# Analysis

Once you have built a list of surfaces, you can study these surfaces using the various tabs in the normal surface list viewer.

Above all of these tabs is a header displaying the total number of surfaces, the original enumeration parameters, and a link to the underlying triangulation. The enumeration parameters include:

- whether you asked for vertex or fundamental surfaces;

- whether you asked for embedded surfaces only, or for immersed and/or branched surfaces also;

- the normal (or almost normal) coordinate system in which you performed the enumeration.

There are some unusual comments that you might see in this header if you are using old data files and/or customised software:

Legacy surfaces

  You might see this instead of vertex or fundamental surfaces. This means that the list of surfaces was created using Regina 4.93 or earlier (which did not save information on whether the surfaces you enumerated were vertex or fundamental).

Custom surfaces

  You might see this instead of vertex or fundamental surfaces. This means that the list of surfaces was created using some specialised or user-designed algorithm, and Regina cannot give you any more specific information about which surfaces these are.

Legacy almost normal coordinates

  This indicates that the list was created using Regina 4.5.1 or earlier, and that surfaces with more than one octagon were deleted. See the discussion on legacy coordinates for details.

As of version 4.94, Regina also remembers details of the specific enumeration algorithm that was used. This information is not shown in the user interface, but can be extracted using Python scripting.

### Summary of Surfaces

The Summary tab breaks down the total number of surfaces into sub-counts for different types of surfaces, as illustrated below. At a broad level, the total is divided into closed surfaces, bounded surfaces (which have only real boundary), and spun-normal surfaces (which have infinitely many triangles). For each category that contains one or more surfaces, a table is given to break this down further according to orientability, 1-or-2-sidedness, and Euler characteristic.

Spun-normal surfaces will only be listed here in the Summary tab if your chosen coordinate system supports them.

### Details of Individual Surfaces

You can view details of the individual surfaces in the Surface Coordinates tab. This brings up a large table in which each row represents a single normal (or almost normal) surface.

Above the table are some drop-down boxes that let you view the list in different ways. These are discussed further in the notes on coordinate systems and filtering surfaces.

In the first column, surfaces are numbered 0,1,2,... (in no particular order) so that you can make note of them for later on. You can also assign arbitrary names to surfaces by typing directly into the second column; these names will be saved with your data file.

#### Properties of surfaces

The next few columns describe various properties of each surface. Some of these columns might be empty or absent in your viewer (for instance, Regina does not compute Euler characteristic for spun-normal surfaces, and it hides the orientability column if your enumeration allowed for immersed or singular surfaces).

The columns and their meanings are:

Euler

  Shows the Euler characteristic of the surface.

Orient

  Contains a tick (✓) if the surface is orientable, or the text Non-or. if it is not.

Sides

  Shows whether the surface is one-sided or two-sided.

Bdry

  Indicates what type of boundary the surface has. This will be one of:

  —

    Indicates a closed, compact surface (i.e., no boundary at all and finitely many discs).

  Real

    Indicates a compact surface with boundary (i.e., finitely many discs, some of which meet the boundary of the triangulation).

  Spun

    Indicates a spun-normal surface (i.e., a non-compact surface with infinitely many discs). These only appear when the enumeration is done in quadrilateral or quadrilateral-octagon coordinates.

    Regina can compute boundary slopes for spun-normal surfaces, but only for quadrilateral coordinates (not quadrilateral-octagon coordinates), only for orientable 3-manifolds, and only for SnapPea triangulation packets (because Regina's native triangulation packets do not provide a meridian and longitude on each cusp).

    If you have a native Regina triangulation and you wish to view boundary slopes for spun-normal surfaces, you must first convert it to a SnapPea triangulation, which will install a default (shortest, second shortest) basis on each cusp. After this you can enumerate vertex or fundamental surfaces relative to the new SnapPea triangulation, and Regina will show you the boundary slopes. If you already began with a SnapPea triangulation (for instance, one that you imported from SnapPy), then Regina will work with whatever basis SnapPea was already using for that triangulation.

    Each boundary slope is presented as a pair (`p`, `q`) for each cusp, indicating that the boundary curve passes `p` times around the meridian and `q` times around the longitude. If there are multiple cusps, these pairs will be ordered by cusp number.

