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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX- </script> <title>GAP (simpcomp) - Chapter 7: Functions and operations for SCNormalSurface</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap7" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <p id="mathjaxlink" class="pcenter"><a href="chap7.html">[MathJax off]</a></p> <p><a id="X8071FAE8806ACAA2" name="X8071FAE8806ACAA2"></a></p> <div class="ChapSects"><a href="chap7_mj.html#X8071FAE8806ACAA2">7 <span class="Heading">F <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7A59063180BD596 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86C51E1F7CFA39F <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X78F4044683236A9 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8266FAFC7C6B868 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X80F13BB484B3E9B <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7A180D487B8941C </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X82C9F57780C0B7F <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X82351AAE793DCB6 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X788BAE187D58410 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X81F8071385FD9C1 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F8B561C823DDDB <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7AE0029985BD077 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X79F60850875BB68 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X806E6A4C7CD30A9 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X78D66254858CE90 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X81AF20DC814B51A <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X860375D980E9A80 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X78C860DC851167F <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8026B46F8236124 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X84ACF7D580FE8B7 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X84CBD2F780A1F63 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X81DA367A813F759 </div></div> </div> <h3>7 <span class="Heading">Functions and operations for <code class="code">SCNormalSurface</c <p><a id="X7C63CE8578FBB0C7" name="X7C63CE8578FBB0C7"></a></p> <h4>7.1 <span class="Heading">Creating an <code class="code">SCNormalSurface</code> object</spa <p>This section contains functions to construct discrete normal surfaces that are slicings fro <p>For a very short introduction to the theory of discrete normal surfaces and slicings see Secti <p><a id="X7A59063180BD5969" name="X7A59063180BD5969"></a></p> <h5>7.1-1 SCNSEmpty</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: discrete normal surface of type <code class="code">SCNormalSurface</code> upon succ <p>Generates an empty complex (of dimension <span class="SimpleMath">\(-1\)</span>), i. e. an obj <div class="example"><pre> gap> SCNSEmpty(); <NormalSurface: empty normal surface | dim = -1> </pre></div> <p><a id="X86C51E1F7CFA39F9" name="X86C51E1F7CFA39F9"></a></p> <h5>7.1-2 SCNSFromFacets</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: discrete normal surface of type <code class="code">SCNormalSurface</code> upon succ <p>Constructor for a discrete normal surface from a facet list, see <code class="func">SCFromFac <div class="example"><pre> gap> sl:=SCNSFromFacets([[1,2,3],[1,2,4,5],[1,3,4,6],[2,3,5,6],[4,5,6]]); <NormalSurface: unnamed complex 114 | dim = 2> </pre></div> <p><a id="X78F4044683236A97" name="X78F4044683236A97"></a></p> <h5>7.1-3 