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\title{TOPCOM User Manual\\ \normalsize version 1.2.0}
\author{J\"org Rambau\\Universt\"at Bayreuth, Germany}



\begin{document}
\maketitle

\begin{verbatim}
\end{verbatim}

\section{Introductory Remark}
\label{sec:remark}

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
\begin{itemize}
\item enumeration of all triangulations,
\item enumeration of circuits from points,
\item enumeration of cocircuits from points.
\end{itemize}
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 \texttt{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
(\url{https://polymake.org/doku.php/mptopcom}).  The enumeration of
\emph{all} triangulations, however, can be done with similar memory
consumption now by the TOPCOM client \texttt{points2(n)alltriangs}.
This client, for the time being, is the fastest method in TOPCOM to
enumerate all triangulations of point configurations.


\section{What is TOPCOM?}
\label{sec:whatis}

TOPCOM is a collection of clients to compute \emph{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.


\section{How do I use TOPCOM?}
\label{sec:howto}

All programs read the input from \texttt{stdin} and write the result to
\texttt{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.

\begin{verbatim}
    [[0,0,1],[0,1,1],[1,0,1],[1,1,1]]
\end{verbatim}

You may specify generators of the combinatorial symmetry of a point
configuration as permutations of the vertex numbers. The symmetry
generators of the square read as follows (observe that the count
starts at 0):

\begin{verbatim}
    [[3,2,1,0],[2,3,0,1],[0,2,1,3]]
\end{verbatim}

Some algorithms also read in a given seed triangulation given by a set
of sets of numbers, where elements of a set are enclosed in curly
brackets and separated by commas.  For example:

\begin{verbatim}
    {{0,1,2},{1,2,3}}
\end{verbatim}

If you want to give a seed but no symmetries, specify \verb|[]| for
the symmetries.

Finally, some algorithms observe prescribed symmetries for the
resulting triangulations.  To generate only triangulations of the
square symmetric w.r.t.{} the diagonal of the first quadrant, add the
following line:

\begin{verbatim}
    [[3,1,2,0]]
\end{verbatim}

If you want to prescribe symmetries but no seed, specify \verb|{}| for
the seed.

\section{Commands}
\label{sec:commands}

The following commands are provided:
\begin{description}
\item[\texttt{points2prettyprint}] (New.) Displays the point and their
  symmetry generators in a more readable form.
\item[\texttt{points2chiro}] Computes the chirotope of a point configuration.

\item[\texttt{chiro2dual}] Computes the dual of a chirotope.
\item[\texttt{chiro2circuits}] Computes the circuits of a chirotope.
\item[\texttt{points2circuits}] Dto.\ for point configurations
  (using a faster method).
\item[\texttt{chiro2cocircuits}] Computes the cocircuits of a chirotope.
\item[\texttt{points2cocircuits}] Dto.\ for point configurations
  (using a faster method).
\item[\texttt{cocircuits2facets}] Computes the facets of a set of cocircuits.
\item[\texttt{points2facets}] Computes the facets of a point configuration.

\item[\texttt{points2gale}] Computes a Gale transform of a point configuration.
\item[\texttt{chiro2circuits}] Computes the circuits of a point configuration.
\item[\texttt{chiro2cocircuits}] Computes the cocircuits of a point configuration.
\item[\texttt{points2facets}] Computes the facets of a point configuration.

\item[\texttt{points2nflips}] Computes the number of flips of a point
  configurations and the seed triangulation.
\item[\texttt{points2flips}] Computes all flips of a point configurations and the
  seed triangulation.

\item[\texttt{chiro2placingtriang}] Computes the placing triangulation of a
  chirotope given by the numbering of the elements.
\item[\texttt{points2placingtriang}] Dto.\ for point configurations.

\item[\texttt{chiro2finetriang}] Computes a fine (i.e., using all vertices)
  triangulation by placing and pushing.
\item[\texttt{points2finetriang}] Dto.\ for point configurations.

\item[\texttt{chiro2triangs}] 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.
\item[\texttt{points2triangs}] Dto.\ for point configurations.

\item[\texttt{chiro2ntriangs}] 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.

\item[\texttt{points2ntriangs}] Dto.\ for point configurations.

\item[\texttt{chiro2finetriangs}] Computes all fine triangulations
  (the ones that use all the points sometimes called ``full'') of a
  chirotope that are connected by bistellar flips to a fine seed
  triangulation.
\item[\texttt{points2finetriangs}] Dto.\ for point configurations.

