Regular triangulations of the 3-dilated 3-simplex

You can find all regular triangulations of the 3-dilated 3-simplex here. There are 910,974,879 triangulations in total.

These output files were produced by mptopcom v1.0. The triangulations are denoted up to group action by the linear symmetry group on the simplex.

After extracting, every line in the output files corresponds to a triangulation orbit. E.g.:

> xzcat 3d3.result.000.xz |head
(0.0):[0->20,4:{{0,3,9,19}}]  gkz: [27,0,0,27,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,27] can(gkz): [27,0,0,27,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,27]
(2.0):[0->20,4:{{3,5,9,19},{0,5,9,19},{0,3,5,19}}]  gkz: [18,0,0,18,0,27,0,0,0,18,0,0,0,0,0,0,0,0,0,27] can(gkz): [27,0,0,18,0,0,0,0,0,18,0,0,0,0,27,0,0,0,0,18]
(2.1):[0->20,4:{{0,5,9,19},{5,9,17,19},{0,5,17,19},{3,5,9,17},{0,3,5,17}}]  gkz: [18,0,0,12,0,27,0,0,0,18,0,0,0,0,0,0,0,18,0,15] can(gkz): [18,0,0,18,0,27,0,0,0,12,0,0,0,0,0,0,0,0,18,15]

Every line contains three things:

  1. A representative from the triangulation orbit in curly braces,
  2. The gkz vector of this representative in square brackets after the gkz,
  3. The lex largest gkz vector from this triangulation orbit in square brackets after the can(gkz).

Note This is the old mptopcom format, the output format is changed in mptopcom v1.1.

This part assumes that you are familiar with computations on large clusters, so we will not go into detail on these here and just focus on the example at hand.

To run this example yourself on a cluster, you will need a build of mptopcom and the following three files:

  1. The cluster script.
  2. 3d3.dat The input file.
  3. perl script for validating checkpoint files produced by mptopcom.

These files are contained in 3d3_cluster.tar.

The cluster script is highly automatized: After 9 hours it will trigger mptopcom to write a checkpoint. It will then automatically validate the checkpoint and, if valid, rename the output files with a number. You can submit several jobs at once, the option -hold_jid will make any new job wait for the previously submitted jobs. After the previous job(s) finish, the new job will automatically be started by your cluster scheduler.

To produce the massive gkz vectors from the output of mptopcom, we used the following perl script. It demonstrates how to deal with .xz files in polymake and how to use massive chains.

  • dequivalence.txt
  • Last modified: 2019/09/19 12:50
  • by lkastner