<h4>7.1 <span class="Heading">Creating an <code class="code">SCNormalSurface</code> object</span></h4>
<p>This section contains functions to construct discrete normal surfaces that are slicings from a list of <span class="SimpleMath">\(2\)</span>-dimensional facets (triangles and quadrilaterals) or combinatorial <span class="SimpleMath">\(3\)</span>-manifolds.</p>
<p>For a very short introduction to the theory of discrete normal surfaces and slicings see Section <a href="chap2_mj.html#X7BE7221B7C38B27D"><span class="RefLink">2.4</span></a> and Section <a href="chap2_mj.html#X86275D5979B4B531"><span class="RefLink">2.5</span></a>, for an introduction to the <strong class="pkg">GAP</strong> object type <code class="code">SCNormalSurface</code> see <a href="chap5_mj.html#X80F19FA07C71EDDC"><span class="RefLink">5.4</span></a>, for more information see the article <a href="chapBib_mj.html#biBSpreer10NormSurfsCombSlic">[Spr11b]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCNSEmpty</code>( )</td><td class="tdright">( function )</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>Generates an empty complex (of dimension <span class="SimpleMath">\(-1\)</span>), i. e. an object of type <code class="code">SCNormalSurface</code> with empty facet list.</p>
<div class="example"><pre>
gap> SCNSEmpty();
<NormalSurface: empty normal surface | dim = -1>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCNSFromFacets</code>( <var class="Arg">facets</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>Constructor for a discrete normal surface from a facet list, see <code class="func">SCFromFacets</code> (<a href="chap6_mj.html#X7B5A874584FF34A7"><span class="RefLink">6.1-1</span></a>) for details.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCNSSlicing</code>( <var class="Arg">complex</var>, <var class="Arg">slicing</var> )</td><td class="tdright">( function )</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 a slicing defined by a partition <var class="Arg">slicing</var> of the set of vertices of the <span class="SimpleMath">\(3\)</span>-dimensional combinatorial pseudomanifold <var class="Arg">complex</var>. In particular, <var class="Arg">slicing</var> has to be a pair of lists of vertex labels and has to contain all vertex labels of <var class="Arg">complex</var>.</p>
<h4>7.2 <span class="Heading">Generating new objects from discrete normal surfaces</span></h4>
<p><strong class="pkg">simpcomp</strong> provides the possibility to copy and / or triangulate normal surfaces. Note that other constructions like the connected sum or the cartesian product do not make sense for (embedded) normal surfaces in general.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCNSTriangulation</code>( <var class="Arg">sl</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 a simplicial subdivision of a slicing <var class="Arg">sl</var> without introducing new vertices. The subdivision is stored as a property of <var class="Arg">sl</var> and thus is returned as an immutable object. Note that symmetry may be lost during the computation.</p>
<h4>7.3 <span class="Heading">Properties of <code class="code">SCNormalSurface</code> objects</span></h4>
<p>Although some properties of a discrete normal surface can be computed by using the functions for simplicial complexes, there is a variety of properties needing specially designed functions. See below for a list.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCConnectedComponents</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of simplicial complexes of type <code class="code">SCNormalSurface</code> upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes all connected components of an arbitrary normal surface.</p>
<p>Computes the dimension of a discrete normal surface (which is always <span class="SimpleMath">\(2\)</span> if the slicing <var class="Arg">sl</var> is not empty).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCFVector</code>( <var class="Arg">sl</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a <span class="SimpleMath">\(1\)</span>, <span class="SimpleMath">\(3\)</span> or <span class="SimpleMath">\(4\)</span> tuple of (non-negative) integer values upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the <span class="SimpleMath">\(f\)</span>-vector of a discrete normal surface, i. e. the number of vertices, edges, triangles and quadrilaterals of <var class="Arg">sl</var>, cf. <code class="func">SCFVector</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCFaceLattice</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of facet lists upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the face lattice of a discrete normal surface <var class="Arg">sl</var> in the original labeling. Triangles and quadrilaterals are stored separately (cf. <code class="func">SCSkel</code> (<a href="chap6_mj.html#X8026B46F8236124D"><span class="RefLink">6.9-54</span></a>)).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCFaceLatticeEx</code>( <var class="Arg">complex</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of face lists upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the face lattice of a discrete normal surface <var class="Arg">sl</var> in the standard labeling. Triangles and quadrilaterals are stored separately (cf. <code class="func">SCSkelEx</code> (<a href="chap6_mj.html#X84ACF7D580FE8B76"><span class="RefLink">6.9-55</span></a>)).