\item[\texttt{chiro2nfinetriangs}] Computes the number of all fine
  triangulations of a chirotope that are connected by bistellar flips to a fine
  seed triangulation.
\item[\texttt{points2nfinetriangs}] Dto.\ for point configurations.

\item[\texttt{chiro2alltriangs}] Computes all triangulations of a chirotope.
\item[\texttt{points2alltriangs}] Dto.\ for point configurations.

\item[\texttt{chiro2nalltriangs}] Computes the number of all triangulations of a
  chirotope.
\item[\texttt{points2nalltriangs}] Dto.\ for point configurations.

\item[\texttt{chiro2allfinetriangs}] Computes all fine triangulations
  (sometimes called ``full'') of a chirotope.
\item[\texttt{points2allfinetriangs}] Dto.\ for point configurations.
  
\item[\texttt{chiro2nallfinetriangs}] Computes the number of all fine
  triangulations of a chirotope.
\item[\texttt{points2nallfinetriangs}] Dto.\ for point configurations.

\item[\texttt{chiro2mintriang}] Computes a triangulation of a
  chirotope with a minimum number of simplices.
\item[\texttt{points2mintriang}] Dto.\ for point configurations.
  
\item[\texttt{B\_S n}] Computes the vertices and symmetry generators of
  the permutation polytope of the symmetric group of degree~$n$, also
  known as the Birkhoff polytope.
\item[\texttt{B\_A n}] Computes the vertices and symmetry generators of
  the permutation polytope of the alternating group of degree~$n$, also
  known as the even Birkhoff polytope.
\item[\texttt{B\_D n}] Computes the vertices and symmetry generators of
  the permutation polytope of the dihedral group of degree~$n$.
\item[\texttt{B\_S\_center n}] Computes \texttt{B\_S n} with an additional
  center point.
\item[\texttt{B\_A\_center n}] Computes \texttt{B\_A n} with an additional
  center point.
\item[\texttt{B\_D\_center n}] Computes \texttt{B\_D n} with an additional
  center point.
\item[\texttt{cube d}] Computes the vertices and symmetry generators of a
  $d$-cube.
\item[\texttt{cyclic n d}] Computes the vertices and symmetry generators of
  the cyclic $d$-polytope with $n$ vertices.
\item[\texttt{cross d}] Computes the vertices and symmetry generators
  of the $d$-dimensional cross-polytope.
\item[\texttt{kDn k n}] Computes the vertices and symmetry generators
  of the $k$-times dilated $n$-simplex with all interior lattice points.
\item[\texttt{Dnxk n k}] Computes the vertices and symmetry generators
  of $k$ copies of the $n$-simplex.
\item[\texttt{lattice n m}] Computes the $nm$ two-dimensional lattice
  points with non-negative coordinates at most $(n - 1, m - 1)$ and
  their symmetry generators.
\item[\texttt{hypersimplex d k $[\ell]$}] Computes the vertices 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~$\ell$.
\item[\texttt{santos\_triang}] Computes the point configuration, the symmetry,
  and the Santos triangulation (without flips).
\end{description}



\section{Command Line Options}
\label{sec:options}

The following command line options are supported.  Note that not all
options are sensible for all clients.

\subsection*{Options concerning input/output from files}
\begin{description}
\item[\texttt{-i [filename]}:] read input from [filename] instead of
  stdin.
\end{description}


\subsection*{Options concerning output of information}
\begin{description}
\item[\texttt{-h} or \texttt{-}\texttt{-help}] Print a usage message.

\item[\texttt{-d}] Debug.
   
\item[\texttt{-v}] Verbose.
  
\item[\texttt{-}\texttt{-reportfrequency [n]} ] In enumerations,
  report intermediate results after (at least) \texttt{[n]}th
  discovered nodes.

\item[\texttt{-}\texttt{-heights}] Output a height vector for every
  regular triangulation (implies \texttt{-}\texttt{-regular}).
  
\item[\texttt{-}\texttt{-flips}] Output all flips in terms of IDs of
  adjacent triangulations.  (Can be used to generate the flip graph.)
  
\item[\texttt{-}\texttt{-asy}] 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 \texttt{asy}
  (\url{https://asymptote.sourceforge.io}) using the
  \texttt{asy}-library \texttt{Combinatorial\_Geometry.asy} and the
  \LaTeX-macroes in \texttt{triangbook\_macroes.sty} inside
  \texttt{share/asy/}.
\end{description}

\subsection*{Options for checking input}
\begin{description}
\item[\texttt{-}\texttt{-checktriang}] Check seed triangulation.
\end{description}

\subsection*{Options for reporting properties of discovered
  triangulations:}
\begin{description}
\item[\texttt{-}\texttt{-flipdeficiency}] Check triangulations for
  flip deficiency during flip-graph exploration.
  
