# Analysis

Regina offers a wealth of information about 3-manifold triangulations, spread across the many different tabs in the triangulation viewer. Here we walk through the different properties and invariants that Regina can compute.

### Validity, Orientability and Other Basic Properties

At the top of each triangulation viewer is a banner listing some basic properties of the triangulation (circled in red above). The following words might appear:

Closed

  Signifies that the triangulation has no boundary triangles and no ideal vertices. In other words, the link of every vertex is a 2-sphere.

Ideal bdry

  Signifies that at least one vertex of the triangulation is ideal. That is, the vertex link is a closed surface but not a 2-sphere.

  You can locate any ideal vertices using the skeleton viewers.

Real bdry

  Signifies that the triangulation contains one or more boundary triangles.

Orientable / non-orientable / oriented / not oriented

  The words orientable or non-orientable indicate whether or not the triangulation represents an orientable 3-manifold.

  If the triangulation is orientable, Regina will also tell you whether or not it is oriented; that is, whether the vertex labels 0, 1, 2 and 3 on each tetrahedron induce a consistent orientation for all tetrahedra in the entire triangulation.

  If you need a consistent orientation for all tetrahedra but you see orientable but not oriented instead, you can fix this by orienting your triangulation.

Connected / disconnected

  The words connected or disconnected indicate whether or not the triangulation forms a single connected piece.

Invalid triangulation

  Signifies that the triangulation is “broken” to the point where Regina cannot do any serious work with it. This can happen for one of two reasons: (i) some vertex link is a surface with boundary but not a disc; or (ii) some edge is identified with itself in reverse.

  You can locate the offending vertex or edge using the skeleton viewers. If the triangulation is invalid, no other information will appear in the banner.

Empty

  Signifies that the triangulation contains no tetrahedra at all. In this case, no other information will appear in the banner.

### Viewing Tetrahedron Face Gluings

The Gluings tab shows how the various tetrahedron faces are glued to each other in pairs. The face gluings are presented in a table: each row represents a tetrahedron, and the four columns on the right represent the four triangular faces of each tetrahedron. Tetrahedra are numbered 0,1,2,..., and the four vertices of each tetrahedron are numbered 0,1,2,3.

Each cell of this table represents a single face of a single tetrahedron. For instance, the cell circled in red above represents face 123 of tetrahedron 5 (that is, the face formed from vertices 1,2,3 of tetrahedron 5).

The contents of the cell show how the face is glued. In the example above, the circled cell contains `0 (230)`, indicating that face 123 of tetrahedron 5 is glued to face 230 of tetrahedron 0 using the affine map that matches vertices 1,2,3 of tetrahedron 5 with vertices 2,3,0 of tetrahedron 0 respectively. The same gluing can be seen from the opposite direction in the row for tetrahedron 0.

An empty cell indicates that a face is not glued to anything at all; that is, the face forms part of the boundary of the 3-manifold. In the table above there are two boundary triangles: face 123 of tetrahedron 0, and face 123 of tetrahedron 4. In our example these join together to form the torus boundary of the figure eight knot complement.

You can modify the triangulation by typing new face gluings directly into this table. See the section on modifying triangulations for details.

### Skeletal Information

The Skeleton tab holds two smaller tabs offering combinatorial information about the skeleton and dual skeleton of the triangulation.

#### Skeletal Components

In the Skeleton→Skeletal Components tab you will see the total number of vertices, edges, triangles, tetrahedra, components and boundary components in the triangulation. Beside each number is a View button that lets you view explicit structural details about each object in the class.

##### Viewing Vertices

If you click on the View button beside the vertex count, you will see a table listing the individual vertices of the triangulation.

The columns in this table are:

Vertex #

  Identifies each vertex with an individual vertex number, starting from 0 and counting upwards.

Type

  Gives some information about the link of the vertex (the boundary of a small regular neighbourhood). Text you might see here includes:

  Bdry

    Appears when the vertex is a standard boundary vertex, i.e., the vertex link is a disc.