SCNS</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: discrete normal surface of type <code class="code">SCNormalSurface</code> upon succ <p>Internally calls <code class="func">SCNSFromFacets</code> (<a href="chap7_mj.html#X86C51E <div class="example"><pre> gap> sl:=SCNS([[1,2,3],[1,2,4,5],[1,3,4,6],[2,3,5,6],[4,5,6]]); <NormalSurface: unnamed complex 115 | dim = 2> </pre></div> <p><a id="X8266FAFC7C6B8685" name="X8266FAFC7C6B8685"></a></p> <h5>7.1-4 SCNSSlicing</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: discrete normal surface of type <code class="code">SCNormalSurface</code> upon succ <p>Computes a slicing defined by a partition <var class="Arg">slicing</var> of the set of vertices o <div class="example"><pre> gap> SCLib.SearchByAttribute("F=[ 10, 35, 50, 25 ]"); [ [ 19, "S^3 (VT)" ] ] gap> c:=SCLib.Load(last[1][1]);; gap> sl:=SCNSSlicing(c,[[1..5],[6..10]]); <NormalSurface: slicing [ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10 ] ] of S^3 (VT) \ | dim = 2> gap> sl.Facets; [ [ [ 1, 6 ], [ 1, 8 ], [ 1, 9 ] ], [ [ 1, 6 ], [ 1, 8 ], [ 3, 6 ], [ 3, 8 ] ] , [ [ 1, 6 ], [ 1, 9 ], [ 4, 6 ], [ 4, 9 ] ], [ [ 1, 6 ], [ 3, 6 ], [ 4, 6 ] ], [ [ 1, 8 ], [ 1, 9 ], [ 1, 10 ] ], [ [ 1, 8 ], [ 1, 10 ], [ 3, 8 ], [ 3, 10 ] ], [ [ 1, 9 ], [ 1, 10 ], [ 2, 9 ], [ 2, 10 ] ], [ [ 1, 9 ], [ 2, 9 ], [ 4, 9 ] ], [ [ 1, 10 ], [ 2, 10 ], [ 3, 10 ] ], [ [ 2, 7 ], [ 2, 9 ], [ 2, 10 ] ], [ [ 2, 7 ], [ 2, 9 ], [ 4, 7 ], [ 4, 9 ] ], [ [ 2, 7 ], [ 2, 10 ], [ 5, 7 ], [ 5, 10 ] ], [ [ 2, 7 ], [ 4, 7 ], [ 5, 7 ] ], [ [ 2, 10 ], [ 3, 10 ], [ 5, 10 ] ], [ [ 3, 6 ], [ 3, 8 ], [ 5, 6 ], [ 5, 8 ] ], [ [ 3, 6 ], [ 4, 6 ], [ 5, 6 ] ] , [ [ 3, 8 ], [ 3, 10 ], [ 5, 8 ], [ 5, 10 ] ], [ [ 4, 6 ], [ 4, 7 ], [ 4, 9 ] ], [ [ 4, 6 ], [ 4, 7 ], [ 5, 6 ], [ 5, 7 ] ] , [ [ 5, 6 ], [ 5, 7 ], [ 5, 8 ] ], [ [ 5, 7 ], [ 5, 8 ], [ 5, 10 ] ] ] gap> sl:=SCNSSlicing(c,[[1,3,5,7,9],[2,4,6,8,10]]); <NormalSurface: slicing [ [ 1, 3, 5, 7, 9 ], [ 2, 4, 6, 8, 10 ] ] of S^3 (VT) \ | dim = 2> gap> sl.Facets; [ [ [ 1, 2 ], [ 1, 4 ], [ 3, 2 ], [ 3, 4 ] ], [ [ 1, 2 ], [ 1, 4 ], [ 9, 2 ], [ 9, 4 ] ], [ [ 1, 2 ], [ 1, 10 ], [ 3, 2 ], [ 3, 10 ] ], [ [ 1, 2 ], [ 1, 10 ], [ 9, 2 ], [ 9, 10 ] ], [ [ 1, 4 ], [ 1, 6 ], [ 3, 4 ], [ 3, 6 ] ], [ [ 1, 4 ], [ 1, 6 ], [ 9, 4 ], [ 9, 6 ] ], [ [ 1, 6 ], [ 1, 8 ], [ 3, 6 ], [ 3, 8 ] ], [ [ 1, 6 ], [ 1, 8 ], [ 9, 6 ], [ 9, 8 ] ], [ [ 1, 8 ], [ 1, 10 ], [ 3, 8 ], [ 3, 10 ] ], [ [ 1, 8 ], [ 1, 10 ], [ 9, 8 ], [ 9, 10 ] ], [ [ 3, 2 ], [ 3, 4 ], [ 5, 2 ], [ 5, 4 ] ], [ [ 3, 2 ], [ 3, 10 ], [ 5, 2 ], [ 5, 10 ] ], [ [ 3, 4 ], [ 3, 6 ], [ 5, 4 ], [ 5, 6 ] ], [ [ 3, 6 ], [ 3, 8 ], [ 5, 6 ], [ 5, 8 ] ], [ [ 3, 8 ], [ 3, 10 ], [ 5, 8 ], [ 5, 10 ] ], [ [ 5, 2 ], [ 5, 4 ], [ 7, 2 ], [ 7, 4 ] ], [ [ 5, 2 ], [ 5, 10 ], [ 7, 2 ], [ 7, 10 ] ], [ [ 5, 4 ], [ 5, 6 ], [ 7, 4 ], [ 7, 6 ] ], [ [ 5, 6 ], [ 5, 8 ], [ 7, 6 ], [ 7, 8 ] ], [ [ 5, 8 ], [ 5, 10 ], [ 7, 8 ], [ 7, 10 ] ], [ [ 7, 2 ], [ 7, 4 ], [ 9, 2 ], [ 9, 4 ] ], [ [ 7, 2 ], [ 7, 10 ], [ 9, 2 ], [ 9, 10 ] ], [ [ 7, 4 ], [ 7, 6 ], [ 9, 4 ], [ 9, 6 ] ], [ [ 7, 6 ], [ 7, 8 ], [ 9, 6 ], [ 9, 8 ] ], [ [ 7, 8 ], [ 7, 10 ], [ 9, 8 ], [ 9, 10 ] ] ] </pre></div> <p><a id="X7E853EF97A3D0220" name="X7E853EF97A3D0220"></a></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 no <p><a id="X80F13BB484B3E9B2" name="X80F13BB484B3E9B2"></a></p> <h5>7.2-1 SCCopy</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: discrete normal surface of type <code class="code">SCNormalSurface</code> upon