\item[\texttt{-}\texttt{-findregular $[k]$}] (New form.) Check every
  $k$-th triangulation for regularity and stop if a regular one is
  found during flip-graph exploration.
\end{description}

\subsection*{Options concerning which triangulations are output (no
  influence on flip-graph exploration)}
\begin{description}
  \item[\texttt{-}\texttt{-noorbitcount}] Only count symmetry classes,
    not the total number.

  \item[\texttt{-}\texttt{-cardinality $[k]$}] Count/output only
    triangulations with exactly $k$ simplices.

  \item[\texttt{-}\texttt{-maxcardinality $[k]$}] Count/oputput only
    triangulations with at most $k$ simplices.

  \item[\texttt{-}\texttt{-unimodular}] (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 \texttt{points2nalltriangs}.
    
  \item[\texttt{-}\texttt{-requirepoint $[k]$}] Count/oputput only
    triangulations with simplices containing point~$k$ (currently only
    supported by \texttt{points2(n)alltriangs}).  Note that in this
    case only symmetries stabilizing that point are considered.

  \item[\texttt{-}\texttt{-nonregular}] 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.  
\end{description}


\subsection*{Options concerning which triangulations are explored}
\begin{description}
  
\item[\texttt{-}\texttt{-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 flip-graph exploration, since regular triangulations
  are connected by themselves.
    
\item[\texttt{-}\texttt{-regularearly}] 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.

\item[\texttt{-}\texttt{-noinsertion}] Never flip-in a point that is
  unused in the seed triangulation.
  
\item[\texttt{-}\texttt{-reducepoints}] Try to greedily minimize the
  number of vertices used while flipping; keep a global upper bound on
  the current minimal number of vertices and do not accept
  triangulations with more vertices.

\item[\texttt{-}\texttt{-keepcard}] Never change the cardinality of
  triangulations by flipping.
\end{description}


\subsection*{Options concerning symmetries}
\begin{description}
  
\item[\texttt{-}\texttt{-affinesymmetries}] (New form.) Assume that
  the symmetries are affine, in particular, that they conserve
  regularity.
  
\item[\texttt{-}\texttt{-isometricsymmetries}] (New.) Assume that the
  symmetries are isometric, in particular, that they preserve volume.
  
\item[\texttt{-}\texttt{-nosymmetries}] Ignore the symmetries.
\item[\texttt{-}\texttt{-triangsymmetries}] (New.) Observe the
  prescribed symmetries for triangulations in the result.
\item[\texttt{-}\texttt{-notriangsymmetries}] (New; currently the
  default.) Ignore the prescribed symmetries for triangulations in the
  result (option supported for backwards-compatibility.
\end{description}


\subsection*{Options controlling the internals of the clients}
\begin{description}
\item[\texttt{-}\texttt{-memopt}] Save memory by using caching
  techniques.
\item[\texttt{-}\texttt{-usegkz}] (New.) Use GKZ vectors as a finger
  print in symmetry handling (only for points with isometric
  symmetries).
\item[\texttt{-}\texttt{-usenaivesymmetries}] (New.) Use naive full
  traversal of all symmetries for symmetry handling.

\item[\texttt{-}\texttt{-useswitchtables}] (New.) Use
  Jordan-Joswig-Kastner switch tables for symmetry handling.
  
\item[\texttt{-}\texttt{-usesymmetrytables}] (New.) Use tables of
  classified symmetries for symmetry handling.  Obsolete, since
  slower than the other options.
  
\item[\texttt{-}\texttt{-symtables [n]}](New.)  Use \texttt{[n]}
  symtables for preprocessing symmetries.  Obsolete, since slower than
  the other options.
  
\item[\texttt{-}\texttt{-preprocesschiro}] (New.) Preprocess the
  chirotope (default for \texttt{points2[n]alltriangs}).
  
\item[\texttt{-}\texttt{-preprocesspoints}] (New.) Heuristically
  transform points (only relevant for (co)circuit enumeration).
  
\item[\texttt{-}\texttt{-simpidxsymmetries}] (New.) Preprocess a
  representation of the symmetry group on simplex indices (only
  relevant for triangulation enumeration).
  
