1
<chapter id="triangulations">
2
<title>Triangulations</title>
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3-manifolds in ®ina; are typically represented by
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<firstterm>triangulations</firstterm>. A triangulation of a 3-manifold
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consists of a set of tetrahedra with instructions on how some or all
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of the faces of these tetrahedra should be glued together in pairs.
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Triangulations in ®ina; are less strict than simplicial complexes:
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you may glue two faces of the same tetrahedron together,
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or you may glue faces so that different edges of the same tetrahedron become
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identified (and likewise for vertices).
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Indeed, the best triangulations for computation are often
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<emphasis>one-vertex triangulations</emphasis>, where all vertices of
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all tetrahedra become identified together.
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The downside of this flexibility is that, if you are not careful,
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your triangulation might not represent a 3-manifold at all.
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If this happens, ®ina; will
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<link linkend="tri-basicprops">tell you about it</link> when you open
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<sect1 id="tri-creation">
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<title>Creation</title>
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<title>New Triangulations</title>
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The simplest way to create a triangulation is through the
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<guimenu>Packet Tree</guimenu>
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<guimenuitem>New Triangulation</guimenuitem>
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menu item (or the corresponding toolbar button),
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which will create a new triangulation from scratch.
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<imagedata fileref="menu-newtri.png"/>
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In addition to the usual information, you are asked what
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<emphasis>type</emphasis> of triangulation to create (see the
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Here we walk through the various options.
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<imagedata fileref="newtri.png"/>
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<sect3 id="tri-new-empty">
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This will create a new triangulation with no tetrahedra at all.
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This is best if you wish to enter a triangulation by hand:
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first create an empty triangulation, and then
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manually <link linkend="tri-editgluings">add tetrahedra
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and edit the face gluings</link>.
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<imagedata fileref="newtri-empty.png"/>
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<sect3 id="tri-new-lens">
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<title>Layered Lens Space</title>
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This will create a layered lens space with the given parameters.
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This involves building two <link linkend="tri-new-layered">layered
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solid tori</link> and gluing them together along their torus boundaries.
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Layered lens spaces were introduced by Jaco and Rubinstein
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<xref linkend="bib-0-efficiency"/>,
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<xref linkend="bib-layeredlensspaces"/> and others.
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<imagedata fileref="newtri-lens.png"/>
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(<replaceable>p</replaceable>, <replaceable>q</replaceable>)
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must be non-negative and coprime, and must satisfy
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<replaceable>p</replaceable>><replaceable>q</replaceable>
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(although the exceptional case (0, 1) is also allowed).
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The resulting 3-manifold will be the lens space
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L(<replaceable>p</replaceable>,<replaceable>q</replaceable>).
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<title>&SFSLong;</title>
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This will create an orientable &sfslong; over the 2-sphere with any
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number of exceptional fibres. ®ina; will choose the simplest
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construction that it can based upon the given parameters.
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<imagedata fileref="newtri-sfs.png"/>
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The parameters for the &sfslong; must be given as a sequence of pairs of
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integers (&a1;,&b1;) (&a2;,&b2;) ... (&an;,&bn;), where each pair
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(&ai;,&bi;) describes a single exceptional fibre.
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An example is (2,-1) (3,4) (5,-4), which represents the
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&poincare; homology sphere.
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The two integers in each pair must be
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relatively prime, and none of &a1;, &a2;, ..., &an; may be zero.
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Each pair (&ai;,&bi;)
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does not need to be normalised; that is, the parameters may be positive or
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negative, and &bi; may lie outside the range [0,&ai;).
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There is no separate twisting
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parameter; each additional twist can be incorporated into the existing
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parameters by replacing some pair
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(&ai;,&bi;) with (&ai;,&ai;+&bi;).
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Pairs of the form (1,<replaceable>k</replaceable>) and even
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(1,0) are acceptable.
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<sect3 id="tri-new-layered">
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<title>Layered Solid Torus</title>
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This will create a layered solid torus with the given parameters.
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This is a solid torus built from a two-triangle &mobius; band by
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repeatedly adding new layers of tetrahedra onto the boundary.
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Layered solid tori were introduced by Jaco and Rubinstein
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<xref linkend="bib-0-efficiency"/>,
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<xref linkend="bib-layeredlensspaces"/> and others.
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<imagedata fileref="newtri-lst.png"/>
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(<replaceable>a</replaceable>, <replaceable>b</replaceable>,
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<replaceable>c</replaceable>) must be non-negative and coprime,
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and one must be the sum of the other two. These parameters
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describe how many times the meridional disc of the solid torus
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intersects the three edges on the boundary of the triangulation.
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<sect3 id="tri-new-loop">
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<title>Layered Loop</title>
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This will create a layered loop of the given length.
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This involves layering <replaceable>n</replaceable> tetrahedra
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(where <replaceable>n</replaceable> is the given length),
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and then gluing the final tetrahedron back around to the first.
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If the <guilabel>Twisted</guilabel> box is checked,
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this final gluing will be done with a
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a 180-degree rotation.
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Full details of the construction can be found in
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<xref linkend="bib-burton-phd"/>.
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<imagedata fileref="newtri-loop.png"/>
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A twisted layered loop of length
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<replaceable>n</replaceable> forms a one-vertex triangulation of
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&sss;/Q<subscript>4<replaceable>n</replaceable></subscript>.
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An untwisted layered loop of length <replaceable>n</replaceable>
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forms a two-vertex triangulation of the lens space
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L(<replaceable>n</replaceable>,1).
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<sect3 id="tri-new-aug">
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<title>Augmented Triangular Solid Torus</title>
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This will create an augmented &trist; with the given parameters.
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An augmented &trist; is created by building
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a three-tetrahedron solid torus and then attaching three
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<link linkend="tri-new-layered">layered solid tori</link> to its
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boundary. Details of the construction can be found in
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<xref linkend="bib-burton-phd"/>.
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<imagedata fileref="newtri-ast.png"/>
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You must provide six parameters, grouped into three
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pairs of integers (&a1;,&b1;) (&a2;,&b2;) (&a3;,&b3;). Each pair
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of integers describes one of the layered solid tori that is attached.
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The two integers in each pair must be
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relatively prime, and both positive and negative integers are allowed.
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If none of &a1;, &a2; or &a3; is zero, the resulting 3-manifold
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will be a &sfslong; over the sphere with at most
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three exceptional fibres. Conversely, any &sfslong; of this type
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can be represented as an augmented &trist;.
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<sect3 id="tri-new-isosig">
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<title>Isomorphism Signature</title>
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This will reconstruct a triangulation from an isomorphism signature.
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An <firstterm>isomorphism signature</firstterm> is a compact sequence
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of letters, digits and/or punctuation that identifies a
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triangulation uniquely up to combinatorial isomorphism (i.e.,
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relabelling tetrahedra and their vertices). An example is
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<literal>cPcbbbiht</literal> (which describes the figure eight knot
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<imagedata fileref="newtri-isosig.png"/>
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Stated precisely: every triangulation has a unique isomorphism signature,
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and two triangulations have the same signature if and only if they
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Isomorphism signatures are introduced in the paper
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<xref linkend="bib-burton-simps3"/>.
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The isomorphism signature for an existing triangulation can be viewed
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through the <link linkend="tri-composition-isosig">triangulation
270
composition</link> tab.
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Isomorphism signatures are <emphasis>case sensitive</emphasis>!
274
Be sure that you are entering upper-case and lower-case correctly
275
(or better, copy and paste the signature using the clipboard if you
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<sect3 id="tri-new-dehydration">
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<title>Dehydration</title>
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This will rehydrate a triangulation from the given dehydration string.
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A <firstterm>dehydration string</firstterm> is a sequence of letters
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that contains enough information to reconstruct a triangulation
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(though tetrahedra and their vertices might be relabelled).
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An example is <literal>dadbcccaqhx</literal>
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(which describes the &snappea; census triangulation
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<literal>m025</literal>).
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Dehydration strings appear in
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census papers such as the hyperbolic cusped census of
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Callahan, Hildebrand and Weeks <xref linkend="bib-cuspedcensus"/>,
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in which the dehydration format is explicitly described.
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<imagedata fileref="newtri-dehydration.png"/>
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Only some triangulations have dehydration strings.
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The dehydration string (if it exists) for an existing triangulation
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through the <link linkend="tri-composition-dehydration">triangulation
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composition</link> tab.
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<sect3 id="tri-new-splitting">
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<title>Splitting Surface</title>
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This will reconstruct a triangulation from a splitting surface signature.
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A <firstterm>splitting surface</firstterm> is a compact normal
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surface consisting of precisely one quadrilateral per tetrahedron
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and no other normal discs.
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A <firstterm>splitting surface signature</firstterm> is a string of
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letters arranged into cycles that describe how these quadrilaterals
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are joined together. From this signature, both the normal surface and
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the enclosing triangulation can be reconstructed.
