<h3>5 <span class="Heading">The GAP object types <code class="code">SCSimplicialComplex</code> and <code class="code">SCNormalSurface</code></span></h3>
<p>Currently, the <strong class="pkg">GAP</strong> package <strong class="pkg">simpcomp</strong> supports data structures for two different kinds of geometric objects, namely simplicial complexes (<code class="code">SCSimplicialComplex</code>) and discrete normal surfaces (<code class="code">SCNormalSurface</code>) which are both subtypes of the <strong class="pkg">GAP</strong> object type <code class="code">SCPolyhedralComplex</code></p>
<h4>5.1 <span class="Heading">The object type <code class="code">SCSimplicialComplex</code></span></h4>
<p>A major part of <strong class="pkg">simpcomp</strong> deals with the object type <code class="code">SCSimplicialComplex</code>. For a complete list of properties that <code class="code">SCSimplicialComplex</code> handles, see Chapter <a href="chap6.html#X82BDBFFC81D080D1"><span class="RefLink">6</span></a>. For a few fundamental methods and functions (such as checking the objectclass, copying objects of this type, etc.) for <code class="code">SCSimplicialComplex</code> see below.</p>
<p>Checks if <var class="Arg">object</var> is of type <code class="code">SCSimplicialComplex</code>. The object type <code class="code">SCSimplicialComplex</code> is derived from the object type <code class="code">SCPropertyObject</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCDetails</code>( <var class="Arg">complex</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a string of type <code class="code">IsString</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>The function returns a list of known properties of <var class="Arg">complex</var> an lists some of these properties explicitly.</p>
<div class="example"><pre>
gap> c:=SC([[1,2,3],[1,2,4],[1,3,4],[2,3,4]]);
<SimplicialComplex: unnamed complex 1 | dim = 2 | n = 4>
gap> Print(SCDetails(c));
[SimplicialComplex
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCCopy</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a copy of <var class="Arg">complex</var> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Makes a ``deep copy'' of <var class="Arg">complex</var> -- this is a copy such that all properties of the copy can be altered without changing the original complex.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ShallowCopy (SCSimplicialComplex)</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a copy of <var class="Arg">complex</var> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Makes a copy of <var class="Arg">complex</var>. This is actually a ``deep copy'' such that all properties of the copy can be altered without changing the original complex. Internally calls <codeclass="func">SCCopy</code> (<a href="chap5.html#X80F13BB484B3E9B2"><span class="RefLink">5.1-3</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCPropertiesDropped</code>( <var class="Arg">complex</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a object of type <code class="code">SCSimplicialComplex</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>An object of the type <code class="code">SCSimplicialComplex</code> caches its previously calculated properties such that each property only has to be calculated once. This function returns a copy of <var class="Arg">complex</var> with all properties (apart from Facets, Dim and Name) dropped, clearing all previously computed properties. See also <code class="func">SCPropertyDrop</code> (<a href="chap18.html#X8304C9BC7E8CF4F8"><span class="RefLink">18.1-8</span></a>) and <code class="func">SCPropertyTmpDrop</code> (<a href="chap18.html#X836F936B7E09E5F1"><span class="RefLink">18.1-13</span></a>).</p>
<h4>5.2 <span class="Heading">Overloaded operators of <code class="code">SCSimplicialComplex</code></span></h4>
<p><strong class="pkg">simpcomp</strong> overloads some standard operations for the object type <code class="code">SCSimplicialComplex</code> if this definition is intuitive and mathematically sound. See a list of overloaded operators below.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation + (SCSimplicialComplex, Integer)</code>( <var class="Arg">complex</var>, <var class="Arg">value</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: the simplicial complex passed as argument upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Positively shifts the vertex labels of <var class="Arg">complex</var> (provided that all labels satisfy the property <code class="code">IsAdditiveElement</code>) by the amount specified in <var class="Arg">value</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation - (SCSimplicialComplex, Integer)</code>( <var class="Arg">complex</var>, <var class="Arg">value</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: the simplicial complex passed as argument upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Negatively shifts the vertex labels of <var class="Arg">complex</var> (provided that all labels satisfy the property <code class="code">IsAdditiveElement</code>) by the amount specified in <var class="Arg">value</var>.</p>
