Provided by: regina-normal_4.95-1build1_amd64 
NAME
tricensus - Form a census of 3-manifold triangulations
SYNOPSIS
tricensus [ -t, --tetrahedra=tetrahedra ] [ -2, --dim2 ] [ -b, --boundary | -i, --internal | -B,
--bdryfaces=triangles ] [ -o, --orientable | -n, --nonorientable ] [ -f, --finite | -d, --ideal ] [ -m,
--minimal | -M, --minprime | -N, --minprimep2 ] [ -s, --sigs | -c, --subcontainers ] [ -p, --genpairs |
-P, --usepairs ] output-file
tricensus --help
DESCRIPTION
Forms a census of all 3-manifold or 2-manifold triangulations that satisfy some set of conditions.
These conditions are specified using various command-line arguments. The only condition that you must
provide is the number of tetrahedra, but there are many other options available.
The default behaviour is to enumerate 3-manifold triangulations. If you wish to enumerate 2-manifold
triangulations instead, you must pass --dim2.
Each triangulation will be output precisely once up to combinatorial isomorphism. Invalid 3-manifold
triangulations (i.e., triangulations with edges identified to themselves in reverse, or vertices whose
links have boundary but are not discs) will not be output at all.
As the census progresses, the state of progress will be written (slowly) to standard output. Once the
census is complete, the full census will be saved to the given output file.
You can use the options --genpairs and --usepairs to split a census into smaller pieces. See also
tricensus-mpi, a more powerful tool that allows you to distribute a census across a high-performance
computing cluster.
Caution:
A census with even a small number of tetrahedra can take an incredibly long time to run, and can
chew up massive amounts of memory. It is recommended that you try very small censuses to begin
with (such as 3 or 4 tetrahedra), and work upwards to establish the limits of your machine.
For very large census runs, it is highly recommended that you use the --sigs option, which will
keep the output file small and significantly reduce the memory footprint.
OPTIONS
-t, --tetrahedra=tetrahedra
Specifies the number of tetrahedra used to build the triangulations.
If --dim2 is passed, this same option must be used to specify the number of triangles instead.
-2, --dim2
Build a census of 2-manifold triangulations, not 3-manifold triangulations.
This is incompatible with several options; for other options it simply translates the relevant
constraint into two dimensions. See each individual option for details on how it interacts with
--dim2.
-b, --boundary
Only produce triangulations with at least one boundary triangle.
If --dim2 is passed, this specifies at least one boundary edge.
-i, --internal
Only produce triangulations with all triangles internal (i.e., with no boundary triangles).
If --dim2 is passed, this indicates that all edges must be internal.
-B, --bdryfaces=triangles
Only produce triangulations with the precise number of boundary triangles specified.
If --dim2 is passed, this specifies the number of boundary edges.
-o, --orientable
Only produce orientable triangulations.
-n, --nonorientable
Only produce non-orientable triangulations.
-f, --finite
Only produce finite triangulations (triangulations with no ideal vertices).
This option cannot be used with --dim2.
-d, --ideal
Only produce triangulations with at least one ideal vertex. There might or might not be internal
vertices (whose links are spheres) as well.
This option cannot be used with --dim2.
-m, --minimal
Do not include triangulations that are obviously non-minimal.
This option uses a series of fast tests that try to eliminate non-minimal triangulations, but that
are not always conclusive. If Regina cannot quickly tell whether a triangulation is non-minimal,
it will place the triangulation in the census regardless.
-M, --minprime
Do not include triangulations that are obviously non-minimal, non-prime and/or disc-reducible.
This can significantly speed up the census and vastly reduce the final number of triangulations
produced.
As above, this option uses a series of fast tests that are not always conclusive. If Regina
cannot quickly tell whether a triangulation is non-minimal, non-prime or disc-reducible, it will
place the triangulation in the census regardless.
This option cannot be used with --dim2.
-N, --minprimep2
Do not include triangulations that are obviously non-minimal, non-prime, P2-reducible and/or disc-
reducible.
This can significantly speed up the census and vastly reduce the final number of triangulations
produced, even more so than --minprime.
As above, this option uses a series of fast tests that are not always conclusive. If Regina
cannot quickly tell whether a triangulation is non-minimal, non-prime, P2-reducible or disc-
reducible, it will place the triangulation in the census regardless.
This option cannot be used with --dim2.
-s, --sigs
Instead of writing a full Regina data file, just output a list of isomorphism signatures.
The output file will be a plain text file. Each line will be a short string of letters, digits
and punctuation that uniquely encodes a triangulation up to combinatorial isomorphism. You can
import this text file from within Regina by selecting File->Import->Isomorphism Signature List
from the menu.
This option is highly recommended for large census enumerations. First, the output file will be
considerably smaller. More importantly, the memory footprint of tricensus will also be much
smaller: triangulations can be written to the output file and forgotten immediately, instead of
being kept in memory to construct a final Regina data file.
-c, --subcontainers
For each face pairing, a new container will be created, and resultant triangulations will be
placed into these containers. These containers will be created even if the face pairing results
in no triangulations.
This option cannot be used with --sigs.
-p, --genpairs
Only generate face pairings, not triangulations.
The outermost layer of the census code involves pairing off the faces of individual tetrahedra
without determining the corresponding gluing permutations. For each face pairing that is
produced, Regina will try many different sets of gluing permutations and generated the
corresponding triangulations.
