TOPCOM
TOPCOM (current version is preview-version 1.2.0-h)
TOPCOM is a package for computing Triangulations Of Point Configurations and Oriented Matroids. It was very much inspired by the maple program PUNTOS, which was written by Jesus de Loera. TOPCOM is entirely written in C++, so there is a significant speed up compared to PUNTOS.
TOPCOM version 1.2.0-h is available prior to the release of 1.2.0 in order to collect possible glitches. It is the version that was used for the results in several papers. I have decided to remove older versions from the download area. If you need an older version, please, contact me.
TOPCOM version 1.2.0-h contains all the new algorithms for the enumeration of
- all triangulations
- all cocircuits
- all circuits
of a point or vector configuration up to symmetry. It uses lock-free multi-threading techniques to speed up computations on multi-core machines. Therefore, a compiler supporting the c++17 standard is needed to build it. TOPCOM 1.2.0-h comes with an accompanying preprint on symmetric lexicographic subset reverse search describing the algorithmic progress and some computational results. The single most important impact on the usage of TOPCOM is that now the extension-based enumeration of all triangulations is both faster and more memory-efficient than the flip-based enumeration of the flip-graph component of the regular triangulations. Moerover, this new method can enumerate symmetric triangulations directly.
The complete ChangeLog can be downloaded here.
The package MPTOPCOM by Skip Jordan, Michael Joswig, and Lars Kastner for the parallel enumeration of triangulations based, among other things, on the TOPCOM C++-library enumerates all so-called subregular triangulations (a subset of all triangulations containing the regular ones) in parallel. It is able to solve larger instances than TOPCOM versions 0.x.x. See their paper for details.
If TOPCOM turns out to be useful to you, I would be happy to hear about why. If you expected TOPCOM to be useful to you and it was not, then I would be happy to hear about why not.
At any rate: I received several questions about how to properly cite TOPCOM in research papers. I prefer a citation to the following publications depending on the version of TOPCOM you used:
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Versions after 1.0.0:
• For the math:
Jörg Rambau. Symmetric lexicographic symmetric-subset reverse search for the enumeration of circuits, cocircuits, and triangulations up to symmetry. arXiv:2607.05967 [math.CO], 2026. (Here is a bibtex-entry.)
• For the code:
Jörg Rambau. TOPCOM - Triangulations of Point Configurations and Oriented Matroids, Zenodo (2026), https://doi.org/10.5281/zenodo.23239982. (Here is a bibtex-entry.)
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Versions before 1.0.0:
Jörg Rambau. TOPCOM: Triangulations of Point Configurations and Oriented Matroids, Mathematical Software - ICMS 2002 (Cohen, Arjeh M. and Gao, Xiao-Shan and Takayama, Nobuki, eds.), World Scientific (2002), pp. 330-340, available as ZIB Report 02-17. (Here is a bibtex-entry.)
Remark: TOPCOM contains a program to produce a point configuration and one of its triangulation without flips as constructed by Francisco Santos in his great paper
Francisco Santos: A Point Configuration whose Space of Triangulations is Disconnected,
Journal of the American Mathematical Society 13 No. 3 (2000), 611-637.
TOPCOM can check the triangulation and compute the number of flips, which in fact is zero.