<h5>5.2-3 Operation mod (SCSimplicialComplex, Integer)</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation mod (SCSimplicialComplex, Integer)</code>( <var class="Arg">complex</var>, <var class="Arg">value</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: the simplicial complex passed as argument upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Takes all vertex labels of <var class="Arg">complex</var> modulo the value specified in <var class="Arg">value</var> (provided that all labels satisfy the property <code class="code">IsAdditiveElement</code>). Warning: this might result in different vertices being assigned the same label or even in invalid facet lists, so be careful.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation ^ (SCSimplicialComplex, Integer)</code>( <var class="Arg">complex</var>, <var class="Arg">value</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: simplicial complex of type <code class="code">SCSimplicialComplex</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Forms the <var class="Arg">value</var>-th simplicial cartesian power of <var class="Arg">complex</var>, i.e. the <var class="Arg">value</var>-fold cartesian product of copies of <var class="Arg">complex</var>. The complex passed as argument is not altered. Internally calls <code class="func">SCCartesianPower</code> (<a href="chap6.html#X8255A2F97A7432F9"><span class="RefLink">6.6-1</span></a>).</p>
<div class="example"><pre>
gap> c:=SCBdSimplex(2)^2; #a torus
<SimplicialComplex: (S^1_3)^2 | dim = 2 | n = 9>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation + (SCSimplicialComplex, SCSimplicialComplex)</code>( <var class="Arg">complex1</var>, <var class="Arg">complex2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: simplicial complex of type <code class="code">SCSimplicialComplex</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Forms the connected sum of <var class="Arg">complex1</var> and <var class="Arg">complex2</var>. Uses the lexicographically first facets of both complexes to do the gluing. The complexes passed as arguments are not altered. Internally calls <code class="func">SCConnectedSum</code> (<a href="chap6.html#X81338CE18195607C"><span class="RefLink">6.6-5</span></a>).</p>
<h4>5.3 <span class="Heading"><code class="code">SCSimplicialComplex</code> as a subtype of <codeclass="code">Set</code></span></h4>
<p>Apart from being a subtype of <code class="code">SCPropertyObject</code>, an object of type <code class="code">SCSimplicialComplex</code> also behaves like a <strong class="pkg">GAP</strong> <code class="code">Set</code> type. The elements of the set are given by the facets of the simplical complex, grouped by their dimensionality, i.e. if <code class="code">complex</code> is an object of type <code class="code">SCSimplicialComplex</code>, <code class="code">c[1]</code> refers to the 0-faces of <code class="code">complex</code>, <code class="code">c[2]</code> to the 1-faces, etc.</p>
<h5>5.3-1 Operation Union (SCSimplicialComplex, SCSimplicialComplex)</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation Union (SCSimplicialComplex, SCSimplicialComplex)</code>( <var class="Arg">complex1</var>, <var class="Arg">complex2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: simplicial complex of type <code class="code">SCSimplicialComplex</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the union of two simplicial complexes by calling <code class="func">SCUnion</code> (<a href="chap7.html#X81DA367A813F7599"><span class="RefLink">7.3-16</span></a>).</p>
<div class="example"><pre>
gap> c:=Union(SCBdSimplex(3),SCBdSimplex(3)+3); #a wedge of two 2-spheres
<SimplicialComplex: S^2_4 cup S^2_4 | dim = 2 | n = 7>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation Difference (SCSimplicialComplex, SCSimplicialComplex)</code>( <var class="Arg">complex1</var>, <var class="Arg">complex2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: simplicial complex of type <code class="code">SCSimplicialComplex</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the ``difference'' of two simplicial complexes by calling <code class="func">SCDifference</code> (<a href="chap6.html#X7FB3D29178076EB4"><span class="RefLink">6.10-5</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation Intersection (SCSimplicialComplex, SCSimplicialComplex)</code>( <var class="Arg">complex1</var>, <var class="Arg">complex2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: simplicial complex of type <code class="code">SCSimplicialComplex</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the ``intersection'' of two simplicial complexes by calling <code class="func">SCIntersection</code> (<a href="chap6.html#X7B4BE2C783E6D0BF"><span class="RefLink">6.10-8</span></a>).</p>
<div class="example"><pre>
gap> c:=SCBdSimplex(3);;
gap> d:=SCBdSimplex(3);;
gap> d:=SCMove(d,[[1,2,3],[]]);;
gap> d:=d+1;;
gap> s1.Facets;
Error, Variable: 's1' must have a value
not in any function at *stdin*:77
</pre></div>
<p>Returns the ``size'' of a simplicial complex. This is <span class="SimpleMath">d+1</span>, where <span class="SimpleMath">d</span> is the dimension of the complex. <span class="SimpleMath">d+1</span> is returned instead of <span class="SimpleMath">d</span>, as all lists in <strong class="pkg">GAP</strong> are indexed beginning with 1 -- thus this also holds for all the face lattice related properties of the complex.</p>
<div class="example"><pre>
gap> SCLib.SearchByAttribute("F=[12,66,108,54]");;
gap> c:=SCLib.Load(last[1][1]);;
gap> for i in [1..Size(c)] do Print(c.F[i],"\n"); od;
12
66
108
54
</pre></div>
<p>Returns the ``size'' of a simplicial complex by calling <code class="code">Size(</code><var class="Arg">complex</var><code class="code">)</code>.</p>
<div class="example"><pre>
gap> SCLib.SearchByAttribute("F=[12,66,108,54]");;
gap> c:=SCLib.Load(last[1][1]);;
gap> for i in [1..Length(c)] do Print(c.F[i],"\n"); od;
12
66
108
54
