Chapter 4. Normal Surfaces

Table of Contents

Enumerating Normal Surfaces
Analysis
Summary of Surfaces
Details of Individual Surfaces
Original Matching Equations
Compatibility Between Surfaces
Crushing and Cutting
Using Filters
Creating Filters
Filtering by Surface Properties
Combining Several Filters

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.

Enumerating Normal Surfaces

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 surface 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 Packet TreeNew Normal Surface List 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:

Triangulation

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.

Coordinate system

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:

Standard normal (tri-quad)

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.

Standard almost normal (tri-quad-oct)

This is a 10n-dimensional system, obtained from standard normal coordinates by adding three octagon coordinates per tetrahedron.

This system supports almost normal surfaces.

Quad normal

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.

Quad-oct almost normal

These are the 6n-dimensional quadrilateral-octagon coordinates, likewise obtained from standard almost normal coordinates by ignoring all triangles. See [Bur10b] for details.

This system supports both almost normal surfaces and spun normal surfaces.

Embedded surfaces only

If this box is checked (the default), this indicates that you are only interested in embedded surfaces. This is consistent with most of the normal surface literature.

If unchecked, this indicates that you are interested not only in embedded normal surfaces, but also immersed and singular surfaces. Regina currently offers only very basic support for such surfaces (it will not even tell you which are immersed and which are singular); moreover, the enumeration of surfaces will become much slower.

Once you are ready, click OK and Regina will enumerate all vertex normal surfaces in the chosen coordinate system.

Once this is done, Regina will package the vertex normal surfaces into a normal surface list and open it for you to view.

If you selected an almost normal coordinate system, Regina will enforce at most one octagon type but it will not enforce precisely one octagon disc (this makes it easier for users to work with convex combinations of vertex almost normal surfaces). As a result, you might see surfaces with multiple octagons (but all of the same type), or surfaces with no octagons at all. The coordinate viewer makes it easy to spot which is which.

Warning

If you have a data file from Regina 4.5.1 or earlier, it will not show almost normal surfaces with more than one octagon. See the discussion on legacy coordinates for details.