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<?xml version="1.0"?>
<regina engine="7.4">
<container label="Sample Data File">
<textdata label="Read Me">Welcome to Regina!
A single Regina data file can store a range of objects, from 3-manifold triangulations and normal surfaces to Python scripts, PDF documents and text notes (such as the one you are reading now).
These objects are called "packets", and are arranged in a tree-like structure as you can see to the left. To view or edit a packet, just click on the packet in the tree (on some platforms, such as macOS, you should double-click instead).
Have a play with the packets in this file to see what Regina can do.
If you ever want to know what some element of the user interface means (such as a button or a text box), just press Shift-F1 and click on the thing that you want to know more about. You can try this now with the tree in the main window, or the buttons in the toolbar. If you ever forget the Shift-F1 shortcut, you can find it in the menu under Help → What's This?
For more detailed help, try the Regina Handbook (in the Help menu, or just press F1). The handbook includes lots of screenshots and talks you through how to do many different things.
Any questions? Just mail the authors; see Help → About for our email addresses.
Enjoy!
- The Regina development team</textdata>
<attachment encoding="base64" filename="welcome.pdf" label="LaTeX Document">
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<surface enc="314" len="40"> 2 2 3 2 5 2 12 2 13 2 19 2 20 2 21 2 22 3 23 3 24 1 30 3 31 3 32 3 33 3
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 6 2 15 2 22 1 23 1 24 1 30 1 31 1 32 1 33 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 2 1 2 6 5 10 2 11 2 15 3 22 1 23 1 28 2 30 2 31 2 32 1 33 1 36 1
<euler value="-2"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 6 1 15 1 22 1 23 1 38 1
<euler value="-1"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 4 1 4 6 2 10 2 11 2 16 2 20 1 21 1 27 1 30 1 31 1 32 1 33 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 2 1 2 6 1 10 1 11 1 16 1 20 1 21 1 35 1
<euler value="0"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 4 1 4 6 3 10 3 11 3 16 1 20 1 21 1 26 2 30 1 31 1 36 1
<euler value="1"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 6 1 6 6 4 10 4 11 4 16 2 20 2 21 2 26 2 30 1 31 1 37 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 4 1 4 6 1 10 1 11 1 16 3 20 1 21 1 25 2 32 1 33 1 39 1
<euler value="-1"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 2 1 2 6 2 10 1 11 1 17 1 20 1 21 1 22 1 23 1 30 1 31 1 32 1 33 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 3 1 3 2 1 3 1 6 1 10 2 11 2 18 1 20 1 21 1 26 2 30 1 31 1 36 1
<euler value="-1"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 1 1 1 7 1 10 1 11 1 12 1 13 1 20 1 21 1 22 1 23 1 30 1 31 1 32 1 33 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 2 2 3 2 8 2 10 2 11 2 12 2 13 2 15 2 20 2 21 2 22 3 23 3 24 1 30 3 31 3 32 3 33 3
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 1 1 1 8 1 10 1 11 1 16 1 20 1 21 1 35 1
<euler value="-2"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 1 1 1 8 3 10 3 11 3 16 1 20 1 21 1 26 2 30 1 31 1 36 1
<euler value="-5"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 2 2 3 2 9 2 10 2 11 2 12 3 13 3 14 1 20 3 21 3 22 3 23 3 30 3 31 3 32 3 33 3