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation [] (SCSimplicialComplex)</code>( <var class="Arg">complex</var>, <var class="Arg">pos</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of faces upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Returns the <span class="SimpleMath">(pos-1)</span>-dimensional faces of <var class="Arg">complex</var> as a list. If <span class="SimpleMath">pos ≥ d+2</span>, where <span class="SimpleMath">d</span> is the dimension of <var class="Arg">complex</var>, the empty set is returned. Note that <var class="Arg">pos</var> must be <span class="SimpleMath">≥ 1</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Iterator (SCSimplicialComplex)</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: an iterator on the face lattice of <var class="Arg">complex</var> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Provides an iterator object for the face lattice of a simplicial complex.</p>
<div class="example"><pre>
gap> c:=SCBdCrossPolytope(4);;
gap> for faces in c do Print(Length(faces),"\n"); od;
8
24
32
16
</pre></div>
<h4>5.4 <span class="Heading">The object type <code class="code">SCNormalSurface</code></span></h4>
<p>The <strong class="pkg">GAP</strong> object type <code class="code">SCNormalSurface</code> is designed to describe slicings (level sets of discrete Morse functions) of combinatorial <span class="SimpleMath">3</span>-manifolds, i. e. discrete normal surfaces. Internally <code class="code">SCNormalSurface</code> is a subtype of <code class="code">SCPolyhedralComplex</code> and, thus, mostly behaves like a <code class="code">SCSimplicialComplex</code> object (see Section <a href="chap5.html#X7E7034FC82152AE6"><span class="RefLink">5.1</span></a>). For a very short introduction to normal surfaces see <a href="chap2.html#X7BE7221B7C38B27D"><span class="RefLink">2.4</span></a>, for a more thorough introduction to the field see <a href="chapBib.html#biBSpreer10NormSurfsCombSlic">[Spr11b]</a>. For some fundamental methods and functions for <code class="code">SCNormalSurface</code> see below. For more functions related to the <code class="code">SCNormalSurface</code> object type see Chapter <a href="chap7.html#X8071FAE8806ACAA2"><spanclass="RefLink">7</span></a>.</p>
<h4>5.5 <span class="Heading">Overloaded operators of <code class="code">SCNormalSurface</code></span></h4>
<p>As with the object type <code class="code">SCSimplicialComplex</code>, <strong class="pkg">simpcomp</strong> overloads some standard operations for the object type <code class="code">SCNormalSurface</code>. See a list of overloaded operators below.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation + (SCNormalSurface, Integer)</code>( <var class="Arg">complex</var>, <var class="Arg">value</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: the discrete normal surface passed as argument upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Positively shifts the vertex labels of <var class="Arg">complex</var> (provided that all labels satisfy the property <code class="code">IsAdditiveElement</code>) by the amount specified in <var class="Arg">value</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation - (SCNormalSurface, Integer)</code>( <var class="Arg">complex</var>, <var class="Arg">value</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: the discrete normal surface passed as argument upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Negatively shifts the vertex labels of <var class="Arg">complex</var> (provided that all labels satisfy the property <code class="code">IsAdditiveElement</code>) by the amount specified in <var class="Arg">value</var>.</p>
<h5>5.5-3 Operation mod (SCNormalSurface, Integer)</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation mod (SCNormalSurface, Integer)</code>( <var class="Arg">complex</var>, <var class="Arg">value</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: the discrete normal surface passed as argument upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Takes all vertex labels of <var class="Arg">complex</var> modulo the value specified in <var class="Arg">value</var> (provided that all labels satisfy the property <code class="code">IsAdditiveElement</code>). Warning: this might result in different vertices being assigned the same label or even invalid facet lists, so be careful.</p>
<h4>5.6 <span class="Heading"><code class="code">SCNormalSurface</code> as a subtype of <code class="code">Set</code></span></h4>
<p>Like objects of type <code class="code">SCSimplicialComplex</code>, an object of type <code class="code">SCNormalSurface</code> behaves like a <strong class="pkg">GAP</strong> <code class="code">Set</code> type. The elements of the set are given by the facets of the normal surface, grouped by their dimensionality and type, i.e. if <code class="code">complex</code> is an object of type <code class="code">SCNormalSurface</code>, <code class="code">c[1]</code> refers to the 0-faces of <code class="code">complex</code>, <code class="code">c[2]</code> to the 1-faces, <code class="code">c[3]</code> to the triangles and <code class="code">c[4]</code> to the quadrilaterals. See below for some examples and Section <a href="chap5.html#X7CB571C97C426BE2"><span class="RefLink">5.3</span></a> for details.</p>
<h5>5.6-1 Operation Union (SCNormalSurface, SCNormalSurface)</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Operation Union (SCNormalSurface, SCNormalSurface)</code>( <var class="Arg">complex1</var>, <varclass="Arg">complex2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: discrete normal surface of type <code class="code">SCNormalSurface</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the union of two discrete normal surfaces by calling <code class="func">SCUnion</code> (<a href="chap7.html#X81DA367A813F7599"><span class="RefLink">7.3-16</span></a>).</p>
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