\item[\texttt{-}\texttt{-userandomorder}] (New.) Sort simplices in
  preprocessed index table randomly (only for points with isometric
  symmetries).
  
\item[\texttt{-}\texttt{-usevolumeorder}] (New.) Sort simplices in
  preprocessed index table by volume (only for points with isometric
  symmetries).
  
\item[\texttt{-}\texttt{-usevolumes}] (New.) Use volumes to check
  extendability of partial triangulations (only for points with
  isometric symmetries).
  
\item[\texttt{-}\texttt{-fullextensioncheck}] (New.) Put more effort
  in the check of extendability of a partial triangulation.
  
\item[\texttt{-}\texttt{-noextensioncheck}] (New.) Skip the check of
  extendability of a partial triangulation.
  
\item[\texttt{-}\texttt{-extensioncheckfirst}] (New.) Check
  extendability prior to symmetry.

\item[\texttt{-}\texttt{-preprocesspoints}] Preprocess the coordinate
  matrix of the points (slightly useful for (co-)circuit enumeration)

\item[\texttt{-}\texttt{-chirocache $[n]$}] Set the chirotope cache to
  $n$ elements.

\item[\texttt{-}\texttt{-localcache $[n]$}] Set the cache for local
  operations.

\item[\texttt{-}\texttt{-qsopt\_ex}] Use \texttt{QSopt\_ex} for
  regularity checks (not thread-safe).
   
\item[\texttt{-}\texttt{-soplex}] Use \texttt{soplex} for regularity
  checks (requires separate installation of \texttt{soplex}).
\end{description}

\subsection*{Options concerning multi-threading}

\begin{description}
\item[\texttt{-}\texttt{-parallelenumeration}] Use multiple threads
  for enumeration.

\item[\texttt{-}\texttt{-workbuffercontrol}] (Currently the default.)
  Control the interrupt of workers by size of the current workbuffer.

\item[\texttt{-}\texttt{-noworkbuffercontrol}] Control the interrupt of
  workers by node budgets.

\item[\texttt{-}\texttt{-parallelsymmetries}] Use multiple threads
  only locally for symmetry checks.

\item[\texttt{-}\texttt{-threads [n]}] Use \texttt{[n]} threads (if possible).

\item[\texttt{-}\texttt{-minnodebudget [n]}] Let each thread process
  at least \texttt{[n]} nodes (to avoid multi-threading overhead).

\item[\texttt{-}\texttt{-maxnodebudget [n]}] Let each thread process
  at most \texttt{[n]} nodes (to avoid thread starving).

\item[\texttt{-}\texttt{-scalenodebudget [n]}] Scale the default node
  budget by \texttt{[n]} percent (n integer)

\item[\texttt{-}\texttt{-minworkbuffer [n]}] (Currently unused.) Try
  to keep the work buffer above \texttt{[n]} nodes (to balance
  overhead and thread starving).

\item[\texttt{-}\texttt{-maxworkbuffer [n]}] (Currently unused.) Try
  to keep the work buffer below \texttt{[n]} node (to balance
  overhead and thread starving).

\end{description}


\subsection*{Options for warm starts from previous calculations}
These options currently work for the clients
\texttt{points2(n)triangs}, \texttt{chiro2(n)triangs},
\texttt{points2(n)alltriangs}, \texttt{chiro2(n)alltriangs}, and
\texttt{points2(n)(co)circuits}.
\label{sec:options-warm-starts}
\begin{description}
\item[\texttt{-}\texttt{-dump}] Write intermediate results into a
  file; a symbolic link with the basename will be placed pointing to
  the most recent actual dump file.
\item[\texttt{-}\texttt{-dumprotations $[k]$}] Dump into $k$ different
  rotating files indicated by numerical postfixes.
\item[\texttt{-}\texttt{-dumpfile $[$dumpfilename$]$}] Write
  intermediate results into files with basename \texttt{dumpfilename}
  (default: \texttt{TOPCOM.dump}); a symbolic link with the basename
  will be placed pointing to the most recent actual dump file.
\item[\texttt{-}\texttt{-dumpfrequency $[k]$}] Dump the results at
  each $k$th ``object'' (what object means, depends on the kind of
  enumeration)
\item[\texttt{-}\texttt{-read}] Read intermediate results from a file.
\item[\texttt{-}\texttt{-readfile $[$readfilename$]$}] Read
  intermediate results from file \texttt{dumpfilename} (default:
  \texttt{TOPCOM.dump}).
\end{description}


\section{Examples}
\label{sec:examples}

In the subdirectory \texttt{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:

\begin{verbatim}
  points2prettyprint < cube_3.dat
\end{verbatim}

It prints the matrix of point coordinates and the symmetry generators in
tabular form.

