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Name

retriangulate — Exhaustively search through triangulations or knot diagrams using local moves

Synopsis

retriangulate [-h, --height=height] [-t, --threads=threads] [[-3, --dim3] | [-4, --dim4] | [-k, --knot]] [-a, --anysig] [--] {signature}

retriangulate {--help}

Description

Given a 3-manifold triangulation, 4-manifold triangulation or knot diagram, this utility uses local moves to exhaustively search for other triangulations/diagrams of the same manifold/knot that are the same size or smaller. Here “local moves” means Pachner moves for triangulations, or Reidemeister moves for knots.

Specifically, suppose the input triangulation or knot diagram contains n tetrahedra/pentachora/crossings (for a 3-manifold, 4-manifold or knot respectively). Then this utility will exhaustively modify the triangulation or knot diagram using local moves, without ever exceeding n + height tetrahedra/pentachora/crossings in total. Moreover, all such triangulations/diagrams are guaranteed to be found, each once and only once (up to an appropriate notion of combinatorial isomorphism).

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 or knot diagrams, it will use all types of Pachner moves or Reidemeister moves respectively.

The input is assumed to represent a 3-manifold triangulation unless one of the options --dim4 or --knot is passed.

The program will output each triangulation or knot diagram that it finds of the same size n (including the original input triangulation/diagram). If it ever finds a smaller triangulation or diagram (thereby proving the original to be non-minimal), it will output that smaller triangulation/diagram and then stop immediately. Otherwise it will continue outputting triangulations or diagrams of size n until no more can be found. Although the program also finds larger triangulations/diagrams as part of its exhaustive search using local moves, these larger triangulations/diagrams (of which there are typically many) will not be output at all.

All triangulations or knot diagrams, both input and output, are described using isomorphism signatures and knot signatures respectively. These are short text strings that identify a triangulation or knot diagram uniquely up to combinatorial isomorphism (which includes relabellings of tetrahedra/pentachora/crossings, relabellings of the vertices of tetrahedra/pentachora vertices, and reflection/reversal of knot diagrams).

To view the isomorphism signature of a triangulation: in Regina's graphical user interface you can find this in the Composition tab in the triangulation viewer, and in Python you can call t.isoSig() for a triangulation t. To view a knot signature: in Regina's graphical user interface this is available through the Codes tab in the knot/link viewer, and in Python you can call k.knotSig() for a knot k.

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.

Tip

Very large triangulations or knots have signatures that begin with a dash (-). Such a signature could be mistaken for an option when passing it on the command line. To avoid this, you can pass the special option -- immediately before it: this indicates that all command-line options have finished, and whatever comes next should be treated as the input signature. 

    example$ retriangulate -h1 -- -b-LLvALwvM...

Options

-h, --height=height

Specifies the number of additional tetrahedra, pentachora or crossings (for a 3-manifold, 4-manifold or knot respectively) that we allow during intermediate stages of retriangulation. That is, if the input triangulation or knot diagram has n tetrahedra/pentachora/crossings, then this utility will exhaustively search through all triangulations or knot diagrams that it can reach via local moves that do not exceed n + height tetrahedra/pentachora/crossings in total.

Note that these larger intermediate triangulations or diagrams will not be written to output; however, a larger height may allow the program to access additional smaller triangulations or diagrams that were otherwise inaccessible.

The given height must be a non-negative integer. In addition, for 3-manifolds it must be strictly positive, and for 4-manifolds it must be even.

If not specified, this option defaults to 1 when working with 3-manifolds or knot diagrams, and it defaults to 2 when working with 4-manifolds.

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 or diagrams of the input manifold or knot.

This program is typically able to use parallelism effectively, and so running with k threads should approximately divide the running time by k.

If not specified, this option defaults to 1 (i.e., single-threaded processing, with no parallelism).

-3, --dim3 (default)

Indicates that the given signature is the isomorphism signature of a 3-manifold triangulation. The local moves used will be 2-3 and 3-2 Pachner moves.

This is the default if none of the arguments --dim3, --dim4 or --knot is passed.

-4, --dim4

Indicates that the given signature is the isomorphism signature of a 4-manifold triangulation. The local moves used will be 1-5, 2-4, 3-3, 4-2 and 5-1 Pachner moves.

-k, --knot

Indicates that the given signature is a knot signature. The local moves used will be the Reidemeister moves.

-a, --anysig

Indicates that the output is not required to be classic isomorphism signatures.

Regina 7.0 introduced alternate types of isomorphism signatures. Like the original isomorphism signatures that were introduced many years earlier, each type of signature uniquely identifies a triangulation up to combinatorial isomorphism. Moreover, Regina can reconstruct a triangulation or link from a signature of any type.

Internally, this utility uses one of the newer, alternate types of signature that is faster to compute. However, it still outputs classic signatures; that is, the same isomorphism signatures that were originally introduced back in 2011. This conversion from alternate to classic signatures adds extra overhead to the running time.

If you pass the option --anysig, Regina will not convert its output back to classic signatures; instead it will output whatever alternate signature type it uses internally. This will be faster, and you can still use these alternate signatures to reconstruct triangulations; the only reason not to do this is if you neeed to ensure compatibility with the original classical signatures (e.g., for matching against a list of signatures that was generated elsewhere).

Warning

The internal signature type is subject to change in future versions of Regina. That is, if you do use --anysig, then you may get different output depending upon which version of Regina you are using.

Note

Currently Regina only uses alternate signature types with triangulations. For knot signatures, it still uses classic signatures, though again this is subject to change in future version of Regina.

--

Indicates that all other options have finished, and whatever comes next on the command line should be treated as the input signature.

This is useful when your signature begins with a dash, to avoid confusing your input signature with a regular command line option.

Examples

The following 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: gLLPQcdcefffqsjpunw
    example$

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: fLAMcbbcdeedhwhxn
    example$

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 7.0\bin\retriangulate.exe.

See Also

regina-gui.

Author

This utility was written by Benjamin Burton . Many people have been involved in the development of Regina; see the acknowledgements page for a full list of credits.

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