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<imagedata fileref="newtri-splitting.png"/>
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When entering a splitting surface signature, you may use
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any block of punctuation to separate cycles of letters. All
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whitespace will be ignored. Examples of valid signatures
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are <literal>(ab)(bC)(Ca)</literal> and <literal>AAb-bc-C</literal>.
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The precise format of splitting surface signatures is described
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in <xref linkend="bib-burton-phd"/>.
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<sect3 id="tri-new-example">
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<title>Example Triangulation</title>
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®ina; also offers a small
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selection of ready-made sample triangulations;
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these include the figure eight knot complement,
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the &poincare; homology sphere, the Weber-Seifert dodecahedral
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space, and several others.
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Simply select one from the list provided and the
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corresponding triangulation will be built for you.
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<imagedata fileref="newtri-example.png"/>
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<sect2 id="tri-imported">
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<title>Importing Triangulations</title>
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You can import triangulations into ®ina; from other
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programs, such as &snappeapylink; or &orburl;. This is done through the
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<guimenu>File</guimenu>
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<guisubmenu>Import</guisubmenu>
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menu. For details, see the chapter on
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<link linkend="foreign-import">importing and exporting data</link>.
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<sect2 id="tri-census-creation">
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<title>Creating a Census</title>
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®ina; can build a census of all 3-manifold triangulations
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satisfying a variety of different constraints. The best way to
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do this is through the command-line tool
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<link linkend="man-tricensus"><command>tricensus</command></link>.
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For very long calculations,
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<link linkend="man-tricensus-mpi"><command>tricensus-mpi</command></link>
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may be used to distribute the computation across a cluster of machines.
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<sect1 id="tri-analysis">
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<title>Analysis</title>
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®ina; offers a wealth of information about 3-manifold
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triangulations, spread across the many different tabs in the
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triangulation viewer. Here we walk through the different properties
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and invariants that ®ina; can compute.
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<sect2 id="tri-basicprops">
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<title>Validity, Orientability and Other Basic Properties</title>
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<imagedata fileref="triheader.png"/>
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At the top of each triangulation viewer is a banner listing some
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basic properties of the triangulation (circled in red above).
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The following words might appear:
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<glossentry id="tri-propclosed">
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<glossterm><guilabel>Closed</guilabel></glossterm>
426
Signifies that the triangulation has no boundary faces and no
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ideal vertices. In other words, the link of every vertex is a
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<glossentry id="tri-propideal">
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<glossterm><guilabel>Ideal bdry</guilabel></glossterm>
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Signifies that at least one vertex of the triangulation is
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<firstterm>ideal</firstterm>. That is, the vertex link is
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a closed surface but not a 2-sphere.
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You can locate any ideal vertices using the
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<link linkend="tri-vertices">skeleton viewers</link>.
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<glossentry id="tri-propbdry">
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<glossterm><guilabel>Real bdry</guilabel></glossterm>
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Signifies that the triangulation contains one or more boundary faces.
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<glossentry id="tri-proporient">
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<glossterm><guilabel>Orientable</guilabel> /
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<guilabel>non-orientable</guilabel> /
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<guilabel>oriented</guilabel></glossterm>
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The words <guilabel>orientable</guilabel>
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or <guilabel>non-orientable</guilabel> indicate
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whether or not the triangulation represents an orientable
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If the words <guilabel>orientable and oriented</guilabel> appear,
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this indicates that the vertex labels 0, 1, 2 and 3 on each
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tetrahedron induce a consistent orientation for all tetrahedra
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in the entire triangulation.
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If you need a consistent orientation for all tetrahedra but you
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only see <guilabel>orientable</guilabel> (not
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<guilabel>orientable and oriented</guilabel>), you can fix this by
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<link linkend="tri-orient">orienting your triangulation</link>.
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<glossentry id="tri-propconn">
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<glossterm><guilabel>Connected</guilabel> /
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<guilabel>disconnected</guilabel></glossterm>
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The words <guilabel>connected</guilabel>
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or <guilabel>disconnected</guilabel> indicate
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whether or not the triangulation forms a single connected piece.
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<glossentry id="tri-propvalid">
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<glossterm><guilabel>Invalid triangulation</guilabel></glossterm>
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Signifies that the triangulation is “broken” to the
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point where ®ina; cannot do any serious work with it.
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This can happen for one of two reasons:
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(i) some vertex link is a surface with boundary
494
(ii) some edge is identified with itself in reverse.
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You can locate the offending vertex or edge using the
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<link linkend="tri-skeleton-skelcomp">skeleton viewers</link>.
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If the triangulation is invalid, no other information will appear
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<glossterm><guilabel>Empty</guilabel></glossterm>
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Signifies that the triangulation contains no tetrahedra at all.
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In this case, no other information will appear in the banner.
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<sect2 id="tri-viewgluings">
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<title>Viewing Tetrahedron Face Gluings</title>
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The <guilabel>Gluings</guilabel> tab shows how the various
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tetrahedron faces are glued to each other in pairs.
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The face gluings are presented in a table:
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each row represents a tetrahedron, and the four columns on the right
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represent the four faces of each tetrahedron.
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Tetrahedra are numbered 0,1,2,..., and
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the four vertices of each tetrahedron are numbered 0,1,2,3.
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<imagedata fileref="tri-viewgluings.png"/>
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Each cell of this table represents a single face of a single
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tetrahedron. For instance, the cell circled in red
537
above represents face 123 of tetrahedron 5 (that is, the face
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formed from vertices 1,2,3 of tetrahedron 5).
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The contents of the cell show how the face is glued. In the example
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above, the circled cell contains <literal>2 (301)</literal>,
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indicating that face 123 of tetrahedron 5 is glued to
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face 301 of tetrahedron 2 using the affine map that
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matches vertices 1,2,3 of tetrahedron 5 with vertices
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3,0,1 of tetrahedron 2 respectively.
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The same gluing can be seen from the opposite direction in the row
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for tetrahedron 2.
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An empty cell indicates that a face is not glued to anything at all;
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that is, the face forms part of the <emphasis>boundary</emphasis> of
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the 3-manifold. In the table above there are two boundary faces:
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face 023 of tetrahedron 2, and face 123 of
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tetrahedron 4. In our example these join together to form the torus
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boundary of the figure eight knot complement.
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You can modify the triangulation by typing new face gluings directly
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into this table. See the section on
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<link linkend="tri-editgluings">modifying triangulations</link> for
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<sect2 id="tri-skeleton">
567
<title>Skeletal Information</title>
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The <guilabel>Skeleton</guilabel> tab holds
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two smaller tabs offering combinatorial information about
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the skeleton and dual skeleton of the triangulation.
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<sect3 id="tri-skeleton-skelcomp">
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<title>Skeletal Components</title>
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<guilabel>Skeleton</guilabel>→<guilabel>Skeletal Components</guilabel>
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tab you will see the total number of vertices, edges, faces, tetrahedra,
581
components and boundary components in the triangulation.
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Beside each number is a <guibutton>View</guibutton> button that
583
lets you view explicit structural details about each object in the class.
588
<imagedata fileref="tri-skeleton.png"/>
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<sect4 id="tri-vertices">
593
<title>Viewing Vertices</title>
595
If you click on the <guibutton>View</guibutton> button beside the vertex
596
count, you will see a table listing the individual vertices of the
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<imagedata fileref="tri-vertices.png"/>
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The columns in this table are:
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<glossterm><guilabel>Vertex #</guilabel></glossterm>
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Identifies each vertex with an
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individual <firstterm>vertex number</firstterm>, starting from 0
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and counting upwards.
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<glossterm><guilabel>Type</guilabel></glossterm>
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Gives some information about the <firstterm>link</firstterm> of the
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vertex (the boundary of a small regular neighbourhood).
622
Text you might see here includes:
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<glossterm><guilabel>Bdry</guilabel></glossterm>
627
Appears when the vertex is a standard boundary vertex,
628
i.e., the vertex link is a disc.
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<glossterm><guilabel>Cusp (torus)</guilabel></glossterm>
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Appears when the vertex is a torus cusp,
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i.e., the vertex link is a torus.
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<glossterm><guilabel>Cusp (klein bottle)</guilabel></glossterm>
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Appears when the vertex is a Klein bottle cusp,
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i.e., the vertex link is a Klein bottle.
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<glossterm><guilabel>Cusp
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(<replaceable>surface</replaceable>)</guilabel></glossterm>
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Appears when the vertex is a non-standard cusp,
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i.e., the vertex link is a closed surface but not a sphere,
651
torus or Klein bottle. Here
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<replaceable>surface</replaceable>
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will describe the orientability and genus of the vertex link.
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An example might be <literal>Cusp (orbl, genus 3)</literal>.
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<glossentry id="tri-vertices-nonstdbdry">
658
<glossterm><guilabel>Non-std bdry</guilabel></glossterm>
660
Appears when the vertex is a non-standard boundary vertex.
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This means the vertex link is a surface with boundary but not
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a disc. If a vertex like this appears, the entire
664
will be <link linkend="tri-propvalid">marked as invalid</link>.