Link

  Indicates if a surface is a vertex link or a thin edge link (i.e., the boundary of a small regular neighbourhood of a vertex or edge). If this is a one-sided surface whose double is a thin edge link, then it will be reported as a thin edge link for these purposes.

  The relevant vertex or edge will be listed also, using the vertex and edge numbers that appear in the first column of the vertex viewer and edge viewer. It is possible for a surface to be the thin edge link for two edges at the same time, in which case both edges will be listed.

  If the surface is not a vertex link or a thin edge link, this cell will be left empty.

Type

  Indicates if this is one of a few special types of surface that Regina identifies. Possible values are:

  Central

    There is at most one normal or almost normal disc per tetrahedron (which may be a triangle, quadrilateral or octagon). The cell will also list the total number of normal discs (i.e., the total number of tetrahedra that this surface meets).

  Splitting

    There is precisely one quadrilateral per tetrahedron and no other normal (or almost normal) discs. Although splitting surfaces are also central, only the word Splitting will be displayed.

  If the surface is not one of these types, this cell will be left empty.

Octagon

  Indicates which coordinate position contains the octagonal discs (if any), and how many octagonal discs there are. (Recall that the enumeration procedure insists that at most one coordinate position can have octagonal discs, but allows any number of octagons of that type.)

  This column only appears if you enumerated using an almost normal coordinate system.

  If this cell is empty, it means the surface does not contain any octagons at all (i.e., you have a normal surface, not an almost normal surface). Otherwise it will state which coordinate position contains the octagons and how many octagons there are (for example, K2: 03/12 (3 octs)).

#### Coordinates and Coordinate Systems

The remaining columns give the precise normal coordinates of the surface.

You can view surfaces in several different coordinate systems (not just the system you used for enumeration). To change the coordinate system, simply select a new system from the drop-down box above the table.

This will not re-enumerate surfaces in the new coordinate system; it will simply re-display the surfaces you already have. For example, if you enumerated in quadrilateral coordinates then your list will not contain any vertex links, and these will not suddenly appear when you view in standard coorinates. If your list contains spun-normal surfaces and you view them in standard coordinates, they will not disappear (instead you will see triangular coordinates of ∞).

The available coordinate systems as follows. Some options might not be available for your surface list (for instance, if you enumerated in standard normal coordinates then the almost normal systems will not appear).

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.

  Triangle coordinates are labelled 0:0, 0:1, 0:2, 0:3, 1:0, etc., where coordinate `t`:`v` counts the number of triangles in tetrahedron `t` that separate vertex `v` of that tetrahedron from the others. Here 0 ≤ `t` < `n` and `v` ∊ {0,1,2,3}.

  Quadrilateral coordinates are labelled 0:01/23, 0:02/13, 0:03/12, 1:01/23, etc., where coordinate `t`:`ab`/`cd` counts the number of quadrilaterals in tetrahedron `t` that separate vertices `a` and `b` of that tetrahedron from vertices `c` and `d`. Here 0 ≤ `t` < `n`, and `a`,`b`,`c`,`d` are some permutation of 0,1,2,3.

Quad normal

  These are the 3`n`-dimensional quadrilateral coordinates, obtained from standard normal (tri-quad) coordinates by ignoring all triangles and considering only the quadrilaterals. See [Tol98] or [Bur09a] for details.

Standard almost normal (tri-quad-oct)

  This is a 10`n`-dimensional system for almost normal surfaces, obtained from standard normal (tri-quad) coordinates by adding three octagon coordinates per tetrahedron.

  Octagon coordinates are again labelled 0:01/23, 0:02/13, 0:03/12, 1:01/23, etc., where coordinate `t`:`ab`/`cd` counts the number of octagons in tetrahedron `t` that separate vertices `a` and `b` from `c` and `d`.

  To avoid ambiguity, all triangle, quadrilateral and octagon coordinate labels are prefixed with T, Q and K respectively. The full coordinates are therefore T0:0, T0:1, T0:2, T0:3, Q0:01/23, Q0:02/13, Q0:03/12, K0:01/23, K0:02/13, K0:03/12, T1:0, etc.

Quad-oct almost normal

  These are the 6`n`-dimensional quadrilateral-octagon coordinates, obtained from standard almost normal (tri-quad-oct) coordinates by ignoring all triangles and considering only the quadrilaterals and octagons. See [Bur10b] for details.