  Cusp (torus)

    Appears when the vertex is a torus cusp, i.e., the vertex link is a torus.

  Cusp (klein bottle)

    Appears when the vertex is a Klein bottle cusp, i.e., the vertex link is a Klein bottle.

  Cusp (`surface`)

    Appears when the vertex is a non-standard cusp, i.e., the vertex link is a closed surface but not a sphere, torus or Klein bottle. Here `surface` will describe the orientability and genus of the vertex link. An example might be `Cusp (orbl, genus 3)`.

  Non-std bdry

    Appears when the vertex is a non-standard boundary vertex. This means the vertex link is a surface with boundary but not a disc. If a vertex like this appears, the entire triangulation will be marked as invalid.

  If the vertex link is a sphere (i.e., the vertex is an ordinary internal vertex of the triangulation), then the second column will be left empty.

Degree

  Lists the degree of each vertex. This is the number of individual tetrahedron vertices that are identified together to make this vertex of the triangulation.

Tetrahedra (Tet vertices)

  Lists precisely which vertices of which tetrahedra come together to form each overall vertex of the triangulation. An example is `3 (0), 7 (1), 3 (2), 5 (0)`, indicating a degree 4 vertex obtained by identifying vertices 0 and 2 of tetrahedron 3, vertex 1 of tetrahedron 7, and vertex 0 of tetrahedron 5.

##### Viewing Edges

If you click on the View button beside the edge count, you will see a table listing the individual edges of the triangulation.

The columns in this table are:

Edge #

  Identifies each edge with an individual edge number, starting from 0 and counting upwards.

Type

  Gives some additional information about the edge. Text you might see here includes:

  Bdry

    Indicates a boundary edge (i.e., an edge that lies on some boundary triangle of the triangulation).

  INVALID

    Indicates an edge glued to itself in reverse (so the midpoint of this edge is a projective plane cusp). If an edge like this appears, the entire triangulation will also be marked as invalid.

  If the edge is valid and an ordinary internal edge (i.e., the relative interior of the edge lies within the interior of the triangulation), then the second column will be left empty.

Degree

  Lists the degree of each edge. This is the number of individual tetrahedron edges that are identified together to make this edge of the triangulation.

Tetrahedra (Tet vertices)

  Lists precisely which edges of which tetrahedra come together to form each overall edge of the triangulation. An example is `0 (31), 1 (01), 0 (02)`, indicating a degree 3 edge obtained by identifying edges 31 and 02 of tetrahedron 0, and edge 01 of tetrahedron 1 (here edge 31 means the edge running from vertex 3 to vertex 1, and so on).

  The order of vertices is important: this example also shows that vertex 3 of tetrahedron 0, vertex 0 of tetrahedron 1, and vertex 0 of tetrahedron 0 all represent the same end of the edge.

  The order of tetrahedra in this list is also important: tetrahera are written in the order in which one sees them when walking around the edge link.

##### Viewing Triangles

If you click on the View button beside the triangle count, you will see a table listing the individual triangles (i.e., 2-faces) of the triangulation.

The columns in this table are:

Triangle #

  Identifies each triangle with an individual triangle number, starting from 0 and counting upwards.

Type

  Gives some information about the shape of the triangle within the triangulation, according to how its edges and vertices are identified together. Text you might see here includes:

  Triangle

    No vertices or edges of the triangle are identified.

  Scarf

    Two vertices of the triangle are identified; all edges are distinct.

  Parachute

    All three vertices of the triangle are identified; all edges are distinct.

  Möbius band

    Two edges of the triangle are identified to form a Möbius band (causing all three vertices to be identified); the third edge remains distinct.

  Cone

    Two edges of the triangle are identified to form a cone (causing two vertices to be identified); the third edge and third vertex remain distinct.

  Horn

    Two edges of the triangle are identified to form a cone and all the third vertex is identified with the others; the third edge remains distinct.