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If the vertex link is a sphere (i.e., the vertex is an ordinary
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internal vertex of the triangulation), then the second column will
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<glossterm><guilabel>Degree</guilabel></glossterm>
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Lists the <firstterm>degree</firstterm> of each vertex.
677
This is the number of individual tetrahedron vertices that are
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identified together to make this vertex of the triangulation.
682
<glossterm><guilabel>Tetrahedra (Tet vertices)</guilabel></glossterm>
684
Lists precisely which vertices
685
of which tetrahedra come together to
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form each overall vertex of the triangulation. An example is
687
<literal>3 (0), 7 (1), 3 (2), 5 (0)</literal>,
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indicating a degree 4 vertex obtained by identifying
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vertices 0 and 2 of tetrahedron 3,
690
vertex 1 of tetrahedron 7, and
691
vertex 0 of tetrahedron 5.
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<sect4 id="tri-edges">
698
<title>Viewing Edges</title>
700
If you click on the <guibutton>View</guibutton> button beside the
701
edge count, you will see a table listing the individual
702
edges of the triangulation.
707
<imagedata fileref="tri-edges.png"/>
712
The columns in this table are:
715
<glossterm><guilabel>Edge #</guilabel></glossterm>
717
Identifies each edge with an
718
individual <firstterm>edge number</firstterm>, starting from 0
719
and counting upwards.
723
<glossterm><guilabel>Type</guilabel></glossterm>
725
Gives some additional information about the edge.
726
Text you might see here includes:
729
<glossterm><guilabel>Bdry</guilabel></glossterm>
731
Indicates a boundary edge (i.e., an edge that lies on some
732
boundary face of the triangulation).
736
<glossterm><guilabel>INVALID</guilabel></glossterm>
738
Indicates an edge glued to itself in reverse (so the midpoint of
739
this edge is a projective plane cusp).
740
If an edge like this appears, the entire triangulation will
741
also be <link linkend="tri-propvalid">marked as invalid</link>.
745
If the edge is valid and an ordinary internal edge (i.e.,
746
the relative interior of the edge lies within the interior
747
of the triangulation), then the second column will be left empty.
751
<glossterm><guilabel>Degree</guilabel></glossterm>
753
Lists the <firstterm>degree</firstterm> of each edge.
754
This is the number of individual tetrahedron edges that are
755
identified together to make this edge of the triangulation.
759
<glossterm><guilabel>Tetrahedra (Tet vertices)</guilabel></glossterm>
762
Lists precisely which edges
763
of which tetrahedra come together to
764
form each overall edge of the triangulation. An example is
765
<literal>0 (31), 1 (01), 0 (02)</literal>,
766
indicating a degree 3 edge obtained by identifying
767
edges 31 and 02 of tetrahedron 0, and
768
edge 01 of tetrahedron 1
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31 means the edge running from vertex 3 to vertex 1,
774
The order of vertices is important: this example also shows that
775
vertex 3 of tetrahedron 0,
776
vertex 0 of tetrahedron 1, and
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vertex 0 of tetrahedron 0 all represent
778
the <emphasis>same end</emphasis> of the edge.
781
The order of tetrahedra in this list is also important: tetrahera
782
are written in the order in which one sees them when walking
783
around the edge link.
790
<sect4 id="tri-faces">
791
<title>Viewing Faces</title>
793
If you click on the <guibutton>View</guibutton> button beside the
794
face count, you will see a table listing the individual
795
faces of the triangulation.
800
<imagedata fileref="tri-faces.png"/>
805
The columns in this table are:
808
<glossterm><guilabel>Face #</guilabel></glossterm>
810
Identifies each face with an
811
individual <firstterm>face number</firstterm>, starting from 0
812
and counting upwards.
816
<glossterm><guilabel>Type</guilabel></glossterm>
818
Gives some information about the <firstterm>shape</firstterm>
819
of the face in the triangulation, according to how its
820
edges and vertices are identified together.
821
Text you might see here includes:
824
<glossterm><guilabel>Triangle</guilabel></glossterm>
826
No vertices or edges of the face are identified.
830
<glossterm><guilabel>Scarf</guilabel></glossterm>
832
Two vertices of the face are identified; all edges are distinct.
836
<glossterm><guilabel>Parachute</guilabel></glossterm>
838
All three vertices of the face are identified; all edges are
843
<glossterm><guilabel>&mobius; band</guilabel></glossterm>
845
Two edges of the face are identified to form a &mobius; band
846
(causing all three vertices to be identified); the third edge
851
<glossterm><guilabel>Cone</guilabel></glossterm>
853
Two edges of the face are identified to form a cone (causing
854
two vertices to be identified); the third edge and third vertex
859
<glossterm><guilabel>Horn</guilabel></glossterm>
861
Two edges of the face are identified to form a cone and all
862
the third vertex is identified with the others; the third edge
867
<glossterm><guilabel>Dunce hat</guilabel></glossterm>
869
All three edges of the face are identified, some with
870
orientable and some with non-orientable gluings.
874
<glossterm><guilabel>L(3,1)</guilabel></glossterm>
876
All three edges of the face are identified using non-orientable
877
gluings; note that this forms a spine for the lens space L(3,1).
881
In addition to the shape, you will also see the text
882
<guilabel>(Bdry)</guilabel> for each boundary face
883
(i.e., each face that lies entirely within the boundary of the
888
<glossterm><guilabel>Degree</guilabel></glossterm>
890
Lists the <firstterm>degree</firstterm> of each face,
891
i.e., the number of individual tetrahedron faces that are
892
identified together to make this face of the triangulation.
893
This is always 1 for a boundary face, or 2 for an internal
898
<glossterm><guilabel>Tetrahedra (Tet vertices)</guilabel></glossterm>
901
Lists precisely which faces
902
of which tetrahedra come together to
903
form each overall face of the triangulation. An example is
904
<literal>2 (123), 3 (120)</literal>,
905
indicating an internal face obtained by gluing
906
faces 123 of tetrahedron 2 with
907
faces 120 of tetrahedron 3.
910
Again, the order of vertices is important: this example also shows
911
that vertex 3 of tetrahedron 2 represents the
912
<emphasis>same corner</emphasis> of the face as
913
vertex 0 of tetrahedron 3.
920
<sect4 id="tri-components">
921
<title>Viewing Components</title>
923
If you click on the <guibutton>View</guibutton> button beside the
924
component count, you will see a table listing the individual
925
connected components of the triangulation.
930
<imagedata fileref="tri-comp.png"/>
935
The columns in this table are:
938
<glossterm><guilabel>Cmpt #</guilabel></glossterm>
940
Identifies each connected component with an
941
individual <firstterm>component number</firstterm>, starting from 0
942
and counting upwards.
946
<glossterm><guilabel>Type</guilabel></glossterm>
948
Gives some additional information about the individual
949
component, similar to the <link linkend="tri-basicprops">basic
950
properties</link> that you can view for each triangulation.
951
Text you might see here includes:
954
<glossterm><guilabel>Real</guilabel> /
955
<guilabel>Ideal</guilabel></glossterm>
957
The text <guilabel>Real</guilabel> indicates that the
958
the component contains no ideal vertices, and the text
959
<guilabel>Ideal</guilabel> indicates that the component
960
contains at least one ideal vertex.
961
An <firstterm>ideal vertex</firstterm> is a vertex whose
962
link is a closed surface but not a 2-sphere.
966
<glossterm><guilabel>Orbl</guilabel> /
967
<guilabel>Non-orbl</guilabel></glossterm>
969
Indicates whether the component is orientable or
977
<glossterm><guilabel>Size</guilabel></glossterm>
979
Gives the number of tetrahedra belonging to each connected
984
<glossterm><guilabel>Tetrahedra</guilabel></glossterm>
987
Lists the individual tetrahedra belonging to each connected
995
<sect4 id="tri-bdrycomponents">
996
<title>Viewing Boundary Components</title>
998
If you click on the <guibutton>View</guibutton> button beside the
999
component count, you will see a table listing the individual
1000
boundary components of the triangulation.
1001
This includes <firstterm>real</firstterm> boundary components
1002
(consisting of several boundary faces), and also
1003
<firstterm>ideal</firstterm> boundary components (each of which
1004
consists of a single ideal vertex).
1009
<imagedata fileref="tri-bc.png"/>
1011
</inlinemediaobject>
1014
The columns in this table are:
1017
<glossterm><guilabel>Cmpt #</guilabel></glossterm>
1019
Identifies each boundary component with an
1020
individual <firstterm>boundary component number</firstterm>,
1021
starting from 0 and counting upwards.
1025
<glossterm><guilabel>Type</guilabel></glossterm>
1027
Either <guilabel>Real</guilabel> or <guilabel>Ideal</guilabel>,
1028
according to whether this is a real or ideal boundary component
1029
(as described above).
1033
<glossterm><guilabel>Size</guilabel></glossterm>
1035
For a real boundary component, this gives the number of
1036
boundary faces that make up the component.
