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A normal surface is a surface within a
3-manifold triangulation that meets each tetrahedron in a collection
of triangles and/or quadrilaterals, as illustrated to the right.
For a nice overview of normal surface theory, see
[HLP99].
Regina typically works with embedded normal surfaces, but it also offers basic support for immersed and singular surfaces. In addition, it can work with almost normal surfaces (which are like normal surfaces but with an extra “exceptional disc”) and spun normal surfaces (with infinitely many triangles spinning out towards the vertices).
For almost normal surfaces, Regina uses the restricted definition of Thompson [Tho94] where the exceptional piece is an octagon>. Regina does not currently support the more general definition of Rubinstein [Rub95] where the exception piece may be either an octagon or a tube.
Normal surfaces are stored in lists, which typically represent all vertex normal (or almost normal) surfaces within a triangulation in some normal coordinate system.
A normal su rface list must remain “connected” to the corresponding triangulation. It always lives immediately beneath the triangulation in the packet tree, and the triangulation cannot be modified unless all of its normal surface lists are deleted. The triangulation will be marked with a small padlock to remind you of this.
To create a new normal surface list, select → from the menu (or press the corresponding toolbar button).
You will be offered the usual new packet window, as shown below.
In addition to the usual label option, there are important details that you must provide:
This is the triangulation that will contain your normal surfaces. The new normal surface list will appear as a child of this triangulation in the packet tree.
This is the coordinate system that Regina will use to enumerate normal surfaces. If you have a favourite system that you use all the time, you can change the default in Regina's normal surface options.
Your choice of coordinate system will affect which surfaces appear in the final solution set. For instance, spun normal surfaces only appear in quadrilateral and quadrilateral-octagon coordinates; other surfaces (such as vertex links) only appear in standard normal and standard almost normal coordinates.
Your options are:
This is the standard 7n
-dimensional
coordinate system that typically appears in papers and textbooks
(where n
is the number of tetrahedra).
Each tetrahedron contributes three triangle and four
quadrilateral coordinates.
This is a 10n
-dimensional system,
obtained from standard normal coordinates by adding three
octagon coordinates per tetrahedron.
This system supports almost normal surfaces.
These are the 3n
-dimensional
quadrilateral coordinates,
obtained from standard normal coordinates by simply ignoring all
triangles. See [Tol98] or
[Bur09a] for details.
This system supports spun normal surfaces.