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A variety of triangulation properties are available for calculation; most of these can be viewed by clicking on the different tabs available in the triangulation viewer.
Above all of the tabs in the triangulation viewer are displayed some basic properties of the triangulation as listed below.
Signifies that the triangulation contains no tetrahedra at all.
Signifies that the triangulation contains an invalid vertex (a vertex whose link has boundary but is not a disc) or an invalid edge (an edge glued to itself in reverse).
Signifies that the triangulation has no boundary faces and no ideal vertices. All vertex links must be 2-spheres.
Signifies that the triangulation contains an ideal vertex (a vertex whose link is closed but not a 2-sphere).
Signifies that the triangulation contains a boundary face.
Shows whether the triangulation is orientable.
Shows whether the triangulation forms a single connected piece.
The Gluings tab in the triangulation viewer shows how individual faces of individual tetrahedra are glued to each other. See the reference for editing face gluings for details on how this information is presented.
The Skeleton tab in the triangulation viewer contains two smaller tabs that offer combinatorial information about the skeleton and dual skeleton of the triangulation.
The Skeletal Components tab in the skeleton viewer offers combinatorial details of the individual vertices, edges, faces, components and boundary components that make up the triangulation (note that each ideal vertex is treated as an individual boundary component).
In the Skeletal Components tab itself appears the total number of vertices, edges and so on. Beside each count is a button that brings up a table providing explicit structural details for each individual object as described below.
The vertex table, the edge table and the face table all have similar structures.
In the first column, the vertices, edges or faces are individually numbered from 0 upwards.
The second column (labelled Type) describes properties of each vertex, edge or face using zero or more of the property labels described below. Note that an object might have no interesting properties at all.
The third column in the table lists the degree of the vertex, edge or face.
The final column lists precisely which vertices
of which tetrahedra form the vertex, edge or face (recall that the
four vertices of an individual tetrahedron are numbered 0, 1, 2 and
3). An example for an edge is
1(20), 4(12), 2(21) which shows that
the edge is the result of identifying edge 20 of tetrahedron 1,
edge 12 of tetrahedron 4 and edge 21 of tetrahedron 2 (where edge
21 is the edge joining vertex 2 with vertex 1 of that tetrahedron
and so on).
The order of vertices is important; this example also shows that
vertex 2 of tetrahedron 1, vertex 1 of tetrahedron 4 and vertex 2
of tetrahedron 2 all represent the same end of the edge.
In the specific case of edges, the order in which the tetrahedra are presented in this column is also important. Tetrahedra are presented in the order in which they are seen when following around the edge link.
The component and boundary component tables are similarly structured.
The first column individually numbers the components or boundary components from 0 upwards.
The second column describes properties of each component or boundary component, again using zero or more of the property labels described below.
The third column lists the number of tetrahedra in the component or the number of faces in the boundary component. Note that a boundary component formed by an ideal vertex will have zero faces.
The fourth column lists the individual tetrahedra forming the component or the individual faces or vertices forming the boundary component. In the case of a boundary component, faces and vertices will be listed using their individual identification numbers as found in the first column of the face and vertex tables described above.
The following information may appear in the Type column for a vertex table.
Indicates a standard boundary vertex; the vertex link is a disc.
Indicates a torus cusp.
Indicates a Klein bottle cusp.
surface)
Indicates a non-standard cusp; surface
will describe the orientability and genus of the cusp surface.
An example is Cusp (orbl, genus 3).
Indicates a non-standard boundary vertex; the vertex link has boundary but is not a disc.
The following information may appear in the Type column for an edge table.
Indicates a boundary edge.
Indicates an edge glued to itself in reverse; the midpoint of this edge is a projective plane cusp.
The following information may appear in the Type column for a face table.
Indicates a boundary face.
No vertices or edges of the face are identified.
Two vertices of the face are identified; all edges are distinct.
All three vertices of the face are identified; all edges are distinct.
Two edges of the face are identified to form a Mobius band (causing all three vertices to be identified); the third edge remains distinct.
Two edges of the face are identified to form a cone (causing two vertices to be identified); the third edge and third vertex remain distinct.
Two edges of the face are identified to form a cone and all the third vertex is identified with the others; the third edge remains distinct.
All three edges of the face are identified, some with orientable and some with non-orientable gluings.
All three edges of the face are identified using non-orientable gluings; note that this forms a spine for the lens space L(3,1).
The following information may appear in the Type column for a component table.
Indicates that the component contains no ideal vertices (see below).
