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Name

dgt — Triangulate a 3-manifold or 4-manifold from a framed link

Synopsis

dgt {[-3, --dim3] | [-4, --dim4]} [-g, --graph] [-r, --real]

dgt {{-v, --version} | {-?, --help}}

Description

This utility builds a triangulation or coloured graph of a 3-manifold or 4-manifold from a framed link.

For 3-manifolds, the manifold constructed is the one obtained by performing integer Dehn surgery on the given link.

For 4-manifolds, the manifold constructed is the one obtained by attaching 4-dimensional 2-handles to the 4-ball along the framed link components.

When you run DGT, it will ask you to input the underlying (unframed) link at the console. This link should be given in the format of a Planar Diagram (PD) code, specifically, in the same format as used by SnapPy. The simplest way to achieve this is to draw the link in SnapPy's PLink editor, and copy the PD code generated by SnapPy via the Info→PD Code menu option in the editor.

Warning

Do not include the PD: text preceding the code generated by the PLink editor in the input to DGT. Only copy and input the code itself, which starts at the left square bracket and terminates with the right square bracket.

For more information, see the full DGT manual, available from https://raburke.github.io/.

Options

-3, --dim3

Build the 3-manifold obtained from integer Dehn surgery on the input link.

One of --dim3 or --dim4 must be given as a command-line argument.

-4, --dim4

Build the 4-manifold obtained by attaching 2-handles along the components of the framed link to the 4-ball.

One of --dim3 or --dim4 must be given as a command-line argument.

-g, --graph

Output an edge list of the edge-coloured graph associated to the manifold. Each node of the graph corresponds to a tetrahedron in the case of 3-manifolds or to a pentachoron in the case of 4-manifolds. Two nodes are connected by a c-coloured edge if the two corresponding top-dimensional simplices of the triangulation have the facets opposite to the vertex labelled c identified.

-r, --real

For 4-manifolds, this option will build the triangulation with real boundary.

By default, if the manifold does not have boundary S3, it will be built with ideal boundary. If the manifold has boundary S3, then the resulting triangulation will be capped off to produce a closed manifold.

This option will be ignored for 3-manifolds, as all 3-manifolds built from this construction are closed.

-v, --version

Show which version of Regina is being used, and exit immediately.

-?, --help

Display brief usage information, and exit immediately.

Examples

The following builds the Poincaré homology 3-sphere obtained by +1 surgery along the right handed trefoil knot. 

    example$ dgt -3
    Enter PD Code of Diagram: [(6,4,1,3),(4,2,5,1),(2,6,3,5)]

    Writhe of
    Component 0: 3
    Enter integer framings for 2-handles (same order as in SnapPy's PLink Editor):
    1
    Self-framing component 0...
    Link should now be self-framed: writhe(component) = framing(component)...
    Writhe of
    Component 0: 1

    1     Generating Negative Curl of Type A (x,x,z,w)...
    2     Generating Negative Curl of Type A (x,x,z,w)...
    3     Generating Positive Crossing...
    4     Generating Positive Crossing...
    5     Generating Positive Crossing...

    Here is the isomorphism signature:
    GLvvQvPvALvzMAQAvAQQQPccgfekjpmswxtvywzrxyDABABCEDBCEFFFaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
    example$

The following builds the complex projective plane by attaching a single 2-handle to the 4-ball along a +1 framed unknot. 

    example$ dgt -4
    Enter PD Code of Diagram: [(1,1,2,2)]

    Writhe of
    Component 0: 1
    Enter integer framings for 2-handles (same order as in SnapPy's PLink Editor):
    1
    Adding additional pair of cancelling curls to component 0 to guarantee existence of a quadricolour...
    Link should now be self-framed: writhe(component) = framing(component)...
    Writhe of
    Component 0: 1

    1     Generating Negative Curl of Type A (x,x,z,w)...
    2     Generating Positive Curl of Type A (x,y,y,w)...
    3     Generating Positive Curl of Type A (x,y,y,w)...

    Performing 1 quadricolour substitution...

    If manifold has (non-spherical) boundary, resulting triangulation will have ideal boundary.
    If manifold has spherical boundary, manifold will be capped off to produce a closed manifold.

    Here is the isomorphism signature:
    mLvAwAQAPQQcfffhijgjgjkkklklllaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
    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/dgt.

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 7.3\bin\dgt.exe.

Author

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

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