Face pairing generation consumes a very small fraction of the total census runtime, and
effectively divides the census into multiple pieces. This option allows you to quickly generate a
complete list of possible face pairings, so that you can feed subsets of this list to different
machines to work on simultaneously. You can coordinate this manually, or you can use tricensus-
mpi to coordinate it for you on a high-performance cluster.
The list of all face pairings will be written to the given output file in text format (though you
may omit the output file from the command line, in which case the face pairings will be written to
standard output). If you are coordinating your sub-censuses manually, you can use the option
--usepairs to generate triangulations from a subset of these face pairings.
Options for orientability, finiteness or minimality cannot be used with --genpairs; instead you
should use them later with --usepairs, or pass them to tricensus-mpi.
This option does not come with progress reporting, though typically it runs fast enough that this
does not matter. You can always track the state of progress by counting lines in the output file.
If --dim2 is passed, this generates edge pairings accordingly.
-P, --usepairs
Use only the given subset of face pairings to build the triangulations.
Each face pairing that is processed must be in canonical form, i.e., must be a minimal
representative of its isomorphism class. All face pairings generated using --genpairs are
guaranteed to satisfy this condition.
Face pairings should be supplied on standard input, one per line. They should be listed in the
format produced by the option --genpairs.
This option effectively lets you run a subset of a larger census. See --genpairs for further
details on how to split a census into subsets that can run simultaneously on different machines,
or tricensus-mpi which can coordinate this process using MPI on a high-performance cluster.
Options for tetrahedra or boundary triangles cannot be used with --usepairs; instead you should
pass them earlier along with --genpairs when you split the original census into pieces.
If --dim2 is passed, this takes a list of edge pairings accordingly.
EXAMPLES
The following command forms a census of all 3-tetrahedron closed non-orientable triangulations and puts
the results in the file results.rga. To ensure that triangulations are closed we use the options -i (no
boundary triangles) and -f (no ideal vertices).
example$ tricensus -t 3 -nif results.rga
Starting census generation...
0:1 0:0 1:0 1:1 | 0:2 0:3 2:0 2:1 | 1:2 1:3 2:3 2:2
0:1 0:0 1:0 2:0 | 0:2 1:2 1:1 2:1 | 0:3 1:3 2:3 2:2
0:1 0:0 1:0 2:0 | 0:2 2:1 2:2 2:3 | 0:3 1:1 1:2 1:3
1:0 1:1 2:0 2:1 | 0:0 0:1 2:2 2:3 | 0:2 0:3 1:2 1:3
Finished.
Total triangulations: 5
example$
The following command forms a census of 4-tetrahedron closed orientable triangulations, where the census
creation is optimised for prime minimal triangulations. Although all prime minimal triangulations will
be included, there may be some non-prime or non-minimal triangulations in the census also.
example$ tricensus -t 4 -oifM results.rga
Starting census generation...
0:1 0:0 1:0 1:1 | 0:2 0:3 2:0 2:1 | 1:2 1:3 3:0 3:1 | 2:2 ...
0:1 0:0 1:0 1:1 | 0:2 0:3 2:0 3:0 | 1:2 2:2 2:1 3:1 | 1:3 ...
...
1:0 1:1 2:0 3:0 | 0:0 0:1 2:1 3:1 | 0:2 1:2 3:2 3:3 | 0:3 ...
Finished.
Total triangulations: 17
example$
The following command generates all face pairings for a 5-tetrahedron census in which all triangulations
have precisely two boundary triangles. The face pairings will be written to pairings.txt, whereupon they
can be broken up and distributed for processing at a later date.
example$ tricensus --genpairs -t 5 -B 2 pairings.txt
Total face pairings: 118
example$
The face pairings generated in the previous example can then be fleshed out into a full census of all
3-manifold triangulations with five tetrahedra, precisely two boundary triangles and no ideal vertices as
follows. The number of tetrahedra and boundary triangles were already specified in the previous command,
and cannot be supplied here. The face pairings will be read from pairings.txt, and the final census will
be written to results.rga.
example$ tricensus --usepairs -f results.rga < pairings.txt
Trying face pairings...
0:1 0:0 1:0 1:1 | 0:2 0:3 2:0 2:1 | 1:2 1:3 3:0 3:1 | 2:2 ...
0:1 0:0 1:0 1:1 | 0:2 0:3 2:0 2:1 | 1:2 1:3 3:0 3:1 | 2:2 ...
...
... (running through all 118 face pairings)
...
1:0 2:0 3:0 4:0 | 0:0 2:1 3:1 4:1 | 0:1 1:1 3:2 4:2 | 0:2 ...
Total triangulations: 5817
example$
MACOS X USERS
If you downloaded a drag-and-drop app bundle, this utility is shipped inside it. If you dragged Regina
to the main Applications folder, you can run it as /Applications/Regina.app/Contents/MacOS/tricensus.
WINDOWS USERS
The command-line utilities are installed beneath the Program Files directory; on some machines this
directory is called Program Files (x86). You can start this utility by running
c:\Program Files\Regina\Regina 4.95\bin\tricensus.exe.
SEE ALSO
sigcensus, tricensus-mpi, regina-gui.
AUTHOR
This utility was written by Benjamin Burton <bab@debian.org>. Many people have been involved in the
development of Regina; see the users' handbook for a full list of credits.
06 November 2013 TRICENSUS(1)