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Several properties of a normal surface list and its individual surfaces are available; most of these properties can be viewed by clicking on the different tabs in the normal surface list viewer.
Above all of the tabs in the surface list viewer, a header is displayed with some basic properties of the normal surface list. These include:
a count of the total number of vertex normal surfaces contained in the list;
a reminder of how the surface list was
originally created, including the coordinate system used and
whether or not the search was restricted
to embedded surfaces only. Note that this does
not actually look at the surfaces in the list
– an unrestricted search might have only produced embedded
surfaces, but this reminder will still say
embedded/immersed/singular.
If the header says that the list was created using legacy almost normal coordinates, this means the list was created using Regina 4.5.1 or earlier, and that all surfaces with more than one octagon were deleted. See the notes below on coordinates systems for details.
The first tab contains a summary of what kinds of surfaces were found. This summary breaks the total count down into several sub-counts, as illustrated in the screenshot below. These counts are first organised by boundary, where we distinguish between closed surfaces, bounded surfaces (which have real boundary) and non-compact surfaces (which have infinitely many normal or almost normal discs). Within each section, the counts are then broken down further by Euler characteristic, orientability, and one-or-two-sidedness. For further information on these properties, see the documentation on individual surface properties.
The summary tab for a normal surface list
The Surface Coordinates tab in the surface list viewer contains a table of all the vertex surfaces that form this list. Each surface is presented as a vector in some underlying coordinate system (this coordinate system is displayed immediately above the table).
Each row of this table represents a single normal surface. The left few columns of the table contain various surface properties (see the individual surface properties section below). The remaining columns contain the actual surface coordinates. This is illustrated in the screenshot below.
The coordinates tab for a normal surface list
To change the coordinate system in which you are viewing the surfaces, simply select a new coordinate system from the drop-down box above the table. Note that this will not regenerate the vertex normal surfaces in the new coordinate system; it will simply redisplay the surfaces that you already have in the new system. So, for instance, if a spun surface is created during a quad space search and the surfaces are viewed in standard (tri-quad) coordinates, this spun surface will be redisplayed in the table with triangular coordinates of infinity. Furthermore, vertex links (which are not found during a quad space search) will not suddenly appear in the list.
The meanings of the individual coordinate columns in the table depend upon the underlying coordinate system as follows.
Standard normal coordinates simply count the number of triangular and quadrilateral discs of each type in each tetrahedron.
Triangular coordinates are labelled
0:0, 0:1,
0:2, 0:3,
1:0, 1:1,
1:2, 1:3,
2:0, etc. Coordinate
t:vt that separate vertex
v of that tetrahedron from the other
three tetrahedron vertices
(v will always be 0, 1, 2 or 3).
Quadrilateral coordinates are labelled
0:01/23, 0:02/13,
0:03/12, 1:01/23,
1:02/13, 1:03/12,
2:01/23, etc. Coordinate
t:ab/cdt that separate vertices
a and b
of that tetrahedron from vertices
c and d
of that tetrahedron
(a, b,
c and d
will always be 0, 1, 2 and 3 in some order).
Quad normal coordinates are identical to standard normal (tri-quad) coordinates except that only quadrilateral discs are considered. See Tollefson [Tol98] for details.
Standard almost normal coordinates are identical to standard normal coordinates except that octagonal discs are also considered.
Octagonal discs are labelled similarly to quadrilateral discs
(each octagonal disc also separates some two vertices of the
corresponding tetrahedron from the other two). To make it clear
which coordinates are which, each triangle label begins with a
T, each quadrilateral label begins with
a Q and each octagon label begins
with a K.
Although an almost normal surface is defined to have precisely one octagonal disc, surfaces here in standard almost normal coordinates are allowed to have no octagons at all, or several octagons (though at most one octagon type can be used). This becomes important if you wish to use vertex almost normal surfaces as a basis for generating all almost normal surfaces.
In contrast, see legacy almost normal coordinates below, where surfaces with more than one octagon are explicitly removed.
Quad-oct almost normal coordinates are identical to standard almost normal (tri-quad-oct) coordinates, except that only quadrilateral and octagonal discs are considered. See [Bur09b] for details.
Like standard almost normal coordinates, surfaces with no octagons or many octagons (but all of the same type) are allowed.
These are identical to standard almost normal (tri-quad-oct) coordinates, except that surfaces with more than one octagon are removed entirely from the list.
This was in fact the behaviour in Regina versions 4.5.1 and earlier. This behaviour was changed in Regina 4.6 because it is important to keep surfaces with multiple octagons if you wish to generate new surfaces as convex combinations of old surfaces.
Unfortunately, if a surface list was created in Regina 4.5.1 or earlier then such surfaces will already have been removed, and there is no way to get them back (except to run a new enumeration of almost normal surfaces). Such lists will always be displayed with the label legacy almost normal coordinates so it is clear what has happened.
You cannot create a new normal surface list in legacy coordinates.
The coordinates are labelled 0,
1, 2, etc. Coordinate
e represents the number of times the
surface crosses edge e of the
triangulation.
Edge numbers and the tetrahedron edges/vertices to which they correspond can be seen in the skeleton section of the triangulation viewer.
Edge weight coordinates are simply offered as a different way of viewing an existing list of normal surfaces. You cannot enumerate a new list of surfaces in edge weight coordinates.