  Dunce hat

    All three edges of the triangle are identified, some with orientable and some with non-orientable gluings.

  L(3,1)

    All three edges of the triangle are identified using non-orientable gluings; note that this forms a spine for the lens space L(3,1).

  In addition to the shape, you will also see the text (Bdry) for each boundary triangle (i.e., each triangle that lies entirely within the boundary of the triangulation).

Degree

  Lists the degree of each triangle, i.e., the number of individual tetrahedron faces that are identified together to make this triangle within the overall triangulation. This is always 1 for a boundary triangle, or 2 for an internal triangle.

Tetrahedra (Tet vertices)

  Lists precisely which faces of which tetrahedra come together to form each overall triangle within the triangulation. An example is `2 (123), 3 (120)`, indicating an internal triangle obtained by gluing faces 123 of tetrahedron 2 with faces 120 of tetrahedron 3.

  Again, the order of vertices is important: this example also shows that vertex 3 of tetrahedron 2 represents the same corner of the triangle as vertex 0 of tetrahedron 3.

##### Viewing Components

If you click on the View button beside the component count, you will see a table listing the individual connected components of the triangulation.

The columns in this table are:

Cmpt #

  Identifies each connected component with an individual component number, starting from 0 and counting upwards.

Type

  Gives some additional information about the individual component, similar to the basic properties that you can view for each triangulation. Text you might see here includes:

  Real / Ideal

    The text Real indicates that the the component contains no ideal vertices, and the text Ideal indicates that the component contains at least one ideal vertex. An ideal vertex is a vertex whose link is a closed surface but not a 2-sphere.

  Orbl / Non-orbl

    Indicates whether the component is orientable or non-orientable.

Size

  Gives the number of tetrahedra belonging to each connected component.

Tetrahedra

  Lists the individual tetrahedra belonging to each connected component.

##### Viewing Boundary Components

If you click on the View button beside the component count, you will see a table listing the individual boundary components of the triangulation. This includes real boundary components (consisting of several boundary triangles), and also ideal boundary components (each of which consists of a single ideal vertex).

The columns in this table are:

Cmpt #

  Identifies each boundary component with an individual boundary component number, starting from 0 and counting upwards.

Type

  Either Real or Ideal, according to whether this is a real or ideal boundary component (as described above).

Size

  For a real boundary component, this gives the number of boundary triangles that make up the component. For an ideal boundary component, this will always state `1 vertex`.

Triangles / Vertex

  For a real boundary component, this lists the individual boundary triangles that it contains. For an ideal boundary component, this lists the individual tetrahedron vertices that are identified to form the overall ideal vertex of the triangulation.

  Triangles and vertices are described in the same manner as in the the individual triangle and for vertex viewers.

#### Face Pairing Graph

The Skeleton→Face Pairing Graph tab offers a visual representation of how the individual tetrahedra are glued together.

The face pairing graph is essentially the dual 1-skeleton of the triangulation: every node of the graph represents a tetrahedron, and every arc represents a pair of tetrahedron faces that are joined together. Each node contains a small label indicating the corresponding tetrahedron number (though these can be switched off). For a closed triangulation the face pairing graph is always 4-valent; for a bounded triangulation there may be nodes of degree three or less.

Regina uses the external application Graphviz to draw the graph. If Graphviz is not installed on your system then the face pairing graph cannot be displayed. Graphviz is a widely-used application, and most GNU/Linux distributions offer Graphviz packages.

If Graphviz is installed but for some reason Regina cannot find it, you can tell Regina where to find Graphviz in the tools options.

### Algebraic Invariants

The Algebra tab holds several smaller tabs that describe different algebraic invariants of the triangulation.

If the triangulation contains ideal vertices, these invariants will be computed assuming the ideal vertices have been truncated, leaving a small boundary component where each ideal vertex used to be.

### Caution

There is no guarantee that invalid edges (edges glued to themselves in reverse) will be handled correctly. In particular, the projective plane cusps they produce may be ignored.

