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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 are displayed some basic properties of the normal surface list. These include:
a count of the total number of vertex normal surfaces contained within;
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.
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.
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 all of a sudden 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.
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.
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.
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.
As described in the surface coordinates section above, the Surface Coordinates tab presents a table of all of the normal surfaces in the list.
The left few columns of this table contain a variety of surface properties as described below. Note that not all of these properties are available in all situations (for instance, Euler characteristic is not yet calculated for non-compact surfaces, and the orientability column will not even appear if the surface list might include non-embedded 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. In particular, 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, 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).
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.
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