Legacy almost normal (pruned tri-quad-oct)

  Legacy coordinates are to support data files created in Regina 4.5.1 or earlier. These are like standard almost normal (tri-quad-oct) coordinates, except that surfaces with more than one octagon are deleted entirely.

  If you created your surfaces in Regina 4.5.1 or earlier, there is no way to recover those surfaces with multiple octagons—they would have been deleted when you originally enumerated them. Instead you will need to enumerate them again. Your list will always be displayed with the label legacy almost normal coordinates to remind you of this.

  From Regina 4.6 onwards, the enumeration process now keeps almost surfaces with multiple octagons (though they must be in the same coordinate position). This is important if you wish to generate new almost normal surfaces by taking convex combinations of old surfaces. If you are only interested in surfaces with one octagon, the Octagon column makes them easy to spot.

Edge weight

  This system has one coordinate for each edge of the triangulation. The coordinates are labelled 0, 1, 2, etc., where coordinate `e` counts the number of times the surface crosses edge number `e`.

  Edge numbers and the tetrahedron edges to which they correspond can be found in the edge viewer, under the triangulation's Skeleton tab.

  Edge weight coordinates are offered for viewing only. You cannot enumerate surfaces in edge weight coordinates.

Triangle arc

  This system has three coordinates for each triangle (2-face) of the triangulation. The coordinates are labelled 0:0, 0:1, 0:2, 1:0, etc., where coordinate `t`:`v` represents the number of times the surface slices through triangle `t` of the triangulation in an arc that truncates vertex `v` of that triangle. Here `v` ∊ {0,1,2}.

  Triangle numbers and the tetrahedron faces to which they correspond can be found in the triangle viewer, under the triangulation's Skeleton tab. The three vertices 0,1,2 of each triangle correspond to the ordering of tetrahedron vertices that you see in the rightmost column of the triangle viewer.

  Triangle arc coordinates are likewise offered for viewing only. You cannot enumerate surfaces in triangle arc coordinates.

### Tip

If you grab and resize one of the coordinate columns, all of the coordinate columns will be resized at once. This is useful if you wish to fit as many columns on the screen as possible.

### Original Matching Equations

The Matching Equations tab shows a table with the individual matching equations that were used when you enumerated this list.

The matching equations will always use the same coordinate system that you used during enumeration. Remember that this coordinate system is always displayed above all of the tabs.

Each row of this table represents an individual matching equation. Each equation is a homogeneous linear equation, and the coefficients for each coordinate position are shown in the individual table cells. See the coordinate viewer for details on how the coordinate columms are labelled.

### Tip

Like the coordinate viewer, if you grab and resize one of the columns then all columns will be resized at once. This is useful if you wish to fit as much of the matrix on screen as possible.

### Compatibility Between Surfaces

The Compatibility tab shows which pairs of surfaces are locally and globally compatible with each other. This means:

Locally compatible

  Two surfaces are locally compatible if they are able to avoid intersection in any given tetrahedron of the triangulation (though not necessarily in all tetrahedra simultaneously).

  In other words, two surfaces are locally compatible if, in each tetrahedron, they together use at most one quadrilateral or octagonal disc type.

Globally compatible

  Two surfaces are globally compatible if they are able to avoid intersection in all tetrahedra of the triangulation simultaneously.

  In other words, two surfaces are globally compatible if they can be made disjoint within the triangulation.

The Compatibility tab displays two matrices (one at a time): one shows local compatibility and one shows global compatibility. You can switch between them using the drop-down box indicated below.

Each matrix has dimensions `S` × `S`, where `S` is the total number of surfaces in the list. Rows and columns are both numbered 0,...,`S`-1. The cell at position (`x`,`y`) is filled if and only if the surfaces numbered `x` and `y` are compatible. Recall that surfaces are numbered in the leftmost column of the coordinate viewer.

For some surfaces, Regina cannot test global compatibility. These include surfaces that are empty, disconnected, or spun. In such cases the corresponding rows and columns will be hashed out, as illustrated below.

If you have too many surfaces in your list, Regina will not generate these matrices automatically. You can still compute them by pressing the Calculate button (indicated below).

### Crushing and Cutting

You can cut along a normal surface, or crush it using the techniques of Jaco and Rubinstein [JR03]. See the chapter on triangulations for details.