1037
For an ideal boundary component, this will always state
1038
<literal>1 vertex</literal>.
1042
<glossterm><guilabel>Faces / Vertex</guilabel></glossterm>
1045
For a real boundary component, this lists the individual
1046
boundary faces that make up the component. For an ideal
1047
boundary component, this lists the specific vertex involved.
1050
Faces are identified using the individual face
1051
numbers that you see in the first column of the
1052
<link linkend="tri-faces">face viewer</link>, and likewise
1053
for <link linkend="tri-vertices">vertices</link>.
1062
<sect3 id="tri-skeleton-facegraph">
1063
<title>Face Pairing Graph</title>
1066
<guilabel>Skeleton</guilabel>→<guilabel>Face Pairing Graph</guilabel>
1067
tab offers a visual representation of how the individual tetrahedra are
1073
<imagedata fileref="tri-fpg.png"/>
1075
</inlinemediaobject>
1078
The <firstterm>face pairing graph</firstterm> is essentially the
1079
dual 1-skeleton of the triangulation: every
1080
node of the graph represents a tetrahedron, and every arc
1081
represents a pair of tetrahedron faces that are joined together.
1082
For a closed triangulation the face pairing graph is always
1083
4-valent; for a bounded triangulation there may be nodes
1084
of degree three or less.
1087
®ina; uses the external application &graphvizurl; to
1088
draw the graph. If &graphviz; is not installed on
1089
your system then the face pairing graph cannot be displayed.
1090
&graphviz; is a widely-used application, and most
1091
&linux; distributions offer &graphviz; packages.
1094
If &graphviz; is installed but for some reason ®ina; cannot find it,
1095
you can tell ®ina; where to find &graphviz; in the
1096
<link linkend="options-triangulation">triangulation options</link>.
1101
<sect2 id="tri-algebra">
1102
<title>Algebraic Invariants</title>
1105
The <guilabel>Algebra</guilabel> tab
1106
holds several smaller tabs that describe different
1107
algebraic invariants of the triangulation.
1110
If the triangulation contains ideal vertices, these invariants
1111
will be computed <emphasis>assuming the ideal vertices have
1112
been truncated</emphasis>, leaving a small boundary component
1113
where each ideal vertex used to be.
1116
There is no guarantee that <link linkend="tri-edges">invalid edges</link>
1117
(edges glued to themselves in reverse) will be handled correctly.
1118
In particular, the projective plane cusps they produce may be
1122
<sect3 id="tri-algebra-homology">
1123
<title>Homology Groups</title>
1125
The <guilabel>Algebra</guilabel>→<guilabel>Homology</guilabel>
1126
tab presents several homology groups of the triangulation.
1128
H1(M), (the first homology group);
1129
H1(M, ∂M),
1130
the relative first homology group with respect to the boundary;
1132
the first homology group of the boundary;
1133
H2(M), the second homology group; and
1134
H2(M ; Z<subscript>2</subscript>), the second homology group
1135
with coefficients in Z<subscript>2</subscript>.
1138
All finite cyclic groups
1139
Z<subscript><replaceable>k</replaceable></subscript>
1140
will be written in the “pidgin &tex;” form
1141
Z_<replaceable>k</replaceable>,
1142
so that the order of each group is easier to read.
1147
<imagedata fileref="tri-homology.png"/>
1149
</inlinemediaobject>
1153
<sect3 id="tri-algebra-fundgroup">
1154
<title>Fundamental Group</title>
1156
The <guilabel>Algebra</guilabel>→<guilabel>Fund. Group</guilabel>
1157
tab displays the fundamental group of the triangulation,
1158
presented as a set of generators and
1162
®ina; will try to recognise the common name of this
1163
group (though the recognition code is fairly naïve).
1164
If it can, the name will be displayed above
1165
the generators and relations. Otherwise the text
1166
<guilabel>Not recognised</guilabel> will be displayed instead.
1171
<imagedata fileref="tri-fundgroup.png"/>
1173
</inlinemediaobject>
1176
If you have &gaplongurl; installed on your system, you can use ⪆
1177
to simplify the group presentation. ®ina; does
1178
try to simplify the presentation on its own, but ⪆ will
1179
typically do a better job.
1182
To simplify the presentation using ⪆,
1183
press the <guibutton>Simplify using GAP</guibutton> button
1184
at the bottom of the panel.
1185
You can try this more than once if you like: sometimes
1186
⪆ finds a better presentation when run a second or third time.
1189
If ®ina; is having trouble starting ⪆, you can tell it how
1190
to start ⪆ in the
1191
<link linkend="options-triangulation">triangulation options</link>.
1194
If you wish to see a full transcript of the conversation between
1195
®ina; and ⪆, start ®ina; from the command-line by running
1196
<command>regina-kde</command>. The entire conversation
1197
will be shown in the text console where you ran
1198
<command>regina-kde</command> command.
1201
<sect3 id="tri-algebra-turaevviro">
1202
<title>Turaev-Viro Invariants</title>
1204
The <guilabel>Algebra</guilabel>→<guilabel>Turaev-Viro</guilabel>
1205
tab allows you to compute Turaev-Viro state sum invariants with
1206
arbitrary parameters.
1211
<imagedata fileref="tri-tv.png"/>
1213
</inlinemediaobject>
1216
Each Turaev-Viro invariant is defined by a set of
1217
<firstterm>initial data</firstterm>:
1218
an integer <replaceable>r</replaceable> ≥ 3 and a
1219
root of unity <replaceable>q</replaceable><subscript>0</subscript>
1220
of degree 2<replaceable>r</replaceable>
1221
(see Section 7 of <xref linkend="bib-turaevviro"/> for details).
1222
In ®ina; you identify the root of unity
1223
<replaceable>q</replaceable><subscript>0</subscript> using an
1224
integer <replaceable>root</replaceable>
1226
0 < <replaceable>root</replaceable> < 2<replaceable>r</replaceable>
1227
(where <replaceable>r</replaceable> and <replaceable>root</replaceable>
1229
To compute a Turaev-Viro invariant, simply enter the two integers
1230
<replaceable>r</replaceable>, <replaceable>root</replaceable>
1231
into the box provided and press <guilabel>Calculate</guilabel>.
1236
<imagedata fileref="tri-tv-entry.png"/>
1238
</inlinemediaobject>
1241
Once computed, the new invariant will appear in the table beneath.
1242
Be aware that these invariants are computing using
1243
floating point arithmetic (with an exponential number of
1244
arithmetical operations), and so ®ina; cannot guarantee the
1245
accuracy of the result.
1250
<imagedata fileref="tri-tv-results.png"/>
1252
</inlinemediaobject>
1255
Turaev-Viro invariants are stored when you save your data file, so they
1256
do not need to be recalculated when a file is closed and reopened.
1259
Only small values of <replaceable>r</replaceable>
1260
should be used, since the time required to calculate the
1261
invariant grows exponentially with <replaceable>r</replaceable>.
1264
<sect3 id="tri-algebra-cellular">
1265
<title>Cellular Information</title>
1267
The <guilabel>Algebra</guilabel>→<guilabel>Cellular Info</guilabel>
1268
tab contains information on the standard and dual CW-decompositions,
1269
a variety of homology groups and mappings, the
1270
Kawauchi-Kojima invariants of the torsion linking form, and
1271
comments on where the triangulation might be embeddable.
1276
<imagedata fileref="tri-cellular.png"/>
1278
</inlinemediaobject>
1281
As with the other algebraic invariants described above,
1282
all information here refers to the <emphasis>compact</emphasis>
1283
manifold obtained by
1284
truncating any ideal vertices and leaving real boundary surfaces
1288
The information here includes:
1291
<glossterm><guilabel>Cells</guilabel></glossterm>
1294
Lists the number of cells of each dimension for a standard
1295
CW-decomposition of the manifold. This is a list of four
1296
numbers, counting the 0-cells, 1-cells, 2-cells and 3-cells
1300
For a closed triangulation (no ideal vertices), this is simply
1301
the number of vertices, edges, faces and tetrahedra. For an
1302
ideal triangulation this takes into account the truncation of
1303
ideal vertices, and is therefore a little more complex.
1308
<glossterm><guilabel>Dual cells</guilabel></glossterm>
1310
Lists the number of cells of each dimension in the
1311
dual CW-decomposition. As before, this is a list of four
1312
numbers that count the 0-cells, 1-cells, 2-cells and 3-cells
1317
<glossterm><guilabel>Euler characteristic</guilabel></glossterm>
1319
Gives the Euler characteristic of the manifold, as computed from
1320
the CW-decompositions.
1324
<glossterm><guilabel>Homology groups</guilabel></glossterm>
1326
Lists the homology groups of the manifold with coefficients in
1327
the integers. The four groups
1328
H<subscript>0</subscript>, H<subscript>1</subscript>,
1329
H<subscript>2</subscript> and H<subscript>3</subscript> are listed
1334
<glossterm><guilabel>Boundary homology groups</guilabel></glossterm>
1336
Lists the homology groups of the boundary of the manifold, again
1337
with coefficients in the integers. The three groups
1338
H<subscript>0</subscript>, H<subscript>1</subscript> and
1339
H<subscript>2</subscript> are listed in order.
