PrevCurrent Data File Format  Next

Name

simplex — A single top-dimensional simplex in a dim-dimensional triangulation

Synopsis

Content Model 
simplex ::= (#PCDATA)
Attributes
NameType
descCDATA

Description

A simplex element stores a single top-dimensional simplex in a dim-dimensional triangulation, along with its associated facet gluings. Since Regina 7.0, this element is now used by all of the dimensions that Regina supports (2 ≤ dim ≤ 15).

The character data of this XML element should be a whitespace-separated list of dim + 1 integer pairs, representing the gluings of facets 0, 1, …, dim of this simplex. Note that facet  i is always opposite vertex i in a top-dimensional simplex.

For each pair, the first integer represents the other top-dimensional simplex to which the facet is glued (recall that simplices in a triangulation are numbered 0, 1, 2, etc.). The second integer represents the permutation of vertices from this simplex to the other simplex describing precisely how this gluing takes place. This permutation will take the current facet number of this simplex to the corresponding facet number of the adjacent simplex, and the other dim vertex numbers of this simplex to the corresponding dim vertex numbers on the adjacent simplex to which they are identified by this gluing.

There are two ways in which Regina can encode permutations as integers: indices into the symmetric group Sdim+1, and image packs. Which of these methods is used is specified by the perm attribute of the enclosing tri element. If no method is specified (e.g., if this is an older second-generation data file), then Regina will assume indices for dimension 2, and image packs for dimensions dim ≥ 3 (following the behaviour of the second-generation file format).

  • An index encodes a permutation as an integer i in the range 0 ≤ i < (dim+1)!. This integer is the index of the permutation into the symmetric group Sdim+1. These indices list the permutations in “almost lexicographical” order; the exception is that some pairs (2i, 2i+1) are swapped to ensure that even indices correspond to even permutations.

    You can access these indices from within Regina using the functions Perm<dim+1>::SnIndex() and Perm<dim+1>::Sn[...].

  • An image pack encodes a permutation as follows. If the permutation maps 0, 1, …, dim to a0, a1, …, aDim respectively (where a0, a1, …, aDim are 0, 1, …, dim in some order), then the corresponding integer is (a0 + B a1 + B2 a2 + … + Bdim aDim). Here the “base” B is B = 4 for dimension dim = 3, B = 8 for dimensions 4 ≤ dim ≤ 7, and B = 16 for dimensions 8 ≤ dim ≤ 15. As an example, the identity permutation for dimension dim = 8 which maps (0,1,…,8) to (0,1,…,8) is represented by the integer (0 + 1×16 + 2×256 + … + 8×4294967296), which is 36344967696.

    You can access these indices from within Regina using the functions Perm<dim+1>::imagePack() and Perm<dim+1>::fromImagePack(...).

If a facet is a boundary facet (i.e., it is not glued to anything), the two corresponding integers stored in the XML character data should be -1 and -1.

Warning

For dimensions dim = 2..7, permutations are represented by 32-bit integers. In contrast, for dimensions dim = 8..15, permutations are represented by 64-bit integers. If you are writing code that reads Regina data files, you may need to use a large integer type (such as long long in C++) accordingly.

Parents

The following elements contain simplex: tri.

Children

Element simplex has no children.

Attributes

desc

A human-readable description of the role that this top-dimensional simplex plays in the overall triangulation.

Example

The following XML snippet represents simplex number 0 in a 6-dimensional triangulation, using image packs to represent permutations. Facet 0 of this simplex is glued to facet 6 of simplex number 1, with a gluing permutation that maps (0,1,2,3,4,5,6) to (6,0,1,2,3,4,5). Facet 6 of this simplex is glued to facet 0 of simplex number 1, with a gluing permutation that maps (0,1,2,3,4,5,6) to (1,2,3,4,5,6,0). The remaining facets 1,…,5 of this simplex are all boundary facets. 

<simplex> 1 14038087 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 2054353 </simplex>
PrevContentsNext
script Up snappeadata