The coordinates are labelled
0:0, 0:1, 0:2,
1:0, 1:1, 1:2,
2:0, etc. Coordinate
f:vf of the triangulation in an arc
that truncates vertex v of that face
(v will always be 0, 1 or 2).
Face numbers and the tetrahedron faces/vertices to which they correspond can be seen in the skeleton section of the triangulation viewer.
Like edge weight coordinates, face arc coordinates are offered as a different way of viewing an existing normal surface list. You cannot enumerate a new list of surfaces in face arc coordinates.
As described in the surface coordinates section above, the Surface Coordinates tab presents a table of all of the normal surfaces in the list.
In the very leftmost column of this table, the individual surfaces
are numbered from 0 to S-1, where
S is the total number of surfaces in the list.
The next few columns describe several different properties of the surfaces, as described below. Note that not all of these properties appear in all situations (for instance, Regina does not yet calculate Euler characteristic for non-compact surfaces, and the orientability column will not be shown for lists that might include immersed or singular surfaces).
Individual surfaces can be optionally named by the user to help keep track of which surface is which. Surface names are not used by Regina, and do not need to be unique.
To rename a surface, just click on the corresponding table cell and type the new name directly into that table cell.
This column shows the Euler characteristic of each surface.
This column shows whether or not each surface is orientable.
This column shows whether each surface is one-sided or two-sided.
This column describes whether each surface is bounded. It will take one of the following values:
Indicates that the surface is compact (contains finitely many discs) and closed (does not have any boundary).
Indicates that the surface is compact (contains finitely many discs) and bounded (meets the boundary of the enclosing triangulation).
Indicates that the surface is non-compact, i.e., contains infinitely many normal or almost normal discs. Examples of such surfaces are spun normal surfaces [Til08], which can be found in quad space for some ideal triangulations.
If a normal surface or a rational multiple of that surface is recognised as being a link of any interesting subcomplexes within the triangulation, these subcomplexes will be listed here. The following links are currently recognised:
Indicates that the surface is a vertex link. The corresponding vertex number in the triangulation will be listed.
Vertex numbers and the tetrahedron vertices to which they correspond can be seen in the skeleton section of the triangulation viewer.
Indicates that the surface is a thin edge link. The corresponding edge number(s) in the triangulation will be listed. Note that a surface might be both the link of one thin edge and also (independently) the link of another thin edge.
Edge numbers and the tetrahedron edges/vertices to which they correspond can be seen in the skeleton section of the triangulation viewer.
If a normal surface is recognised as playing a particular role within the triangulation, that role will be listed in this column. At most one such role will be displayed for any particular surface. The following roles are currently recognised:
Indicates that the surface is a splitting surface (contains precisely one quadrilateral per tetrahedron and no other normal or almost normal discs).
Indicates that although it is not a splitting surface, the surface is still a central surface (contains at most one normal or almost normal disc per tetrahedron). A number will be displayed indicating the number of tetrahedra that this surface meets (i.e., the number of normal or almost normal discs in the surface).
This column only appears when dealing with almost normal surfaces. Its purpose is to indicate how many octagonal discs each surface contains, and where they are.
For a given surface, if the cell in this column is empty then
the surface does not contain any octagonal discs at all (i.e., it is a
regular normal surface).
Otherwise, it will contain a single coordinate label
and the corresponding number of octagons,
such as K2: 03/12 (3 octs).
Note that there can only ever be one coordinate position in any given surface containing octagonal discs (this is a constraint forced upon the enumeration procedure). However, as of Regina 4.6, this coordinate position may contain more than one octagon. See the discussion on legacy almost normal coordinates for further details.
The Matching Equations tab in the surface list viewer contains a table of the individual matching equations that were used to form this list.
The matching equations are presented in the coordinate system that was used to originally create this surface list (see the surface list creation reference). This coordinate system is displayed above all of the tabs in the surface list viewer.
Each row of this table represents an individual matching equation. A matching equation involves setting a linear combination of surface coordinates to zero; the coefficients of this linear combination are shown in the individual table cells. A description of what the individual surface coordinates mean can be found in the individual surface coordinates section above.
The Compatibility tab in the surface list viewer shows which pairs of surfaces are compatible with each other, both locally and globally. These concepts are defined as follows:
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.
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 contains two matrices, one for local compatibility and one for global compatibility. You can switch between these two matrices by using the drop-down box labelled Display matrix.
Each matrix has dimensions
S-by-S,
where S is the total number of surfaces
in the list. Rows and columns are both numbered from 0 to
S-1 inclusive, and the
(x,y) cell of
the matrix is filled if and only if surfaces
x and y are
compatible. Here we use same numbering scheme as in the first column of
the coordinate viewer.
The global compatibility matrix for a normal surface list
Note that the global compatibility test comes with some constraints. Specifically, it cannot be run with surfaces that are empty, disconnected or non-compact (such as spun normal surfaces). For any such surfaces, the corresponding rows and columns will be hashed out in the matrix as illustrated below.
Hashing out rows and columns for spun normal surfaces
For very large lists of surfaces, the compatibility matrices will not be generated automatically (since this could take a long time and/or require a significant amount of memory). In this case, you can still compute the matrices by pressing the Calculate button at the top of the tab. The default threshold for automatic calculation is 100 surfaces, though you can change this in the normal surface preferences.
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