by flipping.
(New form.) Assume that the symmetries are affine, in particular, that they conserve regularity.
(New.) Assume that the symmetries are isometric, in particular, that they preserve volume.
(New; currently the default.) Ignore the prescribed symmetries for triangulations in the result (option supported for backwards-compatibility).
Save memory by using caching techniques.
(New.) Use an iterative version of the depth-first-search enumeration algorithms (avoids potential call-stack overflows on small machines).
(New.) Use a recursive version of the depth-first-search enumeration algorithms (default).
(New.) Use GKZ vectors as a finger print in symmetry handling (only for points with isometric symmetries).
(New.) Use naive full traversal of all symmetries for symmetry handling.
(New.) Use Jordan-Joswig-Kastner switch tables for symmetry handling.
(New.) Use tables of classified symmetries for symmetry handling. Obsolete, since slower than the other options.
(New.) Use [n] symtables for preprocessing symmetries. Obsolete, since slower than the other options.
(New.) Preprocess the chirotope (default for points2[n]alltriangs).
(New.) Heuristically transform points.
(New.) Preprocess a representation of the symmetry group on simplex indices (only relevant for triangulation enumeration).
(New.) Sort simplices in preprocessed index table randomly (only for points with isometric symmetries).
(New.) Sort simplices in preprocessed index table by volume (only for points with isometric symmetries).
(New.) Use volumes to check extendability of partial triangulations (only for points with isometric symmetries).
(New.) Put more effort in the check of extendability of a partial triangulation.
(New.) Skip the check of extendability of a partial triangulation.
(New.) Check extendability prior to symmetry.
Set the chirotope cache to n elements.
Set the cache for local operations.
Use QSopt_ex for regularity checks (not thread-safe).
Use soplex for regularity checks (requires separate installation of soplex).
Use multiple threads for enumeration.
(Currently the default.) Control the interrupt of workers by size of the current workbuffer.
Control the interrupt of workers by node budgets.
Use multiple threads only locally for symmetry checks.
Use [n] threads (if possible).
Let each thread process at least [n] nodes (to avoid multi-threading overhead).
Let each thread process at most [n] nodes (to avoid thread starving).
Scale the default node budget by [n] percent (n integer)
(Currently unused.) Try to keep the work buffer above [n] nodes (to balance overhead and thread starving).
(Currently unused.) Try to keep the work buffer below [n] node (to balance overhead and thread starving).
These options currently work for the clients points2(n)triangs, chiro2(n)triangs, points2(n)alltriangs, chiro2(n)alltriangs, and points2(n)(co)circuits.
Write intermediate results into a file; a symbolic link with the basename will be placed pointing to the most recent actual dump file.
Dump into k different rotating files indicated by numerical postfixes.
Write intermediate results into files with basename dumpfilename (default: TOPCOM.dump); a symbolic link with the basename will be placed pointing to the most recent actual dump file.
Dump the results at each kth “object” (what object means, depends on the kind of enumeration)
Read intermediate results from a file.
Read intermediate results from file dumpfilename (default: TOPCOM.dump).
In the subdirectory examples you find some example inputs for TOPCOM routines.
If you want to see the input in a prettier way, then you can type the following:
points2prettyprint < cube_3.dat
It prints the matrix of point coordinates and the symmetry generators in tabular form.
Or:
points2chiro < lattice_3_3.dat
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
or
points2ntriangs < lattice_3_3.dat
yields the number of triangulations that are connected to the regular ones by bistellar flips.
points2ntriangs --regular --affinesymmetries < moae.dat
counts all regular triangulations of the “mother of all examples”, two nested triangles in the plane.
points2nalltriangs < lattice_3_3.dat
yields the number of all triangulations via a new algorithm symmetric lex-subset reverse-search on partial triangulations. It is mostly faster than flip-graph exploration and much faster than in older versions of TOPCOM; moreover, 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
The facets of a 4-cube can be computed by
cube 4 | points2facets
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.
Most notably, you can check the Santos triangulation by
santos_triang | points2nflips -v --memopt --checktriang
Recall that the options mean:
verbose;
save memory;
check seed triangulation.
A new result as of version 1.0.10 is that TOPCOM can now enumerate all triangulations of larger point configurations like the 4-cube in reasonable time. The speed depends on the computational environment and the options:
cube 4 | points2nalltriangs -v --parallelenumeration
uses verbose output and multi-threaded symmetry processing with load-balancing based on work buffer control, i.e., the workers stop and push the open nodes into the job list whenever the master’s job list contains fewer elements than there are workers.
Since not every point configuration can be represented in rational coordinates, it is useful to enumerate triangulations from the chirotope data. This way you can compute, e.g., all triangulations of the regular icosahedron:
chiro2nalltriangs -i icosahedron.chiro -v
Note the newly supported alternative syntax -i [filename] for the specification of the input file.
The enumeration of circuits and cocircuits up to symmetry is now a non-trivial part of TOPCOM. For example:
cube 4 | points2ncircuits -v --parallelenumeration
enumerates the symmetry classed of all the circuits of the 4-cube.
Moreover, the clients to explore the flip graph for points can now use the GKZ vector as a fingerprint to detect known triangulation classes in case all symmetries are isometric. This is faster, but needs some more memory for storing all the known GKZ vectors. Moreover, a basic multi-threaded symmetry check can be used. You can activate all of this by:
cube 4 | points2ntriangs --parallelsymmetries \ --usegkz --isometricsymmetries
As an example for the graphics output the 2-by-3 lattice in dimension two is presented. Graphics output is supported by the enumeration of all triangulations, circuits and cocircuits. The files Combinatorial_Geometry.asy and triangbook_macroes.sty (to be found under share/asy in the TOPCOM home directory) must be present in the working directory where TOPCOM_graphics.asy is compiled via asy:
lattice 2 3 | points2nalltriangs -v --parallelenumeration --asy asy TOPCOM_graphics.asy
The point configuration can be found in TOPCOM_points.pdf (see Figure 1).
The enumeration tree can be found in TOPCOM_tree_graph.pdf (see Figure 2).
Here, a clover icon means a triangulation in a new symmetry class, a recycling icon means a node in an old symmetry class, a stop-sign icon means a deadend with no possible admissible lex-larger extension, a bulb icon means an early detected deadend with an uncoverable interior facet, and a lightning-sign icon means that no further extension option can prevent an uncoverable interior facet. Statistics can be found in TOPCOM_statistics.pdf (see Figure 3).