Indicates that the component contains an ideal vertex (one whose link is closed but not a 2-sphere).
Indicates that the component is orientable.
Indicates that the component is non-orientable.
The second smaller tab in the skeleton viewer is the Face Pairing Graph tab. This offers a visual representation of the face pairing graph of the triangulation.
The face pairing graph is essentially the dual 1-skeleton of the triangulation — every vertex of the graph represents a tetrahedron, and every edge represents a pair of tetrahedron faces that are joined together. It follows that the face pairing graph of a closed triangulation is 4-valent, whereas for a bounded triangulation there may be vertices of degree three or less.
Regina relies on the external application Graphviz for drawing the graph. As such, if Graphviz is not installed on your system then the face pairing graph cannot be displayed. Graphviz is a widely-used application, and packages for it are offered with most GNU/Linux distributions.
If Graphviz is installed but Regina cannot find it, the location of the Graphviz executable can be specified in Regina's triangulation options.
The Algebra tab in the triangulation viewer contains a variety of smaller tabs describing different algebraic invariants of the triangulation.
Note that if the triangulation contains ideal vertices, the algebraic properties will be calculated as if these vertices had been truncated. These truncated ideal vertices will also be considered part of the boundary.
There is no guarantee that invalid edges (edges glued to themselves in reverse) will be dealt with correctly. In particular, the projective plane cusps they produce may be ignored.
The Homology tab of the algebra viewer presents various homology groups of the triangulation. These include H1(M) (the first homology group), H1(M, Bdry M) (the relative first homology group with respect to the boundary), H1(Bdry M) (the first homology group of the boundary), H2(M) (the second homology group) and H2(M ; Z2) (the second homology group with coefficients in Z2).
The Fund. Group tab of the algebra viewer contains the fundamental group of the triangulation, presented as a set of generators and relations. The generators and relations will be passed through a fairly weak recognition routine, and if the group is recognised then its common name will be given as well.
If you have GAP (Groups, Algorithms and Programming) installed on your system, you can use GAP to simplify the fundamental group presentation. Regina does perform some automatic simplifications of its own, but GAP's simplification routines are far more powerful.
To simplify the group presentation using GAP, press the button beneath the fundamental group. While Regina is talking to GAP, the state of the conversation will be displayed in a separate dialog box.
If you wish to see a full transcript of the conversation between Regina and GAP, start Regina from the command-line by running regina-kde. A full transcript of the conversation will be written to the text console beneath the regina-kde command.
If Regina is having trouble starting GAP, you can tell it where to find GAP in the triangulation options.
You can try to simplify using GAP more than once. Sometimes GAP gets a better presentation for the group when run a second or third time.
The Turaev-Viro tab of the algebra viewer allows the calculation of arbitrary Turaev-Viro state sum invariants.
Each Turaev-Viro invariant corresponds to a particular set of initial
data, as described in the paper of Turaev and Viro in which these
invariants appear [TV92]. In particular,
Section 7 of this paper describes the initial data as determined
by an integer r >= 3 and a
root of unity q0
of degree 2r.
To calculate a Turaev-Viro invariant, a pair of integers
r, root
should be entered into the text box provided. The integer
r is used directly in the initial data
as described above. The integer root
should be strictly between 0 and 2r;
this identifies which particular root of unity
q0 to use.
Note that only small values of r
should be used, since the time required to calculate the
invariant grows exponentially with r.
Once calculated, the invariant will be displayed in the list box provided. Turaev-Viro invariants are saved to file with the triangulation, so they do not need to be recalculated when the file is closed and reopened.
The Cellular Info tab of the algebra viewer displays information on the standard and dual CW-decompositions, a variety of homology groups and mappings, the Kawauchi-Kojima invariants of the torsion linking form, and comments on where the triangulation might be embeddable. Many thanks are due to Ryan Budney for implementing these features.
Like the other algebraic results, all information on the Cellular Information tab refers to the compact manifold that the triangulation represents. In particular, ideal vertices are truncated and considered as real boundary surfaces.
Specific results on this tab include:
Lists the number of cells of each dimension for a standard CW-decomposition of the manifold. This is a list of four numbers, counting the 0-cells, 1-cells, 2-cells and 3-cells respectively.
For a closed triangulation (no ideal vertices), this is simply the number of vertices, edges, faces and tetrahedra. For an ideal triangulation this takes into account the truncation of ideal vertices, and is therefore a little more complex.
Lists the number of cells of each dimension in the dual CW-decomposition. As before, this is a list of four numbers that count the 0-cells, 1-cells, 2-cells and 3-cells in order.