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 4 2 16 2 25 2 32 1 33 1 34 1
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 2 1 3 1 4 1 10 1 11 1 12 1 13 1 20 1 21 1 24 1 35 2
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 5 2 12 1 13 1 14 1 20 1 21 1 24 1 35 2
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 2 1 3 1 5 1 12 1 13 1 19 1 20 1 21 1 24 2 35 3
<euler value="-4"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 6 1 15 1 24 1 35 1
<euler value="-1"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 2 1 2 6 2 10 1 11 1 17 1 20 1 21 1 24 1 35 2
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 1 1 1 7 1 10 1 11 1 12 1 13 1 20 1 21 1 24 1 35 2
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 2 1 3 1 8 1 10 1 11 1 12 1 13 1 15 1 20 1 21 1 24 2 35 3
<euler value="-4"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 2 2 3 2 9 2 10 2 11 2 12 3 13 3 14 1 20 3 21 3 24 3 35 6
<euler value="-8"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="314" len="40"> 0 1 1 1 2 1 3 1 10 1 11 1 12 1 13 1 20 1 21 1 22 1 23 1 30 1 31 1 32 1 33 1
<euler value="1"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
</surfaces>
<surfaces tri="eA6IKgEAAAA=" type="9" algorithm="273" coords="0" label="Fundamental Normal Surfaces">
<surface enc="279" len="28"> 6 1 12 1 18 1 26 1
<euler value="-1"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 6 2 12 2 16 1 17 1 18 1 21 1 22 1 23 1 24 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 5 2 9 1 10 1 11 1 14 1 15 1 18 1 26 2
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 5 2 9 1 10 1 11 1 14 1 15 1 16 1 17 1 21 1 22 1 23 1 24 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 4 2 13 2 19 2 23 1 24 1 25 1
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 2 1 3 1 4 1 7 1 8 1 9 1 10 1 14 1 15 1 18 1 26 2
<euler value="-2"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 2 1 3 1 4 1 7 1 8 1 9 1 10 1 14 1 15 1 16 1 17 1 21 1 22 1 23 1 24 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 0 1 1 1 4 1 13 2 19 2 23 1 24 1 25 1
<euler value="-1"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 0 1 1 1 2 1 3 1 7 1 8 1 9 1 10 1 14 1 15 1 18 1 26 2
<euler value="-1"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 0 1 1 1 2 1 3 1 7 1 8 1 9 1 10 1 14 1 15 1 16 1 17 1 21 1 22 1 23 1 24 1
<euler value="1"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 0 2 1 2 13 2 19 2 23 1 24 1 25 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 0 2 1 2 6 1 7 1 8 1 13 1 14 1 15 1 26 1
<euler value="0"/>
<orbl value="F"/>
<twosided value="F"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
<surface enc="279" len="28"> 0 4 1 4 6 3 7 3 8 3 13 1 14 1 15 1 20 2 21 1 22 1 27 1
<euler value="1"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="T"/>
<compact value="T"/> </surface>
</surfaces>
</tri>
<tri dim="3" size="5" perm="index" label="Poincaré Homology Sphere">
<simplex> 1 2 2 19 3 16 4 21 </simplex>
<simplex> 0 4 2 14 3 19 4 16 </simplex>
<simplex> 0 9 1 20 3 10 4 19 </simplex>
<simplex> 0 16 1 9 2 18 4 8 </simplex>
<simplex> 0 21 1 16 2 9 3 12 </simplex>
<H1><abeliangroup rank="0"> </abeliangroup></H1>
<H1Rel><abeliangroup rank="0"> </abeliangroup></H1Rel>
<H1Bdry><abeliangroup rank="0"> </abeliangroup></H1Bdry>
<H2><abeliangroup rank="0"> </abeliangroup></H2>
<zeroeff value="T"/>
<splitsfce value="F"/>
</tri>
<tri dim="3" size="3" perm="index" label="RP² x S¹">