1343
<glossterm><guilabel>H1(∂M → M)</guilabel></glossterm>
1345
Since the boundary is a submanifold of the original manifold,
1346
there is an induced map on the first homology group. This
1347
item on the <guilabel>Cellular Info</guilabel> tab
1348
describes some properties of this induced map.
1352
<glossterm><guilabel>Torsion form rank vector</guilabel></glossterm>
1355
Given an oriented 3-manifold &varM;,
1356
there is a symmetric bilinear function
1357
t&hom1;(&varM;) x t&hom1;(&varM;) —> Q/Z
1358
where t&hom1;(&varM;) is the torsion subgroup of &hom1;(&varM;).
1359
It is computed in this way: let &varx; and &vary; be 1-dimensional
1360
torsion homology classes. Then &varn;&varx; is the boundary of
1361
some 2-cycle &varz; (transverse to &vary;) for some integer &varn;.
1362
The <firstterm>torsion linking form</firstterm> of
1363
&varx; and &vary; is the
1364
oriented intersection number of &varz; and &vary;, divided by &varn;.
1368
gave a complete classification of such torsion linking forms
1369
<xref linkend="bib-kktorsionlinkingform"/>. ®ina; computes the
1370
torsion linking form, and implements the Kawauchi-Kojima
1374
This item on the <guilabel>Cellular Info</guilabel> tab
1375
is the first of the three Kawauchi-Kojima invariants of the
1376
torsion linking form on the torsion subgroup of &hom1;:
1377
the <firstterm>torsion form rank vector</firstterm>, which
1378
lists the prime power decomposition of the torsion subgroup of
1380
For example, if &hom1;(&varM;) is a direct sum of &varn; copies of
1381
Z<subscript>20</subscript> and &varm; copies of
1382
Z<subscript>18</subscript>, then the torsion form rank vector
1383
would be: 2(&varm; &varn;) 3(0 &varm;) 5(&varn;)
1385
the group is isomorphic to
1386
&varm;Z<subscript>2</subscript> +
1387
&varn;Z<subscript>2^2</subscript> +
1388
0Z<subscript>3</subscript> +
1389
&varm;Z<subscript>3^2</subscript> +
1390
&varn;Z<subscript>5</subscript>.
1393
Note that the Kawauchi-Kojima invariants are only computed for
1394
connected orientable manifolds.
1399
<glossterm><guilabel>Sigma vector</guilabel></glossterm>
1402
This item is the second of the three Kawauchi-Kojima invariants
1403
described above: the <firstterm>2-torsion sigma vector</firstterm>,
1404
which is relevant for manifolds in which H<subscript>1</subscript>
1405
has 2-torsion. It is an orientation-sensitive invariant, where
1406
the orientation is chosen so that the first tetrahedron in the
1407
triangulation is positively-oriented with its standard parametrisation.
1410
As above, the Kawauchi-Kojima invariants are only computed
1411
for connected orientable manifolds.
1416
<glossterm><guilabel>Legendre symbol vector</guilabel></glossterm>
1419
This is the third of the three Kawauchi-Kojima invariants of the
1420
torsion linking form:
1421
the <firstterm>odd p-torsion Legendre symbol
1422
vector</firstterm>, originally constructed by Seifert,
1423
which is relevant for manifolds in which H<subscript>1</subscript>
1427
Again, the Kawauchi-Kojima invariants are only computed for
1428
connected orientable manifolds.
1433
<glossterm><guilabel>Comments</guilabel></glossterm>
1436
This final item on the
1437
<guilabel>Cellular Info</guilabel> tab comments upon
1438
where the manifold might embed. In particular, it attempts to
1439
make deductions about whether the manifold might embed in
1440
R<superscript>3</superscript>, S<superscript>3</superscript>,
1441
S<superscript>4</superscript>, or a homology sphere. If the manifold
1442
is orientable it tests for the hyperbolicity of the torsion linking
1443
form. It also performs the Kawauchi-Kojima 2-torsion test, useful
1444
for determining if a manifold with boundary does not embed in any
1448
The information in this field might change in future releases
1449
of ®ina; (i.e., it might become more detailed
1450
as more tests become available).
1451
Currently it examines the homology, the Kawauchi-Kojima
1452
invariants and some other elementary properties, and uses
1453
C. T. C. Wall's theorem that 3-manifolds embed
1454
in S<superscript>5</superscript>.
1457
These comments are provided for both orientable and
1458
non-orientable manifolds. In the non-orientable case they may
1459
provide additional information about the embeddability of the
1460
<link linkend="tri-cover">orientable double cover</link>.
1467
The paper <xref linkend="bib-budney-emb11"/> illustrates how
1468
the information on this tab can be used in studying embedding problems.
1473
<sect2 id="tri-composition">
1474
<title>Combinatorial Composition</title>
1476
The <guilabel>Composition</guilabel> tab
1477
offers more detailed information about the combinatorial
1478
structure of the triangulation.
1481
<sect3 id="tri-composition-isomorphism">
1482
<title>Isomorphism / Subcomplex Testing</title>
1484
The upper portion of the composition tab is for testing
1485
combinatorial isomorphism, or testing whether one triangulation is a
1486
subcomplex of another. Simply select some other
1487
triangulation <replaceable>T</replaceable> from the drop-down box
1488
(indicated by the arrow in the diagram below).
1493
<imagedata fileref="tri-iso.png"/>
1495
</inlinemediaobject>
1500
<imagedata fileref="tri-iso-select.png"/>
1502
</inlinemediaobject>
1505
Each time you select a different triangulation
1506
<replaceable>T</replaceable> in the drop-down box,
1507
®ina; will immediately test for any of the following relationships:
1510
whether this triangulation and <replaceable>T</replaceable>
1511
are isomorphic (i.e., identical up to a relabelling of tetrahedra
1512
and their vertices);
1515
whether this triangulation is isomorphic to a subcomplex of
1516
<replaceable>T</replaceable> (i.e., <replaceable>T</replaceable>
1517
can be obtained from this triangulation by adding more tetrahedra
1518
and/or gluing more faces together, again with a possible relabelling);
1521
whether <replaceable>T</replaceable> is isomorphic to a subcomplex of
1529
<imagedata fileref="tri-iso-result.png"/>
1531
</inlinemediaobject>
1534
The relationship, if any, will be reported immediately beneath
1535
the drop-down box (as illustrated above). If a relationship is found,
1536
you can click on the <guibutton>Details</guibutton> button for
1537
the precise relabelling (i.e., the mapping between tetrahedron
1538
labels and between vertices in each tetrahedron).
1542
<sect3 id="tri-composition-composition">
1543
<title>High-Level Recognition, Building Blocks,
1544
Isomorphism Signatures and Dehydrations</title>
1546
In the lower portion of the composition tab is a large box
1547
containing details on the combinatorial composition of the triangulation.
1548
Here ®ina; will search for well-structured features within the
1549
triangulation, and deduce from them what it can.
1550
Sometimes it can recognise the construction and completely identify
1551
both the triangulation and the underlying 3-manifold; other times
1552
it yields little or no useful information.
1557
<imagedata fileref="tri-composition.png"/>
1559
</inlinemediaobject>
1562
In this composition box you will find the following information:
1564
<sect4 id="tri-composition-name">
1565
<title>Recognising the Triangulation and the 3-Manifold</title>
1567
®ina; knows about many infinite families of triangulations.
1568
If your triangulation belongs to one of these families then
1569
®ina; will detect this and report the results here.
1570
®ina; is particularly good at recognising
1571
well-structured triangulations of &sfslong;s and graph manifolds.
1576
<imagedata fileref="tri-composition-name.png"/>
1578
</inlinemediaobject>
1581
If it does recognise your triangulation, ®ina; will name the
1582
3-manifold and also the triangulation itself. See
1583
<xref linkend="bib-burton-phd"/> and <xref linkend="bib-burton-nor7"/>
1584
for details on the families of triangulations and what their names
1585
and parameters mean.
1588
<sect4 id="tri-composition-isosig">
1589
<title>Isomorphism Signature</title>
1591
An <firstterm>isomorphism signature</firstterm> is a compact sequence
1592
of letters, digits and/or punctuation that identifies a
1593
triangulation uniquely up to combinatorial isomorphism.
1594
®ina; will report the isomorphism signature for your
1600
<imagedata fileref="tri-composition-isosig.png"/>
1602
</inlinemediaobject>
1605
Every triangulation has an isomorphism signature (even
1606
disconnected triangulations or triangulations with boundary).
1607
The main features of isomorphism signatures are that they are fast
1608
to compute, and that two triangulations have the same signature
1609
<emphasis>if and only if</emphasis> they are isomorphic.
1610
See <xref linkend="bib-burton-simps3"/> for details.
