Chapter 5. Normal Surfaces and Hypersurfaces

In this chapter we deal with normal surfaces within 3-manifold triangulations, and normal hypersurfaces within 4-manifold triangulations. For an overview of normal surface theory in the context of 3-manifolds (where it is much more developed), see [HLP99].

A normal surface within a 3-manifold triangulation T is a surface that meets each tetrahedron of T in a (possibly empty) collection of triangles and/or quadrilaterals, where:

  • each triangle “truncates” one vertex of the tetrahedron, separating it from the opposite triangle;

  • each quadrilateral separates two opposite edges of the tetrahedron.

See the illustration to the right for examples.

Likewise, a normal hypersurface within a 4-manifold triangulation T meets each pentachoron of T in a (possibly empty) collection of tetrahedra and/or prisms, where:

  • each tetrahedron “truncates” one vertex of the pentachoron, separating it from the opposite 3-face;

  • each prism “truncates” one edge of the pentachoron, separating it from the opposite 2-face.

Regina typically works with properly embedded normal surfaces and hypersurfaces, but it also offers basic support for immersed and singular surfaces. In the context of 3-manifolds, Regina can also 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] in which the exceptional piece may be either an octagon or a tube.

Enumerating Normal Surfaces and Hypersurfaces

Normal (hyper)surfaces are stored in lists, which typically represent all vertex or fundamental normal (hyper)surfaces within a triangulation, defined relative to some normal or almost normal coordinate system.

Regina insists on keeping normal (hyper)surface lists together with their corresponding triangulations, since normal (hyper)surfaces are expressed using coordinates relative to these triangulations. A normal surface list will always live immediately beneath the corresponding 3-manifold triangulation in the packet tree, and a normal hypersurface list will always live immediately beneath the corresponding 4-manifold triangulation. Regina will not let you modify a triangulation that has any associated normal surfaces or hypersurfaces (or angle structures), and the triangulation will be marked with a padlock to remind you of this.

To create a new normal surface or hypersurface list, select either Packet TreeNew Normal Surface List (3-D) or Packet TreeNew Normal Hypersurface List (4-D) from the menu (or press the corresponding toolbar button).

You will be offered the usual new packet window, as shown below.

There are several details that you must provide:

Triangulation

This is the triangulation that will contain your normal (hyper)surfaces. For a normal surface list, you may chose either one of Regina's native 3-manifold triangulation packets, or one of its hybrid SnapPea triangulation packets. For a normal hypersurface list, you must choose a 4-manifold triangulation. Either way, the new normal (hyper)surface list will appear within the packet tree as a child of the selected triangulation.

Coordinate system

This is the coordinate system that Regina will use to enumerate normal surfaces or hypersurfaces.

Your choice of coordinate system will affect which kinds of surfaces appear in the final solution set. For instance: within a 3-manifold triangulation, spun-normal surfaces can only appear in quadrilateral and quadrilateral-octagon coordinates, and some other types of surfaces (such as vertex links) will only appear in standard normal and standard almost normal coordinates.

For normal surfaces in 3-manifold triangulations, the available coordinate systems 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 four triangle and three quadrilateral coordinates.

Standard almost normal (tri-quad-oct)

This is a 10n-dimensional system that extends standard normal coordinates by also 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.

For normal hypersurfaces in 4-manifold triangulations, there is currently only one coordinate system available for enumerating hypersurfaces (though Regina can still view hypersurfaces in other coordinate systems):

Standard normal (tet-prism)

This is a 15n-dimensional coordinate system, where n is the number of pentachora in the underlying triangulation. Each pentachoron contributes five tetrahedron and ten prism coordinates, corresponding to the five vertices and ten edges respectively of the pentachoron that these tetrahedra and prisms truncate.

Enumerate

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

Vertex (hyper)surfaces

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

Regina will only compute one (hyper)surface for each extreme ray—specifically, the smallest integer vector along each ray. This means that the coordinates of each vertex (hyper)surface will have greatest common divisor one.

Fundamental (hyper)surfaces

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

Embedded (hyper)surfaces only

If this box is checked (the default), this indicates that you are only interested in properly embedded (hyper)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 (hyper)surfaces, but also immersed and singular (hyper)surfaces. Regina currently offers only very basic support for such (hyper)surfaces—it will not even tell you which are immersed and which are singular; moreover, the enumeration process could become much slower.

Once you are ready, click OK and Regina will enumerate all normal surfaces or hypersurfaces according to your selection. When this is finished, Regina will package the results into a normal surface or hypersurface list and open it for you to view.

Note

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.