<simplex> 1 0 1 0 2 11 2 2 </simplex>
<simplex> 0 0 0 0 2 21 2 12 </simplex>
<simplex> 0 13 0 4 1 21 1 8 </simplex>
<H1><abeliangroup rank="1"> 2 </abeliangroup></H1>
<H1Rel><abeliangroup rank="1"> 2 </abeliangroup></H1Rel>
<H1Bdry><abeliangroup rank="0"> </abeliangroup></H1Bdry>
<H2><abeliangroup rank="0"> 2 </abeliangroup></H2>
<textdata label="Note">There are in fact two triangulations of this 3-manifold with three tetrahedra.</textdata>
</tri>
<link label="Right-Hand Trefoil">
<crossings size="3">
+ + +
</crossings>
<connections>
_1 ^1
_2 ^2
_0 ^0
</connections>
<components size="1">
^0
</components>
</link>
<link label="Borromean Rings">
<crossings size="6">
+ - + - + -
</crossings>
<connections>
_1 ^2
_2 ^3
_3 ^4
_4 ^5
_5 ^0
_0 ^1
</connections>
<components size="3">
^0 ^2 ^4
</components>
</link>
<snappeadata label="Whitehead Link Complement">% Triangulation
m129
geometric_solution 3.66386238
oriented_manifold
CS_unknown
2 0
torus 0.000000000000 0.000000000000
torus 0.000000000000 0.000000000000
4
1 2 3 1
0132 0132 0132 3201
0 1 0 1
0 1 -1 0 -1 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 -1 0 1 0 0 -1 1 -1 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.000000000000 1.000000000000
0 0 3 2
0132 2310 3120 3120
0 1 1 0
0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 -1 1 0 0 0 0 0 0 -1 0 1 1 -1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.500000000000 0.500000000000
1 0 3 3
3120 0132 0213 3120
0 1 1 0
0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 -1 0 0 -1 1 -1 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.500000000000 0.500000000000
2 2 1 0
3120 0213 3120 0132
0 1 1 0
0 -1 0 1 0 0 1 -1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 -1 0 0 1 -1 1 0 0 1 0 -1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.500000000000 0.500000000000
</snappeadata>
<snappeadata label="Figure 8 Knot Complement" id="WCWKKgEAAAA=">% Triangulation
m004
geometric_solution 2.02988321
oriented_manifold
CS_unknown
1 0
torus 0.000000000000 0.000000000000
2
1 1 1 1
0132 1230 2310 2103
0 0 0 0
0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 -1 0 1 1 0 -1 0 0 1 0 -1 -1 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.500000000000 0.866025403784
0 0 0 0
0132 3201 3012 2103
0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 -1 0 1 -1 0 1 0 1 0 0 -1 0 1 -1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.500000000000 0.866025403784
<tri dim="2" size="8" perm="index" label="Vertex Link">
<simplex> 1 0 2 3 2 1 </simplex>
<simplex> 0 0 3 1 3 5 </simplex>
<simplex> 0 3 0 1 4 2 </simplex>
<simplex> 1 5 5 0 1 1 </simplex>
<simplex> 2 4 6 1 6 5 </simplex>
<simplex> 7 3 3 0 7 5 </simplex>
<simplex> 4 5 7 2 4 1 </simplex>
<simplex> 5 5 5 3 6 4 </simplex>
</tri>
<surfaces tri="WCWKKgEAAAA=" type="5" algorithm="34" coords="0" label="Tri-Quad Normal Surfaces">
<surface enc="279" len="14"> 0 1 1 1 2 1 3 1 7 1 8 1 9 1 10 1
<euler value="0"/>
<orbl value="T"/>
<twosided value="T"/>
<connected value="T"/>
<realbdry value="F"/>
<compact value="T"/> </surface>
</surfaces>
<surfaces tri="WCWKKgEAAAA=" type="5" algorithm="16" coords="1" label="Quad Normal Surfaces">
<surface enc="535" len="14"> 0 inf 1 inf 2 inf 3 inf 4 2 7 inf 8 inf 9 inf 10 inf 11 1
<compact value="F"/> </surface>
<surface enc="535" len="14"> 0 inf 1 inf 2 inf 3 inf 6 2 7 inf 8 inf 9 inf 10 inf 11 1
<compact value="F"/> </surface>