1613
To convert an isomorphism signature back into a triangulation,
1614
you can either <link linkend="tri-new-isosig">create a new
1615
triangulation</link> from a signature, or
1616
<link linkend="import-isosiglist">import a list of
1617
isomorphism signatures</link>. Be aware that the resulting
1618
triangulation might not use the same tetrahedron and vertex
1619
labels as the original.
1622
Isomorphism signatures are case-sensitive (i.e., upper-case and
1623
lower-case matters).
1624
To copy the isomorphism signature to the clipboard, simply select
1625
the line in the box and choose
1626
<menuchoice><guimenu>Edit</guimenu><guimenuitem>Copy</guimenuitem></menuchoice>.
1629
<sect4 id="tri-composition-dehydration">
1630
<title>Dehydration</title>
1632
Like isomorphism signatures, a <emphasis>dehydration string</emphasis>
1633
is a short sequence of letters from which you can reconstruct
1634
your triangulation. Only some triangulations have dehydration
1635
strings (they must be connected with no boundary faces and
1636
≤ 25 tetrahedra), and they are not unique up to isomorphism
1637
(so relabelling tetrahedra might change the dehydration string).
1638
If it exists, the dehydration string will be reported here.
1643
<imagedata fileref="tri-composition-dehydration.png"/>
1645
</inlinemediaobject>
1648
Dehydration strings first appeared in early censuses of hyperbolic
1649
3-manifolds. See <xref linkend="bib-cuspedcensus"/> for details.
1652
To convert a dehydration string back into a triangulation,
1653
you can either <link linkend="tri-new-dehydration">create a new
1654
triangulation</link> from its dehydration, or
1655
<link linkend="import-dehydrationlist">import a list of
1656
dehydration strings</link>. Be aware that the resulting triangulation
1657
might not use the same tetrahedron and vertex labels as the
1661
As with isomorphism signatures, you can copy a dehydration string
1662
to the clipboard by selecting the line in the box and choosing
1663
<menuchoice><guimenu>Edit</guimenu><guimenuitem>Copy</guimenuitem></menuchoice>.
1666
<sect4 id="tri-composition-blocks">
1667
<title>Building Blocks</title>
1669
The remainder of the composition box describes combinatorial
1670
building blocks within the triangulation.
1671
®ina; knows about several families of building blocks
1672
(such as <link linkend="tri-new-layered">layered
1673
solid tori</link>), and it will search for these within the
1675
If it finds any building blocks that it
1676
recognises then it will give details here, including any parameters
1677
for the blocks and where they occur within the triangulation.
1682
<imagedata fileref="tri-composition-blocks.png"/>
1684
</inlinemediaobject>
1687
See <xref linkend="bib-burton-phd"/>
1688
and <xref linkend="bib-burton-nor7"/> for details on the various
1689
families of building blocks that ®ina; understands.
1695
<sect2 id="tri-surfaceproperties">
1696
<title>Properties Involving Normal Surfaces</title>
1699
Some properties of a triangulation are defined by the types of
1700
normal surfaces it contains. These properties can be found under
1701
the <guilabel>Surfaces</guilabel> tab.
1706
<imagedata fileref="tri-surfaces.png"/>
1708
</inlinemediaobject>
1711
For large triangulations, some of these properties are
1712
not automatically calculated (since they might take exponential time).
1713
If a property is listed as <literal>Unknown</literal>, press
1714
the corresponding <guibutton>Calculate</guibutton> button
1715
(and be prepared to wait):
1720
<imagedata fileref="tri-s3-unknown.png"/>
1722
</inlinemediaobject>
1725
The result will appear as soon as the calculation is done:
1730
<imagedata fileref="tri-s3-known.png"/>
1732
</inlinemediaobject>
1735
The following properties are listed on the
1736
<guilabel>Surfaces</guilabel> tab.
1740
<glossentry id="tri-prop0eff">
1741
<glossterm><guilabel>Zero-Efficient</guilabel></glossterm>
1743
Indicates whether the triangulation is 0-efficient. A
1744
triangulation is <firstterm>0-efficient</firstterm> if its only
1745
normal spheres and discs are vertex linking, and if it has no 2-sphere
1746
boundary components.
1747
See <xref linkend="bib-0-efficiency"/> for details.
1750
<glossentry id="tri-propsplitting">
1751
<glossterm><guilabel>Splitting Surface</guilabel></glossterm>
1753
Determines whether the triangulation has a splitting
1754
surface. A <firstterm>splitting surface</firstterm> is a compact
1755
normal surface consisting of precisely one quad per tetrahedron
1756
and no other normal (or almost normal) discs.
1757
See <xref linkend="bib-burton-phd"/> for details.
1760
<glossentry id="tri-prop3sphere">
1761
<glossterm><guilabel>3-Sphere</guilabel></glossterm>
1763
Determines whether this is a triangulation
1764
of the 3-sphere. The 3-sphere recognition algorithm is
1765
highly optimised, and incorporates techniques from
1766
<xref linkend="bib-rubin-3sphere1"/>,
1767
<xref linkend="bib-rubin-3sphere2"/>,
1768
<xref linkend="bib-thinposition"/>,
1769
<xref linkend="bib-0-efficiency"/> and
1770
<xref linkend="bib-burton-quadoct"/>.
1773
<glossentry id="tri-prop3ball">
1774
<glossterm><guilabel>3-Ball</guilabel></glossterm>
1776
Determines whether this is a triangulation of the 3-dimensional ball.
1777
The algorithm is based on 3-sphere recognition as described above.
1783
You can change the number of tetrahedra beyond which properties are
1784
not computed automatically. See ®ina;'s
1785
<link linkend="options-triangulation">triangulation options</link>.
1789
<sect2 id="tri-snappea">
1790
<title>&snappea; Calculations</title>
1793
&snappea; is an excellent piece of software written by Jeffrey Weeks
1794
with a strong focus on hyperbolic 3-manifolds; for more information,
1795
see the &snappywebsite;.
1796
Portions of the &snappea; kernel are built into ®ina;, which
1797
allows ®ina; to compute information about geometries on
1798
triangulations. The results are presented in the
1799
<guilabel>&snappea;</guilabel> tab.
1804
<imagedata fileref="tri-snappea.png"/>
1806
</inlinemediaobject>
1809
&snappea; calculations are not available for all triangulations.
1810
Amongst other constraints, your triangulation
1811
must be connected with no boundary
1812
faces, and every vertex must have a torus or Klein bottle link.
1813
If your triangulation is unsuitable, the
1814
<guilabel>&snappea;</guilabel> tab will give you at
1815
least one reason why.
1818
It is possible to bypass some of these constraints and
1819
allow &snappea; to work with closed triangulations.
1820
You do this <emphasis>at your own risk</emphasis>:
1821
see ®ina;'s <link linkend="options-snappea">&snappea; options</link>
1822
for details and the necessary warnings.
1825
When you open the <guilabel>&snappea;</guilabel> tab,
1826
®ina; will ask &snappea; to solve for a complete hyperbolic
1827
structure. The following information is then presented:
1830
<glossterm><guilabel>Solution Type</guilabel></glossterm>
1832
This describes the type of solution that &snappea; found.
1836
<glossterm><guilabel>Tetrahedra positively
1837
oriented</guilabel></glossterm>
1839
All tetrahedra are either positively oriented or flat, though the
1840
entire solution is not flat and no tetrahedra are degenerate.
1844
<glossterm><guilabel>Contains negatively oriented
1845
tetrahedra</guilabel></glossterm>
1847
The volume is positive, but some tetrahedra are negatively oriented.
1851
<glossterm><guilabel>All tetrahedra flat</guilabel></glossterm>
1853
All tetrahedra are flat, but none have shape 0, 1 or infinity.
1857
<glossterm><guilabel>Contains degenerate
1858
tetrahedra</guilabel></glossterm>
1860
At least one tetrahedron has shape 0, 1 or infinity.
1864
<glossterm><guilabel>Unrecognised solution type</guilabel></glossterm>
1866
The volume is zero or negative, but the solution is
1867
neither flat nor degenerate.
1871
<glossterm><guilabel>No solution found</guilabel></glossterm>
1873
The gluing equations could not be solved.
1880
<glossterm><guilabel>Volume</guilabel></glossterm>
1882
This gives the volume of the underlying 3-manifold, along with the
1883
estimated number of decimal places of accuracy.
1884
This accuracy measure is an <emphasis>estimate only</emphasis>
1885
(based on the differences between terms in Newton's method).
1892
<sect2 id="tri-decomposition">
1893
<title>Decomposition</title>
1895
®ina; implements some high-level algorithms for decomposition a
1896
3-manifold triangulation into “atomic pieces”.
1897
These include the following:
1900
<sect3 id="tri-decomposition-component">
1901
<title>Component Decomposition</title>
1904
If your triangulation is
1905
<link linkend="tri-basicprops">disconnected</link>, you may wish to
1906
break it into its connected components. To do this, select
1908
<guimenu>Triangulation</guimenu>
1909
<guimenuitem>Extract Components</guimenuitem>
1911
You must open the triangulation for viewing before you can do this.