Gives the Euler characteristic of the manifold, as computed from the CW-decompositions.
Lists the homology groups of the manifold with coefficients in the integers. The four groups H0, H1, H2 and H3 are listed in order.
Lists the homology groups of the boundary of the manifold, again with coefficients in the integers. The three groups H0, H1 and H2 are listed in order.
Since the boundary is a submanifold of the original manifold, there is an induced map on the first homology group. This item on the Cellular Information tab describes some properties of this induced map.
Given an oriented 3-manifold M,
there is a symmetric bilinear function
tH1(M) x tH1(M) —> Q/Z
where tH1(M) is the torsion subgroup of H1(M).
It is computed in this way: let x and y be 1-dimensional
torsion homology classes. Then nx is the boundary of
some 2-cycle z (transverse to y) for some integer n.
The torsion linking form of
x and y is the
oriented intersection number of z and y, divided by n.
Kawauchi and Kojima gave a complete classification of such torsion linking forms [KK80]. Regina computes the torsion linking form, and implements the Kawauchi-Kojima classification. This is the first of the three Kawauchi-Kojima invariants of the torsion linking form on the torsion subgroup of H1.
This is the torsion form rank vector,
which
lists the prime power decomposition of the torsion subgroup of
H1(M).
For example, if H1(M) is a direct sum of n copies of
Z20 and m copies of
Z18, then the torsion form rank vector
would be: 2(m n) 3(0 m) 5(n)
since
the group is isomorphic to
mZ2^1 +
nZ2^2 +
0Z3 +
mZ3^2 +
nZ5.
Note that the Kawauchi-Kojima invariants are only computed for connected orientable manifolds.
This is the second of the three Kawauchi-Kojima invariants described above [KK80]. This is the 2-torsion sigma vector, and is relevant for manifolds in which H1 has 2-torsion. It is an orientation-sensitive invariant, where the orientation is chosen so that the first tetrahedron in the triangulation is positively-oriented with its standard parametrization.
As noted above, the Kawauchi-Kojima invariants are only computed for connected orientable manifolds.
This is the third of the three Kawauchi-Kojima invariants of the torsion linking form [KK80]. This is the odd p-torsion Legendre symbol vector, originally constructed by Seifert, and is relevant for manifolds in which H1 has odd torsion.
Again, the Kawauchi-Kojima invariants are only computed for connected orientable manifolds.
This final entry in the Cellular Information tab comments upon where the manifold might embed. In particular, it attempts to make deductions about whether the manifold might embed in R3, S3, S4 or a homology sphere. If the manifold is orientable it tests for the hyperbolicity of the torsion linking form. It also performs the Kawauchi-Kojima 2-torsion test, useful for determining if a manifold with boundary does not embed in any homology 4-sphere.
The information in this field might change over time (specifically, it might become more detailed in future releases of Regina as more tests become available). Currently it examines the homology, the Kawauchi-Kojima invariants and some other elementary properties, and uses C. T. C. Wall's theorem that 3-manifolds embed in S5.
These comments are provided for both orientable and non-orientable manifolds. In the non-orientable case they may provide additional information regarding the embeddability of the orientable double cover.
The paper [Bud08] shows how some of this information can be used in studying embedding problems.
The Composition tab in the triangulation viewer presents information regarding the combinatorial structure of the triangulation.
At the top of the composition tab is an area for performing
isomorphism and subcomplex tests. A drop-down box is provided in
which a second triangulation T can be
selected. Each time a new triangulation T
is chosen, an isomorphism / subcomplex test is performed.
Specifically, the program tests for any of the following relationships:
this triangulation and T are isomorphic;
this triangulation is isomorphic to a subcomplex of
T;
T is isomorphic to a subcomplex of
this triangulation.
Forming the lower portion of the composition tab is a large region in which further details of the triangulation composition and the underlying 3-manifold are displayed. Presented here is all the information that Regina can deduce simply by searching for well-structured features within the triangulation that it can recognise. Some triangulations can be completely identified; others (frequently poorly-structured non-minimal triangulations) yield little or no useful information at all.
If the triangulation and/or its underlying 3-manifold can be identified, these are reported. In addition, the program searches for a variety of standard subcomplexes within the triangulation. If any of these subcomplexes are found, these too are reported along with precise details of where they occur. Many of these subcomplexes are described in detail in Burton's PhD thesis [Bur03] and the papers [Bur07c] and [Bur08a]; these descriptions can be quite laborious and will not be repeated here.
