DGT(1)

💖 DGT(1) 💖

okay bestie, welcome to the The Regina Handbook glow-up, section 1, where we talk about dgt and why it's lowkey iconic. buckle up 🎀

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✨ NAME ✨

dgt — triangulate a 3-manifold or 4-manifold from a framed link 🦋

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🌸 SYNOPSIS 🌸

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

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

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💅 DESCRIPTION 💅

so this lil utility builds a triangulation or a coloured graph of a 3-manifold or 4-manifold from a framed link. no cap. it's giving "topology but make it cute" 🔥

for 3-manifolds, the manifold it constructs is the one you get by performing integer Dehn surgery on the given link.

for 4-manifolds, the manifold it constructs is the one you get by attaching 4-dimensional 2-handles to the 4-ball along the framed link components.

when you run DGT, it will literally ask you to input the underlying (unframed) link at the console. this link needs to be in the format of a Planar Diagram (PD) code — specifically, in the same format as used by SnapPy. the simplest way to do this: draw the link in SnapPy's PLink editor, then copy the PD code that SnapPy generates via the InfoPD Code menu option in the editor. periodt. ✨

⚠️ 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. not me warning you twice, but fr fr, just the brackets. 😭

for more info, see the full DGT manual, available from <URL:https://raburke.github.io/>. 🫶

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🎀 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. (yes, it's mandatory bestie.)

• -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. (same rule, no exceptions.)

• -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 *S*³, then the resulting triangulation will be capped off to produce a closed manifold.

this option will be ignored for 3-manifolds, because all 3-manifolds built from this construction are closed. 💀 (not me being ignored, the flag. the flag is being ignored.)

• -v, --version — show which version of Regina is being used, and exit immediately. ⭐
• -?, --help — display brief usage information, and exit immediately. 💜

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🌟 EXAMPLES 🌟

🧚 building the Poincaré homology 3-sphere 🧚

the following builds the Poincare homology 3-sphere obtained by +1 surgery along the right handed trefoil knot. it's giving legendary. 🎀

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$

🦋 building the complex projective plane 🦋

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

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$

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🍓 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. slay. 💅

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🌈 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 7.3.1\bin\dgt.exe. iykyk. 🫶

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👑 AUTHOR 👑

this utility was written by Rhuaidi Burke <rhuaidi.burke@uq.edu.au>. many people have been involved in the development of Regina; see the users' handbook for a full list of credits. besties, the whole crew ate. 💖

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okay that's the whole page!! this one's from the The Regina Handbook, dated 14 March 2023 — mwah, take care of her, she's an icon 💜✨🎀