1916
<imagedata fileref="menucomponents.png"/>
1918
</inlinemediaobject>
1921
®ina; will create several new triangulations, one for each
1922
connected component. These will be added beneath the original in
1923
the packet tree. Your original (disconnected) triangulation will
1929
<imagedata fileref="tri-extract-components.png"/>
1931
</inlinemediaobject>
1935
<sect3 id="tri-decomposition-connsum">
1936
<title>Connected Sum Decomposition</title>
1938
If your triangulation is <link linkend="tri-basicprops">closed,
1939
orientable and connected</link>, ®ina; can decompose it into a
1940
connected sum of prime 3-manifolds (none of which are 3-spheres).
1943
<guimenu>Triangulation</guimenu>
1944
<guimenuitem>Connected Sum Decomposition</guimenuitem>
1946
You must open the triangulation for viewing before you can do this.
1951
<imagedata fileref="menuconnsum.png"/>
1953
</inlinemediaobject>
1956
Again, ®ina; will create several new triangulations, one for
1957
each prime summand. These will be added beneath the original in
1958
the packet tree, and your original triangulation will remain unchanged.
1959
If your original triangulation is a 3-sphere then no prime summands
1960
will be produced at all.
1965
<imagedata fileref="tri-connsum-results.png"/>
1967
</inlinemediaobject>
1970
With two exceptions (RP<superscript>3</superscript> and
1971
S<superscript>2</superscript>×S<superscript>1</superscript>),
1972
each of the new triangulations is guaranteed to be
1973
<link linkend="tri-prop0eff">0-efficient</link> (i.e., they will
1974
have no non-vertex-linking normal spheres).
1975
The underlying algorithm is based on the 0-efficiency results
1976
of Jaco and Rubinstein <xref linkend="bib-0-efficiency"/>,
1977
and uses <link linkend="tri-prop3sphere">3-sphere recognition</link>
1978
to ensure that none of the summands are trivial.
1981
Connected sum decomposition can be very slow for larger
1982
triangulations, since the underlying normal surface algorithms have
1983
worst-case exponential running time.
1987
<sect2 id="tri-censuslookup">
1988
<title>Census Lookup</title>
1991
®ina; ships with several prepackaged censuses of 3-manifold
1993
To search for your triangulation within these censuses, select
1995
<guimenu>Triangulation</guimenu>
1996
<guimenuitem>Census Lookup</guimenuitem>
1998
You must open the triangulation for viewing before you can do this.
2003
<imagedata fileref="menucensuslookup.png"/>
2005
</inlinemediaobject>
2008
Your triangulation may use different tetrahedron and
2009
vertex labels; ®ina; will search for any isomorphic copy.
2010
Any matches will be reported:
2015
<imagedata fileref="tri-census-found-box.png"/>
2017
</inlinemediaobject>
2020
The matches will also be stored in a new text packet
2021
beneath your triangulation:
2026
<imagedata fileref="tri-census-found-text.png"/>
2028
</inlinemediaobject>
2031
By default, ®ina; will search censuses of
2032
closed orientable and non-orientable 3-manifold triangulations
2033
<xref linkend="bib-burton-nor8"/> <xref linkend="bib-burton-nor10"/>
2034
<xref linkend="bib-burton-genus"/>,
2035
cusped and closed hyperbolic 3-manifold triangulations
2036
<xref linkend="bib-cuspedcensus"/> <xref linkend="bib-closedhypcensus"/>,
2037
and knot and link complements (tabulated by Joe Christy).
2038
To add your own censuses to this list, visit ®ina;'s
2039
<link linkend="options-census">census options</link>.
2044
<sect1 id="tri-modification">
2045
<title>Modification</title>
2048
There are many ways of modifying a 3-manifold triangulation.
2049
Many of these can be found in the <guimenu>Triangulation</guimenu> menu,
2050
which appears when you open a triangulation for viewing.
2055
<imagedata fileref="menutri.png"/>
2057
</inlinemediaobject>
2060
If you open one triangulation for viewing but then select another in
2061
the packet tree, all modifications will apply to the triangulation that
2062
you have open for viewing.
2065
<sect2 id="tri-editgluings">
2066
<title>Editing Tetrahedron Face Gluings</title>
2069
The simplest way to modify a triangulation is to open the
2070
<guilabel>Gluings</guilabel> tab and edit the face gluings table directly.
2071
See the notes on <link linkend="tri-viewgluings">viewing tetrahedron
2072
face gluings</link> for details on how to read the table.
2075
You can add and remove tetrahedra using the
2076
<guilabel>Add Tet</guilabel> and <guilabel>Remove Tet</guilabel>
2077
buttons, and you can change the gluings by typing directly into the table.
2078
If you want to remove a gluing (i.e., make a face part of the
2079
triangulation boundary), just delete the contents of the cell.
2084
<imagedata fileref="tri-editgluings.png"/>
2086
</inlinemediaobject>
2089
If you like, you can also <emphasis>name</emphasis> tetrahedra to help
2090
keep track of their roles within the triangulation.
2091
Click on the cell in the leftmost column (containing the tetrahedron
2092
number), and type a new name directly into the cell.
2097
<imagedata fileref="tri-editnames.png"/>
2099
</inlinemediaobject>
2103
<sect2 id="tri-simplification">
2104
<title>Automatic Simplification</title>
2107
®ina; has a rich set of <link linkend="tri-elementarymove">fast and
2108
effective moves</link> for simplifying
2109
a triangulation without changing the underlying 3-manifold.
2110
If you press the <guibutton>Simplify</guibutton> button (or select
2112
<guimenu>Triangulation</guimenu>
2113
<guimenuitem>Simplify</guimenuitem>
2115
then ®ina; will use a combination of these moves to reduce the
2116
triangulation to as few tetrahedra as it can.
2117
This is often very effective, but there is
2118
<emphasis>no guarantee</emphasis> that this will produce the fewest
2119
possible tetrahedra:
2120
®ina; might get stuck at a local minimum from which it cannot escape.
2125
<imagedata fileref="tri-simplify.png"/>
2127
</inlinemediaobject>
2130
If your triangulation has boundary, this routine will also try to
2131
make the number of boundary faces as small as it can (but again
2132
there is no guarantee of reaching a global minimum).
2136
<sect2 id="tri-elementarymove">
2137
<title>Manual Simplification: Elementary Moves</title>
2140
Instead of using automatic simplification, you might wish to modify
2141
your triangulation manually one step at a time. You can do this
2142
using <firstterm>elementary moves</firstterm>, which are small
2144
to the triangulation that preserve the underlying 3-manifold.
2145
To perform an elementary move, select
2147
<guimenu>Triangulation</guimenu>
2148
<guimenuitem>Elementary Move</guimenuitem>
2155
<imagedata fileref="tri-eltmove.png"/>
2157
</inlinemediaobject>
2160
This will bring up a box containing all the elementary moves that
2161
can be performed upon your triangulation. There are many different
2162
types of moves available, and this list may continue to grow with
2163
future releases of ®ina;.
2168
<imagedata fileref="tri-eltmovebox.png"/>
2170
</inlinemediaobject>
2173
For each type of move, you will be offered a drop-down list of
2174
locations at which the move can be performed. If a move is disabled
2175
(greyed out), this means there are no suitable locations in your
2176
triangulation for that move type.
2177
Select the type of move and its location, and press
2178
<guibutton>OK</guibutton> to perform the move.
2181
We do not give full details of the various moves here; see
2182
<xref linkend="bib-burton-phd"/> or the
2183
<classname>NTriangulation</classname> class notes in the ®enginedocs;
2184
for full descriptions of the moves and restrictions on their
2185
possible locations. A brief summary is as follows.
2190
<glossterm><guilabel>3-2 Move</guilabel></glossterm>
2192
Replaces three tetrahedra joined along a degree 3 edge
2193
with two tetrahedra joined along a face.
2197
<glossterm><guilabel>2-3 Move</guilabel></glossterm>
2199
Replaces two tetrahedra joined along a face
2200
with three tetrahedra joined along a degree 3 edge.
2204
<glossterm><guilabel>4-4 Move</guilabel></glossterm>
2206
Replaces four tetrahedra joined along a degree 4 edge
2207
with four tetrahedra joined along a new
2208
degree 4 edge that points in a different direction.
2212
<glossterm><guilabel>2-0 Move (Edge)</guilabel></glossterm>
2214
Takes two tetrahedra joined along a degree 2 edge and
2219
<glossterm><guilabel>2-0 Move (Vertex)</guilabel></glossterm>
2221
Takes two tetrahedra that meet at a degree 2 vertex and
2226
<glossterm><guilabel>2-1 Move</guilabel></glossterm>
2228
Merges the tetrahedron containing a degree 1 edge with an
2229
adjacent tetrahedron.