#### Homology Groups

The Algebra→Homology tab presents several homology groups of the triangulation. These include: H1(M), (the first homology group); H1(M, ∂M), the relative first homology group with respect to the boundary; H1(∂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.

All finite cyclic groups Z`k` will be written in the “pidgin TeX” form Z_`k`, so that the order of each group is easier to read.

#### Fundamental Group

The Algebra→Fund. Group tab displays the fundamental group of the triangulation, presented as a set of generators and relations.

Regina will try to recognise the common name of this group (though the recognition code is fairly naïve). If it can, the name will be displayed above the generators and relations. Otherwise the text Not recognised will be displayed instead.

If you have GAP (Groups, Algorithms and Programming) installed on your system, you can use GAP to simplify the group presentation. Regina does try to simplify the presentation on its own, but GAP will typically do a better job.

To simplify the presentation using GAP, press the Simplify using GAP button at the bottom of the panel. You can try this more than once if you like: sometimes GAP finds a better presentation when run a second or third time.

If Regina is having trouble starting GAP, you can tell it how to start GAP in the tools options.

### Tip

If you wish to see a full transcript of the conversation between Regina and GAP, start Regina from the command-line by running regina-gui. The entire conversation will be shown in the text console where you ran regina-gui command.

#### Turaev-Viro Invariants

The Algebra→Turaev-Viro tab allows you to compute Turaev-Viro state sum invariants with arbitrary parameters.

Each Turaev-Viro invariant is defined by a set of initial data: an integer `r` ≥ 3 and a root of unity `q`0 of degree 2`r` (see Section 7 of [TV92] for details). In Regina you identify the root of unity `q`0 using an integer `root` in the range 0 < `root` < 2`r` (where `r` and `root` must be coprime). To compute a Turaev-Viro invariant, simply enter the two integers `r`, `root` into the box provided and press Calculate.

Once computed, the new invariant will appear in the table beneath. Be aware that these invariants are computing using floating point arithmetic (with an exponential number of arithmetical operations), and so Regina cannot guarantee the accuracy of the result.

Turaev-Viro invariants are stored when you save your data file, so they do not need to be recalculated when a file is closed and reopened.

### Caution

Only small values of `r` should be used, since the time required to calculate the invariant grows exponentially with `r`.

#### Cellular Information

The Algebra→Cellular Info tab contains 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.

As with the other algebraic invariants described above, all information here refers to the compact manifold obtained by truncating any ideal vertices and leaving real boundary surfaces in their place.

The information here includes:

Cells

  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, triangles and tetrahedra. For an ideal triangulation this takes into account the truncation of ideal vertices, and is therefore a little more complex.

Dual cells

  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.

Euler characteristic

  Gives the Euler characteristic of the manifold, as computed from the CW-decompositions.

Homology groups

  Lists the homology groups of the manifold with coefficients in the integers. The four groups H0, H1, H2 and H3 are listed in order.

Boundary homology groups

  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.

H1(∂M → M)

  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 Info tab describes some properties of this induced map.

Torsion form rank vector

  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 `n` `x` 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 item on the Cellular Info tab is the first of the three Kawauchi-Kojima invariants of the torsion linking form on the torsion subgroup of H1: 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 `m`Z2 + `n`Z2^2 + 0Z3 + `m`Z3^2 + `n`Z5.

  Note that the Kawauchi-Kojima invariants are only computed for connected orientable manifolds.

Sigma vector

  This item is the second of the three Kawauchi-Kojima invariants described above: the 2-torsion sigma vector, which 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 parametrisation.

  As above, the Kawauchi-Kojima invariants are only computed for connected orientable manifolds.

Legendre symbol vector

  This is the third of the three Kawauchi-Kojima invariants of the torsion linking form: the odd p-torsion Legendre symbol vector, originally constructed by Seifert, which is relevant for manifolds in which H1 has odd torsion.