succ <p>Copies a <strong class="pkg">GAP</strong> object of type <code class="code">SCNormalSurface</c <div class="example"><pre> gap> sl:=SCNSSlicing(SCBdSimplex(4),[[1],[2..5]]); <NormalSurface: slicing [ [ 1 ], [ 2, 3, 4, 5 ] ] of S^3_5 | dim = 2> gap> sl_2:=SCCopy(sl); <NormalSurface: slicing [ [ 1 ], [ 2, 3, 4, 5 ] ] of S^3_5 | dim = 2> gap> IsIdenticalObj(sl,sl_2); false </pre></div> <p><a id="X7A180D487B8941C5" name="X7A180D487B8941C5"></a></p> <h5>7.2-2 SCNSTriangulation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: simplicial complex of type <code class="code">SCSimplicialComplex</code> upon succ <p>Computes a simplicial subdivision of a slicing <var class="Arg">sl</var> without introducing n <div class="example"><pre> gap> SCLib.SearchByAttribute("F=[ 10, 35, 50, 25 ]"); [ [ 19, "S^3 (VT)" ] ] gap> c:=SCLib.Load(last[1][1]);; gap> sl:=SCNSSlicing(c,[[1,3,5,7,9],[2,4,6,8,10]]);; gap> sl.F; [ 25, 50, 0, 25 ] gap> sc:=SCNSTriangulation(sl);; gap> sc.F; [ 25, 75, 50 ] </pre></div> <p><a id="X83A1885E876D7483" name="X83A1885E876D7483"></a></p> <h4>7.3 <span class="Heading">Properties of <code class="code">SCNormalSurface</code> objects</ <p>Although some properties of a discrete normal surface can be computed by using the functions f <p><a id="X82C9F57780C0B7F8" name="X82C9F57780C0B7F8"></a></p> <h5>7.3-1 SCConnectedComponents</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a list of simplicial complexes of type <code class="code">SCNormalSurface</code> upo <p>Computes all connected components of an arbitrary normal surface.</p> <div class="example"><pre> gap> sl:=SCNSSlicing(SCBdCrossPolytope(4),[[1,2],[3..8]]); <NormalSurface: slicing [ [ 1, 2 ], [ 3, 4, 5, 6, 7, 8 ] ] of Bd(\beta^4) | di\ m = 2> gap> cc:=SCConnectedComponents(sl); [ <NormalSurface: Connected component #1 of slicing [ [ 1, 2 ], [ 3, 4, 5, 6, \ 7, 8 ] ] of Bd(\beta^4) | dim = 2>, <NormalSurface: Connected component #2 of slicing [ [ 1, 2 ], [ 3, 4, 5, 6, \ 7, 8 ] ] of Bd(\beta^4) | dim = 2> ] </pre></div> <p><a id="X82351AAE793DCB68" name="X82351AAE793DCB68"></a></p> <h5>7.3-2 SCDim</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: an integer upon success, <code class="keyw">fail</code> otherwise.</p> <p>Computes the dimension of a discrete normal surface (which is always <span class="SimpleMath <div class="example"><pre> gap> sl:=SCNSEmpty();; gap> SCDim(sl); -1 gap> sl:=SCNSFromFacets([[1,2,3],[1,2,4,5],[1,3,4,6],[2,3,5,6],[4,5,6]]);; gap> SCDim(sl); 2 </pre></div> <p><a id="X788BAE187D584103" name="X788BAE187D584103"></a></p> <h5>7.3-3 SCEulerCharacteristic</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: an integer upon success, <code class="keyw">fail</code> otherwise.</p> <p>Computes the Euler characteristic of a discrete normal surface <var class="Arg">sl</var>, cf. <c <div class="example"><pre> gap> list:=SCLib.SearchByName("S^2xS^1");; gap> c:=SCLib.Load(list[1][1]);; gap> sl:=SCNSSlicing(c,[[1..5],[6..10]]);; gap> SCEulerCharacteristic(sl); 4 </pre></div> <p><a id="X81F8071385FD9C1D" name="X81F8071385FD9C1D"></a></p> <h5>7.3-4 SCFVector</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a <span class="SimpleMath">\(1\)</span>, <span class="SimpleMath">\(3\)</span> or <spa <p>Computes the <span class="SimpleMath">\(f\)</span>-vector of a discrete normal surface, i. e. <div class="example"><pre> gap> list:=SCLib.SearchByName("S^2xS^1");; gap> c:=SCLib.Load(list[1][1]);; gap> sl:=SCNSSlicing(c,[[1..5],[6..10]]);; gap> SCFVector(sl); [ 