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Name
retriangulate — Exhaustively retriangulate a given manifold
Synopsis
retriangulate [-h, --height=
height
] [-t, --threads=
threads
] [-4, --dim4
] {isosig
}
retriangulate {--help
}
Description
Given a 3-manifold or 4-manifold triangulation, this utility uses Pachner moves to exhaustively search for other triangulations of the same manifold that are the same size or smaller.
Specifically, suppose the input triangulation contains
n
tetrahedra (for a 3-manifold)
or pentachora (for a 4-manifold). Then this utility will
exhaustively retriangulate the manifold using repeated Pachner moves,
without ever exceeding
n
+ height
tetrahedra/pentachora in total.
Moreover, all such retriangulations are guaranteed to be found, each
once and only once.
For 3-manifold triangulations, this utility will only attempt 2-3 and 2-3 Pachner moves, never 1-4 or 4-1 moves. For 4-manifold triangulations, all types of Pachner moves are used (1-5, 2-4, 3-3, 4-2, and 5-1).
The program will output each triangulation that it finds of the same
size n
(including the original input
triangulation). If it ever finds a
smaller triangulation (thereby proving the
original to be non-minimal), it will output that smaller
triangulation and then stop immediately. Otherwise it will continue
outputting triangulations of size n
until
no more can be found. Although the program also finds larger
triangulations as part of its exhaustive search using Pachner moves,
these larger triangulations (of which there are typically many) will
not be output at all.
All triangulations, both input and output, are described using
isomorphism signatures. These are short text
strings that identify a triangulation uniquely up to combinatorial
isomorphism (i.e., up to relabellings of the tetrahedra/pentachora
and their vertices).
The input triangulation will be interpreted as the isomorphism
signature of a 3-manifold unless you pass the option
--dim4
(in which case it will be treated as a
4-manifold instead).
From Regina's graphical user interface,
you can view the isomorphism signature
of a triangulation by switching to the Composition
tab in the triangulation viewer. From Python you can fetch the
isomorphism signature of the triangulation x
by calling x.isoSig()
.
For a full and precise specification of isomorphism signatures for
3-manifolds, see
Simplification paths in the Pachner graphs of closed
orientable 3-manifold triangulations, Burton, 2011,
arXiv:1110.6080.
Options
-h, --height=
height
Specifies the number of additional tetrahedra (for a 3-manifold) or pentachora (for a 4-manifold) that we allow during intermediate stages of retriangulation. That is, if the input triangulation has
n
tetrahedra (or pentachora), then this utility will exhaustively search through all triangulations that it can reach via Pachner moves that do not exceedn
+height
tetrahedra (or pentachora) in total.Note that these larger intermediate triangulations will not be written to output; however, a larger height may allow the program to access additional smaller triangulations that were otherwise inaccessible.
The given height must be positive, and for 4-manifolds it must also be even. If not specified, this option defaults to 1 for 3-manifolds (if
--dim4
is not passed), or 2 for 4-manifolds (if--dim4
is passed).Warning
In general, the running time can grow superexponentially with
height
. It is strongly suggested that you begin with--height=1
(or 2 for 4-manifolds) and increase it one step at a time until the performance becomes unacceptable.-t, --threads=
threads
Specifies the degree to which this utility uses parallel processing. Specifically, this program will use
threads
simultaneous threads of execution as it works its way through the different retriangulations of the input manifold.This program is typically able to use parallelism effectively, and so running with
k
threads should approximately divide the running time byk
.If not specified, this option defaults to 1 (i.e., single-threaded processing, with no parallelism).
-4, --dim4
Indicates that the given isomorphism signature describes a 4-manifold triangulation. If this argument is not given, then by default the argument is assumed to describe a 3-manifold triangulation.
Examples
The 3-manifold triangulation is non-minimal, but it requires a bit of work to see this:
example$
retriangulate -h2 hLLAAkbdceefggdonxdjxn
hLLAAkbdceefggdonxdjxn hLALPkbcbefgfghxwnxark Found 2 triangulation(s).example$
retriangulate -h3 hLLAAkbdceefggdonxdjxn
hLLAAkbdceefggdonxdjxn hLALPkbcbefgfghxwnxark hLLMMkbcdfefgglcghtchj gLLPQcdcefffqsjpunw Triangulation is non-minimal! Smaller triangulation: gLLPQcdcefffqsjpunwexample$
Although the program stopped as soon as it found a smaller triangulation, this can be simplified even further:
example$
retriangulate gLLPQcdcefffqsjpunw
gLLPQcdcefffqsjpunw fLAMcbbcdeedhwhxn Triangulation is non-minimal! Smaller triangulation: fLAMcbbcdeedhwhxnexample$
A little more probing shows this to be the cusped hyperbolic
manifold m112
:
example$
censuslookup fLAMcbbcdeedhwhxn
fLAMcbbcdeedhwhxn: 1 hit m112 : #2 -- Cusped hyperbolic census (orientable)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/retriangulate
.
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 5.2\bin\retriangulate.exe
.
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 acknowledgements
page for a full list of credits.
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