If it exists, the dehydration string for the triangulation is also reported here. Dehydration strings, described by Callahan, Hildebrand and Weeks [CHW99], allow complex triangulations to be represented by a simple strings of letters. Regina can convert a dehydration string back into a real triangulation, either by creating a new triangulation or by importing a list of dehydration strings.
Note that dehydration strings only exist for connected triangulations with no boundary faces and at most 25 tetrahedra. Furthermore, they implicitly describe a canonical ordering of the triangulation tetrahedra and vertices — dehydrating and then recreating a triangulation may not give you the same triangulation back, though it will certainly give you an isomorphic triangulation.
Viewing the triangulation composition
You can copy text from the composition box by right-clicking on a line of text and selecting from the pop-up menu that appears. This is particularly useful for copying dehydration strings.
Certain properties of a triangulation are defined by the types of normal surfaces it contains. These properties can be found under the Surfaces tab in the triangulation viewer.
For sufficiently large triangulations, some of these properties will
not be calculated by default since the calculations could be quite
slow. If a property is listed as Unknown, press
the corresponding button
(and be prepared to wait). Just how large is “sufficiently
large” can be adjusted in the
triangulation options.
The available properties pertaining to normal surfaces are as follows.
Determines whether the triangulation is 0-efficient. A triangulation is 0-efficient if its only normal spheres and discs are vertex linking, and if it has no 2-sphere boundary components.
Determines whether the triangulation has a splitting surface. A splitting surface is a compact normal surface consisting of precisely one quad per tetrahedron and no other normal (or almost normal) discs.
Determines whether this is in fact a triangulation of the 3-sphere. The 0-efficiency algorithms of Jaco and Rubinstein [JR03] are used, which in turn include components of Rubinstein's original 3-sphere recognition algorithm [Rub95], [Rub97].
Determines whether this is a triangulation of the 3-dimensional ball. This computation is based on the 3-sphere recognition algorithm, as described above.
SnapPea is an excellent piece of software written by Jeffrey Weeks with a strong focus on hyperbolic 3-manifolds. For further information on SnapPea, please visit its website at http://www.geometrygames.org/SnapPea/.
Since version 4.2 (July 2005), portions of the SnapPea kernel have been built into Regina. As a result, more information about the geometries of triangulations and their underlying 3-manifolds is available. This information is presented in the SnapPea tab in the triangulation viewer.
SnapPea calculations are not available for all triangulations. In fact, a triangulation must satisfy several constraints before Regina will even try to pass it to SnapPea for processing. This is a conservative decision on Regina's part, and in no way reflects a bug in SnapPea's code.
Amongst other constraints, a triangulation must have no boundary faces and every vertex must have a torus or Klein bottle link. If SnapPea calculations are not available, the SnapPea tab will include a note giving at least one reason why.
Note that you can bypass some of these restrictions — by configuring Regina appropriately you can allow some closed triangulations to be passed. See the SnapPea options reference for details and relevant warnings.
When the SnapPea tab is selected, the SnapPea kernel attempts to solve for a complete hyperbolic structure. The following results are then presented.
This summarises the type of solution that was found. Possible solution types are as follows.
All tetrahedra are either positively oriented or flat, though the entire solution is not flat and no tetrahedra are degenerate.
The volume is positive, but some tetrahedra are negatively oriented.
All tetrahedra are flat, but none have shape 0, 1 or infinity.
At least one tetrahedron has shape 0, 1 or infinity.
The volume is zero or negative, but the solution is neither flat nor degenerate.
The gluing equations could not be solved.
Note that the details above are available by selecting -> from the menu and clicking on the particular solution type being displayed.
This lists the volume of the underlying 3-manifold, along with the estimated number of decimal places of accuracy. Note that the accuracy measure is an estimate only (based on the differences between terms in Newton's method).
Regina ships with a number of prepackaged censuses of 3-manifold triangulations. The -> menu item will search for the current triangulation within all of the available censuses. Any matches will be reported and noted in a new text packet that will be created directly beneath the current triangulation.
For a match to occur, the triangulation does not need to use the same tetrahedron and face numbers as in the census; any isomorphic copy will suffice.
The list of censuses that will be searched can be configured in the census options. All triangulations within each of the configured census files will be examined.
By default, the censuses that are searched include censuses of closed orientable and non-orientable 3-manifold triangulations [Bur07b] [Bur07a], cusped and closed hyperbolic 3-manifold triangulations [CHW99] [HW94], and knot and link complements (tabulated by Joe Christy).
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