20, 40, 16, 8 ] </pre></div> <p><a id="X7F8B561C823DDDBA" name="X7F8B561C823DDDBA"></a></p> <h5>7.3-5 SCFaceLattice</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <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 <div class="example"><pre> gap> c:=SCBdSimplex(4);; gap> sl:=SCNSSlicing(c,[[1,2],[3..5]]);; gap> SCFaceLattice(sl); [ [ [ [ 1, 3 ] ], [ [ 1, 4 ] ], [ [ 1, 5 ] ], [ [ 2, 3 ] ], [ [ 2, 4 ] ], [ [ 2, 5 ] ] ], [ [ [ 1, 3 ], [ 1, 4 ] ], [ [ 1, 3 ], [ 1, 5 ] ], [ [ 1, 3 ], [ 2, 3 ] ], [ [ 1, 4 ], [ 1, 5 ] ], [ [ 1, 4 ], [ 2, 4 ] ], [ [ 1, 5 ], [ 2, 5 ] ], [ [ 2, 3 ], [ 2, 4 ] ], [ [ 2, 3 ], [ 2, 5 ] ], [ [ 2, 4 ], [ 2, 5 ] ] ] , [ [ [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ], [ [ 2, 3 ], [ 2, 4 ], [ 2, 5 ] ] ], [ [ [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ] ], [ [ 1, 3 ], [ 1, 5 ], [ 2, 3 ], [ 2, 5 ] ], [ [ 1, 4 ], [ 1, 5 ], [ 2, 4 ], [ 2, 5 ] ] ] ] gap> sl.F; [ 6, 9, 2, 3 ] </pre></div> <p><a id="X7AE0029985BD0775" name="X7AE0029985BD0775"></a></p> <h5>7.3-6 SCFaceLatticeEx</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <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 <div class="example"><pre> gap> c:=SCBdSimplex(4);; gap> sl:=SCNSSlicing(c,[[1,2],[3..5]]);; gap> SCFaceLatticeEx(sl); [ [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ] ], [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 5 ], [ 3, 6 ], [ 4, 5 ], [ 4, 6 ], [ 5, 6 ] ], [ [ 1, 2, 3 ], [ 4, 5, 6 ] ], [ [ 1, 2, 4, 5 ], [ 1, 3, 4, 6 ], [ 2, 3, 5, 6 ] ] ] gap> sl.F; [ 6, 9, 2, 3 ] </pre></div> <p><a id="X79F60850875BB683" name="X79F60850875BB683"></a></p> <h5>7.3-7 SCFpBettiNumbers</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a list of non-negative integers upon success, <code class="keyw">fail</code> otherwi <p>Computes the Betti numbers modulo <var class="Arg">p</var> of a slicing <var class="Arg">sl</var>. <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> sl:=SCNSSlicing(c,[[1..13],[14..27]]);; gap> SCFpBettiNumbers(sl,2); [ 2, 14, 2 ] </pre></div> <p><a id="X806E6A4C7CD30A96" name="X806E6A4C7CD30A96"></a></p> <h5>7.3-8 SCGenus</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a non-negative integer upon success, <code class="keyw">fail</code> otherwise.</p> <p>Computes the genus of a discrete normal surface <var class="Arg">sl</var>.</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> sl:=SCNSSlicing(c,[[1..12],[13..27]]);; gap> SCIsConnected(sl); true gap> SCGenus(sl); 7 </pre></div> <p><a id="X78D66254858CE901" name="X78D66254858CE901"></a></p> <h5>7.3-9 SCHomology</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <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</v <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> sl:=SCNSSlicing(c,[[1..12],[13..27]]);; gap> sl.Homology; [ [ 0, [ ] ], [ 14, [ ] ], [ 1, [ ] ] ] gap> sl:=SCNSSlicing(c,[[1..13],[14..27]]);; gap> sl.Homology; [ [ 1, [ ] ], [ 14, [ ] ], [ 2, [ ] ] ] </pre></div> <p><a id="X81AF20DC814B51A6" name="X81AF20DC814B51A6"></a></p> <h5>7.3-10 SCIsConnected</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code> upon success, <code cl <p>Checks if a normal surface <var class="Arg">complex</var> is connected.