What does Regina do?

Regina's greatest strengths are in 3-manifold topology. Here, the primary objects that a user works with are 3-manifold triangulations. A large part of Regina is devoted to the creation, analysis and manipulation of these triangulations.

Regina also offers rich support for triangulated 4-manifolds.

You can create 4-manifold triangulations using similar methods to 3-manifolds, including manual gluings of pentachora, automatic generation of standard families, reconstruction from isomorphism signatures, building from framed links, and constructing ready-made examples.

To analyse 4-manifold triangulations, Regina offers a similar suite of algebraic and combinatorial tools, including detailed information about the skeleton and edge/vertex links, various homology and homotopy groups, analysis and tree decompositions of the facet pairing graph, isomorphism testing, and isomorphism signatures.

To manipulate 4-manifold triangulations, Regina offers a similar combinatorial toolbox to that for 3-manifolds, including elementary moves, automated simplification, barycentric subdivision, double covers, and converting between ideal and real boundary components.

Beyond 4-manifolds, Regina offers very basic support support for triangulations of dimensions 5–8 (and, if you make your own custom build, 9–15). This support is not available through the GUI: for this you will need to use either Python or C++.

Currently this support is limited to basic combinatorial tools, such analysis of the skeleton and face structure of the triangulation, double covers, creating ideal vertices, isomorphism testing, and isomorphism signatures.

The level of support for higher-dimensional triangulations is expected to grow in future releases.

Since version 5.96, Regina now offers native support for knots and links.

You can create knots and links from Gauss, Dowker-Thistlethwaite or planar diagram codes, knot signatures, parameterised families of knots and links, or construction of ready-made examples.

Regina computes several link invariants including the Jones and HOMFLY-PT polynomials and the link group, allows you to build and study the underlying 4-valent graph and its tree decompositions, offers tools for isomorphism testing and isomorphism signatures, and allows you to build the knot/link complement as a 3-manifold triangulation.

To manipulate knots and links, you can edit individual crossings, perform Reidemeister moves, take cablings, or run sophisticated simplification tools.

Regina can be used to form censuses of all triangulations that satisfy various constraints, through the command-line tool tricensus. Census enumeration is currently available for triangulations in dimensions 2, 3 and 4.

The census algorithms are heavily optimised; see [Bur04], [Bur07a] and [Bur11a] for details.

Regina ships with several census databases, including closed 3-manifold triangulations [Bur07b], [Bur07a], [Bur11a], hyperbolic 3-manifolds [Bur14c], [CHW99], [HW94], and knots and links [Bur20].

Regina can also search for a given triangulation across these databases (and indeed across much larger extensions of these databases), using isomorphism signatures [Bur11c], and fast hashing techniques.

Regina is capable of enumerating all vertex or fundamental normal surfaces within a 3-manifold triangulation, or all vertex or fundamental normal hypersurfaces within a 4-manifold triangulation. For 3-manifolds, almost normal surfaces^[1] are supported also.

These operations—enumerating vertex and fundamental normal surfaces—are required by many high-level topological algorithms for 3-manifolds. The underlying enumeration algorithms are highly optimised, and are described in detail in [Bur10a], [Bur09a], [Bur10b], [Bur14a], and [BO13].

For 3-manifolds, this enumeration can be performed in a variety of coordinate systems. For an n-tetrahedron triangulation this includes the 7n standard triangle and quadrilateral coordinates, as well as the smaller set of 3n quadrilateral-only coordinates [Tol98]. Likewise, for almost normal surfaces this includes the 10n standard coordinates as well as the 6n quadrilateral-octagon coordinates [Bur10b].

The enumeration can be restricted to embedded normal surfaces/hypersurfaces, or can be expanded to include immersed and singular surfaces/hypersurfaces. Regina also supports spun-normal surfaces [Til08], which are non-compact surfaces with infinitely many discs found in ideal 3-manifold triangulations.

To analyse normal surfaces within a 3-manifold triangulation, you can:

For normal hypersurfaces in a 4-manifold triangulation, some analogous operations are available. In particular, you can view hypersurfaces in different coordinate systems, calculate basic properties, and test for local (but not yet global) compatibility.

Angle structures, studied originally by Casson and then developed by Lackenby [Lac00a], [Lac00b] and Rivin [Riv94], [Riv03], represent a purely algebraic generalisation of hyperbolic structures. An angle structure on an ideal 3-manifold triangulation is formed by assigning an interior dihedral angle to each edge of every tetrahedron in such a way that a variety of linear equations and inequalities are satisfied.

Using similar techniques to normal surface enumeration, Regina can enumerate all vertex angle structures in a triangulation, or all taut structures (for which each dihedral angle is precisely 0 or π). Regina can also identify veering structures, which are taut structures that induce a specific type of combinatorial structure on the underlying triangulation [Ago11], [HRST11].

For strict angle structures (in which all angles are strictly positive) or generalised angle structures (where negative angles are allowed), Regina can test for existence very quickly and find an explicit example if one exists, using techniques from linear programming and linear algebra.

A splitting surface within a 3-manifold triangulation contains precisely one quadrilateral disc within each tetrahedron and no other normal or almost normal discs. These surfaces have some interesting combinatorial and topological properties, described in detail in [Bur03].

Regina can detect whether splitting surfaces occur within a triangulation. It also provides support for splitting surface signatures, which are compact text-based representations from which splitting surfaces and their enclosing 3-manifold triangulations can be reconstructed. In addition, Regina can generate a census of all possible splitting surface signatures of a given size.

Users can interact with Regina at a lower level by writing and running scripts in Python. These scripts are essentially high-level programs with immediate access to the mathematical core of Regina, and are ideal for performing repetitive tasks over large sets of data. Such tasks might include performing a sequence of tests upon all triangulations in a census, or testing a prototype for a new algorithm. Scripts can be embedded in Regina data files or run from the command line, and custom libraries of routines can be written to share code between files.

On many systems, default installations of Regina and SnapPy can communicate with each other inside the same Python session. See the Python chapter for details.

If you build Regina from source, you have the option of working with either Python 2 or Python 3. (If you download a ready-made package, then the package maintainer will have made this choice for you.)

The usual method of running Regina is through a full point-and-click graphical user interface. Alternatively, for those requiring immediate access to the mathematical core of the software, an interactive command-line interface is available for users to control the program using Python scripting as described above. Various other specialised command-line utility programs are also available.

The mathematical core of Regina is provided as a C++ library, which means that programmers are able to access its low-level mathematical routines directly from within their own code.

Significant effort has been spent on documentation for this software. A full users' handbook is available for Regina (which you are reading now). In addition, the graphical user interface offers extensive assistance through tooltips and "What's This?" texts. Just hit Shift+F1 and click on any element of the user interface for immediate assistance.

For users writing Python scripts or for C++ programmers seeking to modify or extend the software, the full API for Regina's mathematical engine is also extensively documented. You can access the API documentation by selecting Help→Python API Reference.

Regina stores data in a well-organised hierarchical structure, allowing a single data file to contain multiple triangulations, normal surface lists and other topological structures, along with supporting data such as text notes, Python scripts and PDF documents. The file format is based on compressed XML, an open and widely-supported text-based data format. International characters are fully supported.

Again, this XML file format is extensively documented. For details, select Help→File Format Reference.