  Again, the Kawauchi-Kojima invariants are only computed for connected orientable manifolds.

Comments

  This final item on the Cellular Info 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 in future releases of Regina (i.e., it might become more detailed 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 about the embeddability of the orientable double cover.

The paper [Bud08] illustrates how the information on this tab can be used in studying embedding problems.

### Combinatorial Composition

The Composition tab offers more detailed information about the combinatorial structure of the triangulation.

#### Isomorphism / Subcomplex Testing

The upper portion of the composition tab is for testing combinatorial isomorphism, or testing whether one triangulation is a subcomplex of another. Simply select some other triangulation `T` from the drop-down box (indicated by the arrow in the diagram below).

Each time you select a different triangulation `T` in the drop-down box, Regina will immediately test for any of the following relationships:

- whether this triangulation and `T` are isomorphic (i.e., identical up to a relabelling of tetrahedra and their vertices);

- whether this triangulation is isomorphic to a subcomplex of `T` (i.e., `T` can be obtained from this triangulation by adding more tetrahedra and/or gluing more faces together, again with a possible relabelling);

- whether `T` is isomorphic to a subcomplex of this triangulation.

The relationship, if any, will be reported immediately beneath the drop-down box (as illustrated above). If a relationship is found, you can click on the Details button for the precise relabelling (i.e., the mapping between tetrahedron labels and between vertices in each tetrahedron).

#### Combinatorial Recognition, Building Blocks, Isomorphism Signatures and Dehydrations

In the lower portion of the composition tab is a large box containing details on the combinatorial composition of the triangulation. Here Regina will search for well-structured features within the triangulation, and deduce from them what it can. Sometimes it can recognise the construction and completely identify both the triangulation and the underlying 3-manifold; other times it yields little or no useful information.

In this composition box you will find the following information:

##### Recognising the Triangulation and the 3-Manifold

Regina knows about many infinite families of triangulations. If your triangulation belongs to one of these families then Regina will detect this and report the results here. Regina is particularly good at recognising well-structured triangulations of Seifert fibred spaces and graph manifolds.

If it does recognise your triangulation, Regina will name the 3-manifold and also the triangulation itself. See [Bur03] and [Bur07c] for details on the families of triangulations and what their names and parameters mean.

### Tip

If your aim is to determine the underlying 3-manifold, see the recognition tab instead. The recognition tab combines the results of this combinatorial recognition with slower but stronger routines such as 3-sphere, 3-ball and solid torus recognition.

##### Isomorphism Signature

An isomorphism signature is a compact sequence of letters, digits and/or punctuation that identifies a triangulation uniquely up to combinatorial isomorphism. Regina will report the isomorphism signature for your triangulation here.

Every triangulation has an isomorphism signature (even disconnected triangulations or triangulations with boundary). The main features of isomorphism signatures are that they are fast to compute, and that two triangulations have the same signature if and only if they are isomorphic. See [Bur11b] and [Bur11c] for details.

To convert an isomorphism signature back into a triangulation, you can either create a new triangulation from a signature, or import a list of isomorphism signatures. Be aware that the resulting triangulation might not use the same tetrahedron and vertex labels as the original.

Isomorphism signatures are case-sensitive (i.e., upper-case and lower-case matters). To copy the isomorphism signature to the clipboard, simply select the line in the box and choose Edit->Copy.

##### Dehydration

Like isomorphism signatures, a dehydration string is a short sequence of letters from which you can reconstruct your triangulation. Only some triangulations have dehydration strings (they must be connected with no boundary triangles and ≤ 25 tetrahedra), and they are not unique up to isomorphism (so relabelling tetrahedra might change the dehydration string). If it exists, the dehydration string will be reported here.

Dehydration strings first appeared in early censuses of hyperbolic 3-manifolds. See [CHW99] for details.

To convert a dehydration string back into a triangulation, you can either create a new triangulation from its dehydration, or import a list of dehydration strings. Be aware that the resulting triangulation might not use the same tetrahedron and vertex labels as the original.

