Analysis

Shows the Euler characteristic of the surface.

Shows the first homology group of the hypersurface.

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

Shows whether the (hyper)surface is one-sided or two-sided.

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

Indicates if a (hyper)surface is the link of some face of the underlying triangulation. In general, the link of a face is created by forming the boundary b of a small regular neighbourhood of the face. We call this link thin if the (hyper)surface b is already normal; otherwise we refer to the normalised link as the result of applying the standard normalisation process to b (which could change the topology, and in some extreme scenarios could even reduce b to the empty surface).

If this (hyper)surface is the link of a face then the relevant faces will be listed here, using the vertex, edge, triangle and/or tetrahedron numbers that appear in the first column of the skeletal face viewers. Note that it is possible for a (hyper)surface to be the link of several faces at the same time, possibly even of different dimensions, in which case they will all be reported here.

Any thin links will be reported first (indicated by Thin:), followed by any normalised links (indicated by Normal:). Vertices, edges, triangles and tetrahedra will be indicated by the abbreviations vtx, edge, tri and tet. In 3-manifold triangulations, Regina will only detect vertex, edge and triangle links; in 4-manifold triangulations, Regina will detect vertex, edge, triangle and tetrahedron links.

Some examples of things you might see in this column:

A one-sided (hyper)surface whose double is the link of a face will also be reported as such in this table (though this can never happen with vertex links).

If the (hyper)surface is not the link of a face as described above, then this cell will be left empty.

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

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

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)).

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.

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.

These are the 3n-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.

This is a 10n-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.

These are the 6n-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 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.

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.

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.

This is the 15n-dimensional coordinate system that counts all tetrahedra and prisms of all types (where n is the number of pentachora in the underlying triangulation). Each pentachoron contributes five triangle and ten prism coordinates.

Tetrahedron coordinates are labelled 0:0, 0:1, 0:2, 0:3, 0:4, 1:0, etc., where coordinate p:v counts the number of tetrahedra in pentachoron p that truncate vertex v of that pentachoron. Here 0 ≤ p < n and v ∊ {0,1,2,3,4}.

Prism coordinates are labelled 0:01, 0:02, 0:03, 0:04, 0:12, …, 0:34, 1:01, etc., where coordinate p:ab counts the number of prisms in pentachoron t that truncate the edge joining vertices a and b of that pentachoron. Here 0 ≤ p < n, and a,b ∊ {0,1,2,3,4} with a < b.

These are the 10n-dimensional prism coordinates, obtained from standard normal (tet-prism) coordinates by ignoring all tetrahedra and considering only the prisms.

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 hypersurface crosses edge number e.

Edge numbers and the pentachoron 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 hypersurfaces in edge weight coordinates.