Provided by: regina-normal_7.0-2build1_amd64 
NAME
tricensus - Form a census of triangulations
SYNOPSIS
tricensus [ -t, --tetrahedra=tetrahedra ] [ -2, --dim2 | -4, --dim4 ] [ -b, --boundary |
-i, --internal | -B, --bdryfaces=triangles ] [ -o, --orientable | -n, --nonorientable ] [
-f, --finite | -d, --ideal ] [ -m, --minimal | -M, --minprime | -N, --minprimep2 | -h,
--minhyp ] [ -s, --sigs | -c, --subcontainers ] [ -p, --genpairs | -P, --usepairs ]
output-file
tricensus --help
DESCRIPTION
Forms a census of all 2-, 3- or 4-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 top-dimensional simplices (e.g., the number of
tetrahedra for 3-manifolds), but there are many other options available.
The default behaviour is to enumerate 3-manifold triangulations. If you wish to enumerate
2-manifold or 4-manifold triangulations instead, you must pass --dim2 or --dim4
respectively.
Each triangulation will be output precisely once up to combinatorial isomorphism. Invalid
triangulations (for 3-manifolds, this means 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.
Caution:
A census with even a small number of top-dimensional simplices 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
simplices), 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 top-dimensional simplices used to build the triangulations.
For 2-manifolds, 3-manifolds and 4-manifolds, this specifies the number of
triangles, tetrahedra or pentachora respectively.
-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.
This option cannot be used with --dim4.
-4, --dim4
Build a census of 4-manifold triangulations, not 3-manifold triangulations.
This is incompatible with several options; for other options it simply translates
the relevant constraint into four dimensions. See each individual option for
details on how it interacts with --dim4.
This option cannot be used with --dim2.
-b, --boundary
Only produce triangulations with at least one boundary triangle.
For 2-manifolds or 4-manifolds, this option ensures at least one boundary edge or
boundary tetrahedron respectively.
-i, --internal
Only produce triangulations with all triangles internal (i.e., with no boundary
triangles).
For 2-manifolds or 4-manifolds, this option ensures that all edges or tetrahedra
respectively are internal.
-B, --bdryfaces=triangles
Only produce triangulations with the precise number of boundary triangles
specified.
For 2-manifolds or 4-manifolds, this specifies the number of boundary edges or
boundary tetrahedra respectively.
-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.
This option cannot be used with --dim4.
-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 or --dim4.
-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 or --dim4.
-h, --minhyp
Do not include triangulations that are obviously not minimal ideal triangulations
of cusped finite-volume hyperbolic 3-manifolds.
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 a minimal ideal
triangulation of a cusped finite-volume hyperbolic 3-manifold, it will place the
triangulation in the census regardless.
This option is designed for use with ideal triangulations only (so, for instance,
combining it with --finite or --boundary will produce an error message). This
option also cannot be used with --dim2 or --dim4.
-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 facet pairing, a new container will be created, and resultant
triangulations will be placed into these containers. These containers will be
created even if the facet pairing results in no triangulations.
See --genpairs below for further information on facet pairings.
This option cannot be used with --sigs.
-p, --genpairs
Only generate facet pairings, not triangulations. A facet pairing stores which
facets of top-dimension simplices are glued to which others, but it does not store
the precise rotations and/or reflections that are used for each gluing. For
3-manifolds a facet pairing represents a pairing of tetrahedron faces, for
2-manifolds it represents a pairing of triangle edges, and for 4-manifolds it
represents a pairing of pentachoron facets.
The outermost layer of the census code involves pairing off the facets of
individual top-dimensional simplices without determining the corresponding gluing
permutations. For each such facet pairing that is produced, Regina will try many
different sets of gluing permutations and generated the corresponding
triangulations.
Facet 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 facet pairings, so that
you can feed subsets of this list to different machines to work on simultaneously.
The list of all facet 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 facet 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 facet pairings.
Options for orientability, finiteness or minimality cannot be used with --genpairs;
instead you should use them later with --usepairs.
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.
-P, --usepairs
Use only the given subset of facet pairings to build the triangulations.
Each facet pairing that is processed must be in canonical form, i.e., must be a
minimal representative of its isomorphism class. All facet pairings generated
using --genpairs are guaranteed to satisfy this condition.
Facet 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.
Options for the number of top-dimensional simplices (i.e., --tetrahedra) or
boundary facets (i.e., --boundary or --bdryfaces) cannot be used with --usepairs.
Instead you should pass these options earlier along with --genpairs when you split
the original census into pieces.
EXAMPLES
The following command forms a census of all 3-tetrahedron closed non-orientable 3-manifold
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 3-manifold
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 of 3-manifold
triangulation 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 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 7.0\bin\tricensus.exe.
SEE ALSO
censuslookup, sigcensus, regina-gui.
AUTHOR
This utility was written by Benjamin Burton <bab@maths.uq.edu.au>. Many people have been
involved in the development of Regina; see the users' handbook for a full list of credits.
17 December 2021 TRICENSUS(1)