As with isomorphism signatures, you can copy a dehydration string to the clipboard by selecting the line in the box and choosing Edit->Copy.

##### Building Blocks

The remainder of the composition box describes combinatorial building blocks within the triangulation. Regina knows about several families of building blocks (such as layered solid tori), and it will search for these within the triangulation. If it finds any building blocks that it recognises then it will give details here, including any parameters for the blocks and where they occur within the triangulation.

See [Bur03] and [Bur07c] for details on the various families of building blocks that Regina understands.

### High-Level Recognition Routines

The Recognition tab attempts to identify the underlying 3-manifold, and also computes other high-level properties of the triangulation. It offers a combination of slow but exact procedures (such as 3-sphere, 3-ball and solid torus recognition), and fast "opportunistic" combinatorial recognition procedures.

For large triangulations, many of these properties are not automatically calculated (since the algorithms run in worst-case exponential time). If a property is listed as `Unknown`, press the corresponding Calculate button (and be prepared to wait):

The result will appear as soon as the calculation is done:

The following properties are listed on the Recognition tab.

3-Sphere

  Determines whether this is a triangulation of the 3-sphere. This uses a complete, exact 3-sphere recognition algorithm, i.e., it guarantees to terminate with the correct result. The algorithm is highly optimised, and incorporates techniques from [Rub95], [Rub97], [Tho94], [JR03], [Bur10b], and [BO12].

3-Ball

  Determines whether this is a triangulation of the 3-dimensional ball. Again this uses a complete, exact algorithm that guarantees to terminate with the correct result. The algorithm is a simple modification of the 3-sphere recognition algorithm as described above.

Solid Torus

  Determines whether this is a triangulation of the solid torus, or equivalently, the unknot complement. This will work with ideal triangulations as well as triangulations with real boundary triangles (if the triangulation is ideal, any ideal vertices will be treated as though they were truncated). Once again this uses a complete, exact algorithm that guarantees to terminate with the correct result.

Zero-Efficient

  Indicates 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. If a closed orientable triangulation is not 0-efficient (and has more than two tetrahedra), this indicates that either the triangulation is non-minimal or the underlying 3-manifold is non-prime. See [JR03] for details on 0-efficiency.

Splitting Surface

  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. See [Bur03] for details.

Irreducible

  Determines whether the triangulation represents an irreducible manifold. A closed 3-manifold is irreducible if every embedded sphere bounds a ball.

  This test is only available for valid triangulations of closed, orientable and connected 3-manifolds.

Haken

  Determines whether the triangulation represents an Haken manifold. A closed orientable irreducible 3-manifold is Haken if it contains an embedded closed two-sided incompressible surface.

  This test is only available for valid triangulations of closed, orientable, connected and irreducible 3-manifolds.

Manifold

  This last field combines the exact algorithms above with the combinatorial recognition routines from the composition tab, in a multi-pronged attempt to conclusively identify the underlying 3-manifold. If the 3-manifold can be determined by any of these methods, it will be listed here.

  ### Tip

  This will not attempt a connected sum decomposition: if the manifold is non-prime then it will probably not be recognised. Try running a connected sum decomposition first, and then recognising each of the prime summands.

  ### Tip

  Unlike the exact algorithms above (which may be slow but will work in all settings), the "opportunistic" combinatorial recognition benefits from a well-structured triangulation. If the 3-manifold is not recognised, try simplifying the triangulation, or performing elementary moves.

### SnapPea Calculations

SnapPea is an excellent piece of software written by Jeffrey Weeks with a strong focus on hyperbolic 3-manifolds; for more information, see the SnapPy website. Portions of the SnapPea kernel are built into Regina, which allows Regina to compute information about geometries on triangulations. The results are presented in the SnapPea tab.

