| Normal Surfaces |
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The primary motivation for originally writing Regina was to create a
tool for calculating vertex normal surfaces within a triangulation.
A normal surface inside a 3-manifold triangulation is a surface composed entirely of normal discs. A normal disc is a properly embedded disc within a single tetrahedron, and must be either a triangle (whose boundary runs across three of the four tetrahedron faces) or a quadrilateral (whose boundary runs across all four of the tetrahedron faces).
The Regina logo shown on this page illustrates all four possible triangle types and one of the three possible quadrilateral types within a tetrahedron.
Regina will also calculate almost normal surfaces, which have the same restrictions as normal surfaces except that they must contain precisely one additional octagonal disc. The boundary of an octagonal disc runs across each tetrahedron face twice, and there are three possible types of octagonal disc.
More general definitions of almost normal surfaces allow for an annular piece instead of an octagonal disc. Regina works with the more restrictive octagon-only definition as described by Thompson [Tho94] (which is much simpler to deal with computationally).
Normal surfaces are represented as integer vectors indicating how many discs of each type in each tetrahedron are used to make up the surface.
For a summary of enough normal surface theory to use this program, see a reference such as [Bur09a] or [HLP99].
A normal surface list in the packet tree must be a child packet of the triangulation in which the surfaces are contained.
Furthermore, since each normal surface is stored as a vector relative to the triangulation's tetrahedra, a triangulation cannot be modified once it has a normal surface list as a child. If you wish to edit such a triangulation, try cloning the packet (excluding descendants so the normal surfaces are not cloned as well) and edit the clone instead.
Triangulations containing normal surface lists are marked with a small padlock in the packet tree to remind you that they cannot be edited.
A new list of normal surfaces can be created through the -> menu item (or the corresponding toolbar button).
Only vertex normal surfaces will be placed in the new list; all other normal surfaces can be formed as convex combinations of these vertex surfaces.
For almost normal surfaces, older versions of Regina used to ignore any vertex surfaces with more than one octagonal disc. As of Regina 4.6, this behaviour has changed — now Regina will keep surfaces with more than one octagon, though it still insists on at most one octagon type. This is important if you wish to use vertex almost normal surfaces as a basis for generating all almost normal surfaces, as suggested above.
If you have a data file from Regina 4.5.1 or earlier, then any vertex surfaces with more than one octagon will have already been removed. The only way to get them back is to re-run the almost normal surface enumeration using Regina 4.6 or later.
Before the normal surface list is created, you will be asked for some additional details as follows.
Here you must select the triangulation that will contain the new normal surfaces (this must be a triangulation already in the packet tree).
Here you must select the coordinate system in which the vertex normal surfaces will be enumerated.
Whether or not almost normal surfaces are enumerated (in addition to just normal surfaces) will depend upon this choice of coordinate system. Currently almost normal surfaces are only enumerated when using standard almost normal coordinates or quadrilateral-octagon coordinates.
Note that this coordinate system refers only to the initial enumeration; you will be able to view these normal surfaces in a variety of different coordinate systems later on.
Note also that this choice of coordinate system will affect which surfaces are produced. For instance, quad normal coordinates produce spun normal surfaces which standard normal coordinates do not, and standard normal coordinates produce vertex links which quad normal coordinates do not.
The available coordinate systems are described in detail in the documentation for the coordinates tab. If there is a particular coordinate system that you regularly use, the default offering can be changed in the normal surface preferences.
Here you must specify whether the search should be restricted to embedded surfaces only (this is the default and is all that is considered in most of the published theory).
Be aware that a search that is broadened to all surfaces (embedded, immersed and singular) could well take much longer to run than the corresponding embedded-only search. Enumeration of immersed and singular surfaces is only supported for normal surfaces, not almost normal surfaces.
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