tri

5reg

tri

A triangulation packet

Content Model
    tri ::= (&xml-simplex;*,
             &xml-fundgroup;?, &xml-H1;?, &xml-H1Bdry;?, &xml-H1Rel;?, &xml-H2;?,
             &xml-splitsfce;?, &xml-zeroeff;?, &xml-oneeff;?,
             &xml-threesphere;?, &xml-handlebody;?, &xml-txi;?,
             &xml-irreducible;?, &xml-compressingdisc;?, &xml-haken;?,
             &xml-tag;*,
             &packetsyn1;
             &packetsyn2;
             &packetsyn3;
             &packetsyn4;
             &packetsyn5;*)
Attributes
Name | Type
`dim` | `NUMBER` (required)
`perm` | `CDATA` (required)
`size` | `NUMBER` (required)
&packetattrlist2col;

&eltaddedver;7.0

## Description

This represents a triangulation packet, one of the many types of packet that together form the packet tree that a user sees when they open a ®ina; data file.

A triangulation packet stores a `d`-dimensional triangulation (typically of a `d`-manifold), for any dimension 2 ≤ `d` ≤ 15.

The total number of top-dimensional simplices in the triangulation (e.g., the total number of tetrahedra in the case `d` = 3) must be specified by the `size` attribute. The individual simplices are numbered 0, 1, 2, etc., and the gluings between them are described by the corresponding child &xml-simplex; elements. There must be precisely `size` child &xml-simplex; elements in total.

There are additional optional child elements (e.g., &xml-fundgroup;, &xml-H1;, etc.) that can be used to store known properties of this triangulation. Some of these are only used in particular dimensions, as noted in the Children section below.

&packetchilddesc;

### Parents

&packetparents;

### Children

The following elements occur in `tri`, and are used for triangulations in all dimensions:

- &xml-fundgroup;
- &xml-H1;
- &xml-simplex;

.

For triangulations in dimensions 3 and 4, the following child element may also occur:

- &xml-H2;

.

In addition, for triangulations in dimension 3, there are many more optional child elements:

- &xml-compressingdisc;
- &xml-H1Bdry;
- &xml-H1Rel;
- &xml-haken;
- &xml-handlebody;
- &xml-irreducible;
- &xml-oneeff;
- &xml-splitsfce;
- &xml-threesphere;
- &xml-txi;
- &xml-zeroeff;

. &packetchildren;

## Attributes

The following attributes are specific to triangulation packets:

`dim`

- Indicates the dimension of the triangulation. This must be an integer between 2 and 15 inclusive.

`perm`

- Indicates how gluing permutations are encoded within each &xml-simplex; element. Possible values are `index` (where permutations are denoted by indices into the symmetric group S`dim`+1) or `imagepack` (where permutations are denoted by “image packs”). See the &xml-simplex; element for more information on each of these encodings.

`size`

- Indicates the total number of top-dimensional simplices in the triangulation (e.g., the total number of tetrahedra in the case `dim` = 3). This must be a non-negative integer, and must be equal to the number of child &xml-simplex; elements.

&packetattrdesc;

## Example

The following &xml; snippet represents a three-tetrahedron triangulation of the 3-manifold RP2 × S1, along with some pre-computed homology groups.

    <tri dim="3" size="3" perm="index" label="RP² × S¹">
      <simplex> 1 0 1 0 2 11 2 2 </simplex>
      <simplex> 0 0 0 0 2 21 2 12 </simplex>
      <simplex> 0 13 0 4 1 21 1 8 </simplex>
      <H1><abeliangroup rank="1"> 2 </abeliangroup></H1>
      <H1Rel><abeliangroup rank="1"> 2 </abeliangroup></H1Rel>
      <H1Bdry><abeliangroup rank="0"> </abeliangroup></H1Bdry>
      <H2><abeliangroup rank="0"> 2 </abeliangroup></H2>
    </tri>