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCFpBettiNumbers</code>( <var class="Arg">sl</var>, <var class="Arg">p</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of non-negative integers upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the Betti numbers modulo <var class="Arg">p</var> of a slicing <var class="Arg">sl</var>. Internally, <var class="Arg">sl</var> is triangulated (using <code class="func">SCNSTriangulation</code> (<a href="chap7_mj.html#X7A180D487B8941C5"><span class="RefLink">7.2-2</span></a>)) and the Betti numbers are computed via <code class="func">SCFpBettiNumbers</code> using the triangulation.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCHomology</code>( <var class="Arg">sl</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a list of homology groups upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes the homology of a slicing <var class="Arg">sl</var>. Internally, <var class="Arg">sl</var> is triangulated (cf. <code class="func">SCNSTriangulation</code> (<a href="chap7_mj.html#X7A180D487B8941C5"><span class="RefLink">7.2-2</span></a>)) and simplicial homology is computed via <code class="func">SCHomology</code> using the triangulation.</p>
<p>Checks if a normal surface <var class="Arg">complex</var> is the empty complex, i. e. a <code class="code">SCNormalSurface</code> object with empty facet list.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCSkel</code>( <var class="Arg">sl</var>, <var class="Arg">k</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a face list (of <var class="Arg">k+1</var>tuples) or a list of face lists upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes all faces of cardinality <var class="Arg">k+1</var> in the original labeling: <var class="Arg">k</var> <span class="SimpleMath">\(= 0\)</span> computes the vertices, <var class="Arg">k</var> <span class="SimpleMath">\(= 1\)</span> computes the edges, <var class="Arg">k</var> <span class="SimpleMath">\(= 2\)</span> computes the triangles, <var class="Arg">k</var> <span class="SimpleMath">\(= 3\)</span> computes the quadrilaterals.</p>
<p>If <var class="Arg">k</var> is a list (necessarily a sublist of <code class="code">[ 0,1,2,3 ]</code>) all faces of all cardinalities contained in <var class="Arg">k</var> are computed.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCSkelEx</code>( <var class="Arg">sl</var>, <var class="Arg">k</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a face list (of <var class="Arg">k+1</var>tuples) or a list of face lists upon success, <code class="keyw">fail</code> otherwise.</p>
<p>Computes all faces of cardinality <var class="Arg">k+1</var> in the standard labeling: <var class="Arg">k</var> <span class="SimpleMath">\(= 0\)</span> computes the vertices, <var class="Arg">k</var> <span class="SimpleMath">\(= 1\)</span> computes the edges, <var class="Arg">k</var> <span class="SimpleMath">\(= 2\)</span> computes the triangles, <var class="Arg">k</var> <span class="SimpleMath">\(= 3\)</span> computes the quadrilaterals.</p>
<p>If <var class="Arg">k</var> is a list (necessarily a sublist of <code class="code">[ 0,1,2,3 ]</code>) all faces of all cardinalities contained in <var class="Arg">k</var> are computed.</p>
<p>Determines the topological type of <var class="Arg">sl</var> via the classification theorem for closed compact surfaces. If <var class="Arg">sl</var> is not connected, the topological type of each connected component is computed.</p>
<div class="example"><pre>
gap> SCLib.SearchByName("(S^2xS^1)#20");
[ [ 633, "(S^2xS^1)#20" ] ]
gap> c:=SCLib.Load(last[1][1]);;
gap> c.F;
[ 27, 298, 542, 271 ]
gap> for i in [1..26] do sl:=SCNSSlicing(c,[[1..i],[i+1..27]]); Print(sl.TopologicalType,"\n"); od;
S^2
S^2
S^2
S^2
S^2 U S^2
S^2 U S^2
S^2
(T^2)#3
(T^2)#5
(T^2)#4
(T^2)#3
(T^2)#7
(T^2)#7 U S^2
(T^2)#7 U S^2
(T^2)#7 U S^2
(T^2)#8 U S^2
(T^2)#7 U S^2
(T^2)#8
(T^2)#6
(T^2)#6
(T^2)#5
(T^2)#3
(T^2)#2
T^2
S^2
S^2
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SCUnion</code>( <var class="Arg">complex1</var>, <var class="Arg">complex2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: normal surface of type <code class="code">SCNormalSurface</code> upon success, <codeclass="keyw">fail</code> otherwise.</p>
<p>Forms the union of two normal surfaces <var class="Arg">complex1</var> and <var class="Arg">complex2</var> as the normal surface formed by the union of their facet sets. The two arguments are not altered. Note: for the union process the vertex labelings of the complexes are taken into account, see also <code class="func">Operation Union (SCNormalSurface, SCNormalSurface)</code> (<a href="chap5_mj.html#X8462E960847F8B83"><span class="RefLink">5.6-1</span></a>). Facets occurring in both arguments are treated as one facet in the new complex.</p>
<div class="example"><pre>
gap> list:=SCLib.SearchByAttribute("Dim=3 and F[1]=10");;
gap> c:=SCLib.Load(list[1][1]);
<SimplicialComplex: S^3 (VT) | dim = 3 | n = 10>
gap> sl1:=SCNSSlicing(c,[[1..5],[6..10]]);;
gap> sl2:=sl1+10;;
gap> sl3:=SCUnion(sl1,sl2);;
gap> SCTopologicalType(sl3); "S^2 U S^2"
</pre></div>
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