# Normal Surfaces

## Chapter 5. Normal Surfaces

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 or fundamental normal surfaces within a triangulation in some normal coordinate system. Regina also supports octagonal almost normal surfaces.

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 Tree->New 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.

  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 7`n`-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 10`n`-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 3`n`-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 6`n`-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.

Enumerate

  Here you indicate whether you wish to enumerate all vertex normal surfaces, or all fundamental normal surfaces. Fundamental surfaces are much slower to enumerate than vertex surfaces, but in some settings can offer significantly more information.

  Vertex surfaces

    These correspond to the extreme rays of the normal surface solution cone: in the chosen coordinate system, a vertex normal surface cannot be expressed as a non-negative linear combination of normal surfaces other than multiples of itself.

    Regina will only compute one surface for each extreme ray: specifically, the smallest integer vector along that ray. In other words, the coordinates of each surface will have greatest common divisor one.

  Fundamental surfaces

    These correspond to the Hilbert basis of the normal surface solution cone: in the chosen coordinate system, a fundamental normal surface cannot be expressed as a sum of normal surfaces other than zero and itself.

Embedded surfaces only

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

  If unchecked, this indicates that you are interested not only in properly 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 or fundamental normal surfaces in the chosen coordinate system.

Once this is done, Regina will package the results 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.