</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> sl:=SCNSSlicing(c,[[1..5],[6..10]]); <NormalSurface: slicing [ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10 ] ] of S^3 (VT) \ | dim = 2> gap> SCIsConnected(sl); true </pre></div> <p><a id="X860375D980E9A801" name="X860375D980E9A801"></a></p> <h5>7.3-11 SCIsEmpty</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code> upon success, <code cl <p>Checks if a normal surface <var class="Arg">complex</var> is the empty complex, i. e. a <code class <div class="example"><pre> gap> sl:=SCNS([]);; gap> SCIsEmpty(sl); true </pre></div> <p><a id="X78C860DC851167F7" name="X78C860DC851167F7"></a></p> <h5>7.3-12 SCIsOrientable</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code> upon success, <code cl <p>Checks if a discrete normal surface <var class="Arg">sl</var> is orientable.</p> <div class="example"><pre> gap> c:=SCBdSimplex(4);; gap> sl:=SCNSSlicing(c,[[1,2],[3,4,5]]); <NormalSurface: slicing [ [ 1, 2 ], [ 3, 4, 5 ] ] of S^3_5 | dim = 2> gap> SCIsOrientable(sl); true </pre></div> <p><a id="X8026B46F8236124D" name="X8026B46F8236124D"></a></p> <h5>7.3-13 SCSkel</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a face list (of <var class="Arg">k+1</var>tuples) or a list of face lists upon success, <c <p>Computes all faces of cardinality <var class="Arg">k+1</var> in the original labeling: <var clas <p>If <var class="Arg">k</var> is a list (necessarily a sublist of <code class="code">[ 0,1,2,3 ]</cod <div class="example"><pre> gap> c:=SCBdSimplex(4);; gap> sl:=SCNSSlicing(c,[[1],[2..5]]);; gap> SCSkel(sl,1); [ [ [ 1, 2 ], [ 1, 3 ] ], [ [ 1, 2 ], [ 1, 4 ] ], [ [ 1, 2 ], [ 1, 5 ] ], [ [ 1, 3 ], [ 1, 4 ] ], [ [ 1, 3 ], [ 1, 5 ] ], [ [ 1, 4 ], [ 1, 5 ] ] ] </pre></div> <div class="example"><pre> gap> c:=SCBdSimplex(4);; gap> sl:=SCNSSlicing(c,[[1],[2..5]]);; gap> SCSkel(sl,3); [ ] gap> sl:=SCNSSlicing(c,[[1,2],[3..5]]);; gap> SCSkelEx(sl,3); [ [ 1, 2, 4, 5 ], [ 1, 3, 4, 6 ], [ 2, 3, 5, 6 ] ] </pre></div> <p><a id="X84ACF7D580FE8B76" name="X84ACF7D580FE8B76"></a></p> <h5>7.3-14 SCSkelEx</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a face list (of <var class="Arg">k+1</var>tuples) or a list of face lists upon success, <c <p>Computes all faces of cardinality <var class="Arg">k+1</var> in the standard labeling: <var clas <p>If <var class="Arg">k</var> is a list (necessarily a sublist of <code class="code">[ 0,1,2,3 ]</cod <div class="example"><pre> gap> c:=SCBdSimplex(4);; gap> sl:=SCNSSlicing(c,[[1],[2..5]]);; gap> SCSkelEx(sl,1); [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ] </pre></div> <div class="example"><pre> gap> c:=SCBdSimplex(4);; gap> sl:=SCNSSlicing(c,[[1],[2..5]]);; gap> SCSkelEx(sl,3); [ ] gap> sl:=SCNSSlicing(c,[[1,2],[3..5]]);; gap> SCSkelEx(sl,3); [ [ 1, 2, 4, 5 ], [ 1, 3, 4, 6 ], [ 2, 3, 5, 6 ] ] </pre></div> <p><a id="X84CBD2F780A1F63C" name="X84CBD2F780A1F63C"></a></p> <h5>7.3-15 SCTopologicalType</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a string upon success, <code class="keyw">fail</code> otherwise.</p> <p>Determines the topological type of <var class="Arg">sl</var> via the classification theorem fo <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 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> <p><a id="X81DA367A813F7599" name="X81DA367A813F7599"></a></p> <h5>7.3-16 SCUnion</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: normal surface of type <code class="code">SCNormalSurface</code> upon success, <code <p>Forms the union of two normal surfaces <var class="Arg">complex1</var> and <var class="Arg">comp <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> <div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAP </body> </html> ¤ Dauer der Verarbeitung: 0.25 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28