<surface enc="535" len="14"> 0 inf 1 inf 2 inf 3 inf 5 1 7 inf 8 inf 9 inf 10 inf 12 2
<compact value="F"/> </surface>
<surface enc="535" len="14"> 0 inf 1 inf 2 inf 3 inf 5 1 7 inf 8 inf 9 inf 10 inf 13 2
<compact value="F"/> </surface>
</surfaces>
<angles tri="WCWKKgEAAAA=" tautonly="F" algorithm="32" label="Angle Structures">
<struct len="7"> 1 1 3 1 6 1 </struct>
<struct len="7"> 2 1 5 1 6 1 </struct>
<struct len="7"> 0 1 4 1 6 1 </struct>
<struct len="7"> 0 1 2 1 3 2 6 2 </struct>
<struct len="7"> 1 2 4 1 5 1 6 2 </struct>
<spanstrict value="T"/>
<spantaut value="T"/>
</angles>
</snappeadata>
</container>
<container label="4-Manifolds">
<tri dim="4" size="2" perm="index" label="Cappell-Shaneson Knot Complement">
<simplex> 0 73 1 12 1 30 0 33 1 2 </simplex>
<simplex> 0 48 1 19 0 4 0 8 1 9 </simplex>
<fundgroup>
<group generators="2">
<reln> 0^1 1^2 0^-2 1^-3 </reln>
<reln> 1^2 0^3 1^-1 0^-2 </reln>
</group>
</fundgroup>
</tri>
<tri dim="4" size="2" perm="index" label="Twisted S³ x~ S¹" id="+DiKKgEAAAA=">
<simplex> 0 96 1 0 1 0 1 0 0 32 </simplex>
<simplex> 1 96 0 0 0 0 0 0 1 32 </simplex>
<hypersurfaces tri="+DiKKgEAAAA=" type="5" algorithm="32" coords="0" label="Normal Hypersurfaces">
<hypersurface enc="287" len="30"> 10 1 25 1 </hypersurface>
<hypersurface enc="287" len="30"> 2 1 7 1 11 1 17 1 22 1 26 1 </hypersurface>
<hypersurface enc="287" len="30"> 1 1 2 1 3 1 8 1 16 1 17 1 18 1 23 1 </hypersurface>
<hypersurface enc="287" len="30"> 0 1 4 1 5 1 14 1 15 1 19 1 20 1 29 1 </hypersurface>
<hypersurface enc="287" len="30"> 0 1 1 1 2 1 3 1 4 1 15 1 16 1 17 1 18 1 19 1 </hypersurface>
</hypersurfaces>
</tri>
</container>
<container label="2-Manifolds">
<tri dim="2" size="8" perm="index" label="Octahedron Boundary">
<simplex> 4 0 1 1 3 1 </simplex>
<simplex> 5 0 2 1 0 1 </simplex>
<simplex> 6 0 3 1 1 1 </simplex>
<simplex> 7 0 0 1 2 1 </simplex>
<simplex> 0 0 5 1 7 1 </simplex>
<simplex> 1 0 6 1 4 1 </simplex>
<simplex> 2 0 7 1 5 1 </simplex>
<simplex> 3 0 4 1 6 1 </simplex>
</tri>
<tri dim="2" size="2" perm="index" label="Klein Bottle, version 1">
<simplex> 1 1 1 5 1 0 </simplex>
<simplex> 0 1 0 5 0 0 </simplex>
</tri>
<tri dim="2" size="2" perm="index" label="Klein Bottle, version 2">
<simplex> 0 4 1 0 0 2 </simplex>
<simplex> 1 4 0 0 1 2 </simplex>
</tri>
</container>
<container label="Normal Surface Filters">
<filtercomb op="or" label="Tori, Annuli and Discs">
<filterprop orbl="T-" compact="T-" realbdry="TF" euler="0" label="Tori and Annuli">
</filterprop>
<filterprop orbl="T-" compact="T-" realbdry="T-" euler="1" label="Discs">
</filterprop>
</filtercomb>
</container>
<script label="Python Script">
<var name="tri" valueid="APLkAABgAAA=" value="3-Manifolds and Knots"/>
<code># This Python script runs through all 3-manifold triangulations and links
# in this file. It prints the first homology of each Regina triangulation,
# the volume of each SnapPea triangulation, and the HOMFLY-PT polynomial of each link.
#
# See the Regina handbook for more elaborate sample Python sessions.
for t in tri.children():
print(t.label() + ": ", end='')
if t.type() == PacketType.Triangulation3:
print(t.homology())
elif t.type() == PacketType.SnapPea:
print(t.volume())
elif t.type() == PacketType.Link:
print(t.homfly())
else:
print('Unknown packet type')
</code>
</script>
</container>
</regina>
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