SnapPea calculations are not available for all triangulations. Amongst other constraints, your triangulation must be connected with no boundary triangles, and every vertex must have a torus or Klein bottle link. If your triangulation is unsuitable, the SnapPea tab will give you at least one reason why.

It is possible to bypass some of these constraints and allow SnapPea to work with closed triangulations. You do this at your own risk: see Regina's SnapPea options for details and the necessary warnings.

When you open the SnapPea tab, Regina will ask SnapPea to solve for a complete hyperbolic structure. The following information is then presented:

Solution Type

  This describes the type of solution that SnapPea found. Possible types are:

  Tetrahedra positively oriented

    All tetrahedra are either positively oriented or flat, though the entire solution is not flat and no tetrahedra are degenerate.

  Contains negatively oriented tetrahedra

    The volume is positive, but some tetrahedra are negatively oriented.

  All tetrahedra flat

    All tetrahedra are flat, but none have shape 0, 1 or infinity.

  Contains degenerate tetrahedra

    At least one tetrahedron has shape 0, 1 or infinity.

  Unrecognised solution type

    The volume is zero or negative, but the solution is neither flat nor degenerate.

  No solution found

    The gluing equations could not be solved.

Volume

  This gives the volume of the underlying 3-manifold, along with the estimated number of decimal places of accuracy. This accuracy measure is an estimate only (based on the differences between terms in Newton's method).

### Decomposition

Regina implements some high-level algorithms for decomposition a 3-manifold triangulation into “atomic pieces”. These include the following:

#### Component Decomposition

If your triangulation is disconnected, you may wish to break it into its connected components. To do this, select 3-D Triangulation->Extract Components. You must open the triangulation for viewing before you can do this.

Regina will create several new triangulations, one for each connected component. These will be added beneath the original in the packet tree. Your original (disconnected) triangulation will remain unchanged.

#### Connected Sum Decomposition

If your triangulation is closed, orientable and connected, Regina can decompose it into a connected sum of prime 3-manifolds (none of which are 3-spheres). To do this, select 3-D Triangulation->Connected Sum Decomposition. You must open the triangulation for viewing before you can do this.

Again, Regina will create several new triangulations, one for each prime summand. These will be added beneath the original in the packet tree, and your original triangulation will remain unchanged. If your original triangulation is a 3-sphere then no prime summands will be produced at all.

With two exceptions (RP3 and S2×S1), each of the new triangulations is guaranteed to be 0-efficient (i.e., they will have no non-vertex-linking normal spheres). The underlying algorithm is based on the 0-efficiency results of Jaco and Rubinstein [JR03], and uses 3-sphere recognition to ensure that none of the summands are trivial.

### Caution

Connected sum decomposition can be very slow for larger triangulations, since the underlying normal surface algorithms have worst-case exponential running time.

### Vertex Links

If you wish to examine a 3-manifold vertex link in detail, you can explicitly construct it by selecting 3-D Triangulation->Vertex Links. This will build the vertex link as a new 2-manifold triangulation.

You will be asked which vertex link you wish to construct, as illustrated below. For each available vertex, the drop-down list shows the vertex number (as seen in the skeleton viewer), along with details of the individual tetrahedron vertices that combine to form that vertex of the triangulation.

The new 2-manifold trianguation will appear beneath the original 3-manifold triangulation in the packet tree.

### Census Lookup

Regina ships with several prepackaged censuses of 3-manifold triangulations. To search for your triangulation within these censuses, select 3-D Triangulation->Census Lookup. You must open the triangulation for viewing before you can do this.

Your triangulation may use different tetrahedron and vertex labels; Regina will search for any isomorphic copy. Any matches will be reported:

The matches will also be stored in a new text packet beneath your triangulation:

By default, Regina will search censuses of closed orientable and non-orientable 3-manifold triangulations [Bur07b] [Bur07a] [Bur11a], cusped and closed hyperbolic 3-manifold triangulations [CHW99] [HW94], and knot and link complements (tabulated by Joe Christy). To add your own censuses to this list, visit Regina's census options.