Provided by: regina-normal_5.1-2build2_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. 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 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. 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 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, 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.
-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,
or tricensus-mpi which can coordinate this process using MPI on a high-performance cluster.
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 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 5.1\bin\tricensus.exe.
SEE ALSO
censuslookup, sigcensus, tricensus-mpi, 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. 14 December 2016 TRICENSUS(1)