Or:

\begin{verbatim}
  points2chiro < lattice_3_3.dat
\end{verbatim}

outputs the sign string of the chirotope of the sub-lattice of integer
points $(i,j)$ with $i, j = 0, 1, 2$.

\begin{verbatim}
  points2chiro < lattice_3_3.dat | chiro2ntriangs
\end{verbatim}

or 

\begin{verbatim}
  points2ntriangs < lattice_3_3.dat
\end{verbatim}

yields the number of triangulations that are connected to the regular
ones by bistellar flips.

\begin{verbatim}
  points2ntriangs --regular --affinesymmetries < moae.dat
\end{verbatim}

counts all regular triangulations of the ``mother of all examples'',
two nested triangles in the plane.

\begin{verbatim}
  points2nalltriangs < lattice_3_3.dat
\end{verbatim}

yields the number of all triangulations via a new algorithm
\emph{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 \texttt{r12.chiro} is the chirotope of the oriented
matroid $R12$ with disconnected realization space, constructed by
J\"urgen Richter-Gebert.  If you want to compute, e.g., a placing
triangulation of $R12$ then type

\begin{verbatim}
  chiro2placingtriang < r12.chiro
\end{verbatim}

The facets of a $4$-cube can be computed by

\begin{verbatim}
  cube 4 | points2facets
\end{verbatim}

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

\begin{verbatim}
  santos_triang | points2nflips -v --memopt --checktriang
\end{verbatim}

Recall that the options mean:

\begin{description}
\item[\texttt{-v}] verbose;
\item[\texttt{-}\texttt{-memopt}] save memory;
\item[\texttt{-}\texttt{-checktriang}] check seed triangulation.
\end{description}

A new result as of version 1.0.10 is that TOPCOM can now
enumerate \emph{all} triangulations of larger point configurations
like the $4$-cube in reasonable time.  The speed depends on the
computational environment and the options:

\begin{verbatim}
  cube 4 | points2nalltriangs -v --parallelenumeration
\end{verbatim}

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:

\begin{verbatim}
  chiro2nalltriangs -i icosahedron.chiro -v 
\end{verbatim}

Note the newly supported alternative syntax \texttt{-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:

\begin{verbatim}
  cube 4 | points2ncircuits -v --parallelenumeration
\end{verbatim}

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:

\begin{verbatim}
  cube 4 | points2ntriangs --parallelsymmetries \
  --usegkz --isometricsymmetries
\end{verbatim}

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
\texttt{Combinatorial\_Geometry.asy} and
\texttt{triangbook\_macroes.sty} (to be found under \texttt{share/asy}
in the TOPCOM home directory) must be present in the working directory
where \texttt{TOPCOM\_graphics.asy} is compiled via \texttt{asy}:

\begin{verbatim}
  lattice 2 3 | points2nalltriangs -v --parallelenumeration --asy
  asy TOPCOM_graphics.asy
\end{verbatim}

The point configuration can be found in \texttt{TOPCOM\_points.pdf}
(see Figure~\ref{fig:points}).
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=0.2\linewidth,keepaspectratio=true]{TOPCOM_points.pdf}
  \end{center}
  \caption{The points}\label{fig:points}
\end{figure}
The enumeration tree can be found in \texttt{TOPCOM\_tree\_graph.pdf}
(see Figure~\ref{fig:tree}).
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=\linewidth,keepaspectratio=true]{TOPCOM_tree_graph.pdf}
  \end{center}
  \caption{The enumeration tree with classified nodes -- here with
    three threads and workbuffercontrol}\label{fig:tree}
\end{figure}
Here, a \emph{clover icon} means a triangulation in a new symmetry
class, a \emph{recycling icon} means a node in an old symmetry class,
a \emph{stop-sign icon} means a deadend with no possible admissible
lex-larger extension, a \emph{bulb icon} means an early detected
deadend with an uncoverable interior facet, and a
\emph{lightning-sign icon} means that no further extension option
can prevent an uncoverable interior facet.  Statistics can be found in
\texttt{TOPCOM\_statistics.pdf} (see Figure~\ref{fig:stats}).
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=0.5\linewidth,keepaspectratio=true]{TOPCOM_statistics.pdf}
  \end{center}
  \caption{\label{fig:stats}The statistics of classified nodes}
\end{figure}


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