Using FatGHoL

FatGHoL comes with a front-end script to compute the (co)homology of the moduli space of marked smooth Riemann surfaces (using Kontsevich’ “graph homology” complex).

The front-end script is called; you can invoke it with the --help command line option to get a recap of its functionality:

$ ./ --help
usage: mgn [-h] [-D [DEBUG]] [-l LOGFILE] [-o OUTFILE] [-s CHECKPOINT_DIR]
           [-u] [-v] [-V]
           ACTION [ARG [ARG ...]]


      graphs G N
        Generate the graphs occurring in M_{g,n}.

      homology G N
        Print homology ranks of M_{g,n}.

      latex G N [-s DIR] [-o FILE]
        Read the listings of M_{g,n} fatgraphs (from directory DIR)
        and output a pretty-print catalogue of the graphs as LaTeX documents.

      valences G N
        Print the vertex valences occurring in M_{g,n} graphs.

        Start an interactive PyDB shell.

        Run internal code tests and report failures.

positional arguments:
  ACTION                Action to perform, see above.
  ARG                   Arguments depend on the actual action, see above.

optional arguments:
  -h, --help            show this help message and exit
  -D [DEBUG], --debug [DEBUG]
                        Enable debug features:
                            * pydb -- run Python debugger if an error occurs
                            * profile -- dump profiler statistics in a .pf file.
                            Several features may be enabled by separating them
                            with a comma, as in '-D pydb,profile'.
  -l LOGFILE, --logfile LOGFILE
                        Redirect log messages to the named file
                            (by default log messages are output to STDERR).
  -o OUTFILE, --output OUTFILE
                        Save results into named file.
                        Directory for saving computation state.
  -u, --afresh          Do NOT restart computation from the saved state in checkpoint directory.
  -v, --verbose         Print informational and status messages as the computation goes on.
  -V, --version         show program's version number and exit

That should be pretty much self-explanatory; however a bit more detail is given below.

The graphs action

With ./ graphs 0 4 (for example) you can print a list of all the fatgraphs having genus 0 and 4 boundary cycles. If you want to save the list to a file, use the -o option, followed by the file name.

The list of fatgraphs is also saved in directory M0, in several .list files, depending on the number of vertices. For instance, the M0,4-MgnGraphsIterator3.list file is the one collecting fatgraphs with 3 vertices.

The homology action

With ./ homology 0 4 (for example), you can print the Betti numbers of the moduli space of smooth marked Riemann surfaces having genus 0 and 4 marked points. If you want to save the result to a file, use the -o option, followed by the file name.

Internally, the homology function uses the graphs function, so it generates all by-products of that function. In particular, graphs lists are generated and saved in the checkpoint directory (M0, in this example).

In addition to graph lists, the boundary operator matrices are computed and saved in the checkpoint directory. After their ranks have been computed, they are saved as well, so invoking the homology action with a fully-populated checkpoint directory gives the result almost istantaneously.

The latex action

This reads the contents of an existing checkpoint directory and generates a LaTeX report on all the graphs: what graphs are there, their automorphisms and markings, etc.

To save the report into a file, use the -o option followed by the file name, e.g.:

./ latex 0 4 -o report.tex

The name of the checkpoint directory is automatically generated from the parameters G and N; use the -s option to use a different directory.

The valence action

This prints the valences of vertices of fatgraphs with given genus and number of boundary cycles.

As this is a very simple computation, nothing is saved to the checkpoint directory.

Checkpoint directory

This is a directory where FatGHoL saves result of computationally-expensive functions. When FatGHoL is invoked again at a later time, it loads the results from the checkpoint directory instead of calculating them again; this results in a substantial speedup. However, you can use the -u command-line option to tell FatGHoL to ignore the contents of a checkpoint directory.

The name of the checkpoint directory is automatically generated from the parameters G and N; use the -s option to use a different directory.

There is no way of avoiding that FatGHoL creates a checkpoint directory and populates it.