2233
<glossterm><guilabel>Open Book</guilabel></glossterm>
2235
Takes an internal face with two boundary edges and
2236
“unglues” that face, creating two new boundary
2237
faces and exposing the tetrahedra inside to the boundary.
2241
<glossterm><guilabel>Close Book</guilabel></glossterm>
2243
Folds together two adjacent boundary faces around a common
2244
boundary edge, with the result of simplifying the boundary.
2248
<glossterm><guilabel>Shell Boundary</guilabel></glossterm>
2250
Removes an “unnecessary tetrahedron” that sits along
2251
the boundary of the triangulation.
2255
<glossterm><guilabel>Collapse Edge</guilabel></glossterm>
2257
Takes an edge between two distinct vertices and collapses it to a point.
2258
Any tetrahedra that contained the edge will be “flattened
2266
<sect2 id="tri-make0eff">
2267
<title>0-Efficiency</title>
2270
A triangulation is <firstterm>0-efficient</firstterm> if its only
2271
normal spheres and discs are vertex linking, and if it has no 2-sphere
2272
boundary components <xref linkend="bib-0-efficiency"/>.
2273
0-efficient triangulations have significant theoretical and
2274
practical advantages, and often use relatively few tetrahedra.
2277
If your triangulation is
2278
<link linkend="tri-basicprops">closed, orientable and connected</link>,
2279
you can convert it into a 0-efficient triangulation of the same
2280
3-manifold by selecting
2282
<guimenu>Triangulation</guimenu>
2283
<guimenuitem>Make 0-Efficient</guimenuitem>
2289
<imagedata fileref="tri-make0eff.png"/>
2291
</inlinemediaobject>
2294
If your triangulation represents a composite 3-manifold then it
2295
cannot be made 0-efficient—in this case a full connected sum
2296
decomposition will be inserted beneath your triangulation in the
2297
packet tree, and your original triangulation will be left unchanged.
2302
<imagedata fileref="tri-make0eff-connsum.png"/>
2304
</inlinemediaobject>
2307
There are also two exceptional prime manifolds that cannot be made
2308
0-efficient: RP<superscript>3</superscript> and
2309
S<superscript>2</superscript>×S<superscript>1</superscript>.
2310
®ina; will notify you if your triangulation represents one of
2314
The algorithm to make a triangulation 0-efficient
2315
runs in worst-case exponential time.
2316
If your triangulation is large, you should consider whether
2317
<link linkend="tri-simplification">automatic simplification</link>
2318
will suffice (which is much faster at reducing the number of
2319
tetrahedra, but which does not guarantee a 0-efficient result).
2323
<sect2 id="tri-real-ideal">
2324
<title>Switching Between Real and Ideal</title>
2327
You can convert between <firstterm>real</firstterm> boundary
2328
components (formed from boundary faces of tetrahedra) and
2329
<firstterm>ideal</firstterm> boundary components (formed from
2330
individual vertices with closed non-spherical vertex links).
2333
If you have an ideal triangulation, you can select
2335
<guimenu>Triangulation</guimenu>
2336
<guimenuitem>Truncate Ideal Vertices</guimenuitem>
2338
to convert your ideal vertices into real boundary components.
2339
®ina; will subdivide the triangulation and slide off a small
2340
neighbourhood of each ideal vertex.
2341
Any <link linkend="tri-vertices-nonstdbdry">non-standard boundary
2342
vertices</link> will be truncated also.
2347
<imagedata fileref="tri-truncate.png"/>
2349
</inlinemediaobject>
2352
Because of the subdivision, this operation will greatly
2353
increase the number of tetrahedra. After you truncate ideal
2355
<link linkend="tri-simplification">simplifying your triangulation</link>.
2358
Conversely: if your triangulation has real boundary components and you
2359
wish to convert this into an ideal triangulation, select
2361
<guimenu>Triangulation</guimenu>
2362
<guimenuitem>Make Ideal</guimenuitem>
2368
<imagedata fileref="tri-makeideal.png"/>
2370
</inlinemediaobject>
2373
Each real boundary component will be “coned” using new
2374
tetrahedra (one for each boundary face). Your boundary components
2375
will all become ideal, but there are some caveats:
2378
Your triangulation will contain ideal vertices, but also
2379
standard <emphasis>internal vertices</emphasis> whose links are
2380
spheres. To get rid of these internal vertices, try
2381
<link linkend="tri-simplification">simplifying your triangulation</link>.
2384
Any <emphasis>spherical</emphasis> boundary components will
2385
disappear entirely; that is, they will be filled in with balls.
2391
<sect2 id="tri-barycentric">
2392
<title>Subdivision</title>
2395
You can perform a barycentric subdivision on your triangulation by
2398
<guimenu>Triangulation</guimenu>
2399
<guimenuitem>Barycentric Subdivision</guimenuitem>
2401
This involves splitting each original tetrahedron into 24 smaller
2402
tetrahedra, adding new vertices at the
2403
centroid of each tetrahedron, the centroid of each face, and the
2404
midpoint of each edge.
2409
<imagedata fileref="tri-barycentric.png"/>
2411
</inlinemediaobject>
2415
<sect2 id="tri-orient">
2416
<title>Orienting the Triangulation</title>
2419
If your triangulation is
2420
<link linkend="tri-basicprops">orientable but not oriented</link>,
2421
you may wish to reorder the vertices 0,1,2,3 of each tetrahedron so that
2422
adjacent tetrahedra have consistent orientations.
2423
To do this, press the <guibutton>Orient</guibutton> button (or select
2425
<guimenu>Triangulation</guimenu>
2426
<guimenuitem>Orient</guimenuitem>
2433
<imagedata fileref="tri-orient.png"/>
2435
</inlinemediaobject>
2439
<sect2 id="tri-cover">
2440
<title>Double Cover</title>
2443
To convert a non-orientable triangulation into its orientable double cover,
2446
<guimenu>Triangulation</guimenu>
2447
<guimenuitem>Double Cover</guimenuitem>
2453
<imagedata fileref="tri-doublecover.png"/>
2455
</inlinemediaobject>
2458
If your triangulation has any orientable components, they will
2459
simply be duplicated.
2463
<sect2 id="tri-cutcrush">
2464
<title>Cutting Along and Crushing Normal Surfaces</title>
2467
If you have a normal surface in your triangulation, you can either cut
2468
along your surface or crush it to a point.
2472
<emphasis>Cutting along</emphasis> a surface involves carefully
2473
slicing along the surface and retriangulating the resulting
2474
polyhedra, so that the original
2475
surface becomes one or more real boundary components.
2478
This has the advantages that it will never change the topology of the
2479
3-manifold beyond the simple act of slicing along the surface,
2480
and it will never introduce ideal vertices or invalid edges.
2483
The main drawback is that it can
2484
<emphasis>vastly</emphasis> increase the total number of tetrahedra.
2485
This has severe implications if you need to do anything
2486
computationally intensive with the resulting triangulation.
2491
<emphasis>Crushing a surface</emphasis> is a potentially destructive
2492
operation, but when used carefully can be extremely powerful.
2493
The crushing operation is described by
2494
Jaco and Rubinstein <xref linkend="bib-0-efficiency"/>:
2495
in essence, the surface is crushed to a point and any tetrahedron
2496
that contains a quadrilateral disc is “flattened away”.
2499
One key advantage of crushing is that it always
2500
<emphasis>reduces</emphasis> the number of tetrahedra
2501
(unless you crush vertex links, in which case the triangulation
2505
The main disadvantage is that will typically change the topology
2506
of your triangulation, sometimes dramatically.
2507
For example, it can create ideal vertices, undo connected sums,
2508
change the genus of boundary components, and delete entire summands.
2509
In some cases it can even make your triangulation invalid
2510
(for instance, edges might become identified with themselves in
2514
You should only crush a surface when you have theoretical
2515
arguments that tell you exactly what might change and how to detect it.
2516
Examples of such arguments appear in
2517
<xref linkend="bib-0-efficiency"/>, where crushing is used to
2524
To cut along or crush a normal surface: open the list of normal
2525
surfaces, select your surface in the list, and then choose either
2527
<guimenu>Normal Surfaces</guimenu>
2528
<guimenuitem>Cut Along Surface</guimenuitem>
2532
<guimenu>Normal Surfaces</guimenu>
2533
<guimenuitem>Crush Surface</guimenuitem>
2539
<imagedata fileref="tri-cutcrush.png"/>
2541
</inlinemediaobject>
2544
®ina; will create a new triangulation where the surface has been
2545
cut along or crushed accordingly. This new trianguation will appear
2546
beneath the normal surfaces in the packet tree.
2547
Your original triangulation will not be changed.
2552
<imagedata fileref="tri-cutcrush-results.png"/>
2554
</inlinemediaobject>
2557
When cutting along or crushing a normal surface, you might end up
2558
with a disconnected triangulation. You can
2559
<link linkend="tri-decomposition-component">extract connected
2560
components</link> to work with one at a time.
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kdeui/doc/regina/triangulations.docbook
