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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7EDA334983025D3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7F967A717D4E41C <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8373598C7FF5D28 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7987AAE481C31F3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7AAA4669793C57D <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X81DA367A813F759 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D5639CB87A0D3F <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7E7B17317D1B618 </div></div> </div> <h3>6 <span class="Heading">Functions and operations for <code class="code">SCSimplicialCompl <p><a id="X7A93E4B08536E2C8" name="X7A93E4B08536E2C8"></a></p> <h4>6.1 <span class="Heading">Creating an <code class="code">SCSimplicialComplex</code> object <p>This section contains functions to generate or to construct new simplicial complexes. Some o <p><a id="X7B5A874584FF34A7" name="X7B5A874584FF34A7"></a></p> <h5>6.1-1 SCFromFacets</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>Constructs a simplicial complex object from the given facet list. The facet list <var class="A <p>Internally the vertices are mapped to the standard labeling <span class="SimpleMath">\(1..n <div class="example"><pre> gap> c:=SCFromFacets([[1,2,5], [1,4,5], [1,4,6], [2,3,5], [3,4,6], [3,5,6]]); <SimplicialComplex: unnamed complex 12 | dim = 2 | n = 6> gap> c:=SCFromFacets([["a","b","c"], ["a","b",1], ["a","c",1], ["b","c",1]]); <SimplicialComplex: unnamed complex 13 | dim = 2 | n = 4> </pre></div> <p><a id="X7B5470FD7E2320DE" name="X7B5470FD7E2320DE"></a></p> <h5>6.1-2 SC</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>A shorter function to create a simplicial complex from a facet list, just calls <code class="fu <div class="example"><pre> gap> c:=SC(Combinations([1..6],5)); <SimplicialComplex: unnamed complex 14 | dim = 4 | n = 6> </pre></div> <p><a id="X827D29DD79A82CFA" name="X827D29DD79A82CFA"></a></p> <h5>6.1-3 SCFromDifferenceCycles</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>Creates a simplicial complex object from the list of difference cycles provided. If <var class <p>A <span class="SimpleMath">\(n\)</span>-dimensional difference cycle <span class="SimpleMa <p>Note that modulo operations in <strong class="pkg">GAP</strong> are often a little bit cumberso <div class="example"><pre> gap> c:=SCFromDifferenceCycles([[1,1,6],[2,3,3]]);; gap> c.F; [ 8, 24, 16 ] gap> c.Homology; [ [ 0, [ ] ], [ 2, [ ] ], [ 1, [ ] ] ] gap> c.Chi; 0 gap> c.HasBoundary; false gap> SCIsPseudoManifold(c); true gap> SCIsManifold(c); true </pre></div> <p><a id="X804A0B1F85B333C2" name="X804A0B1F85B333C2"></a></p> <h5>6.1-4 SCFromGenerators</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>Constructs a simplicial complex object from the set of <var class="Arg">generators</var> on whi <div class="example"><pre> gap> #group: AGL(1,7) of order 42 gap> G:=Group([(2,6,5,7,3,4),(1,3,5,7,2,4,6)]);; gap> c:=SCFromGenerators(G,[[ 1, 2, 4 ]]); <SimplicialComplex: complex from generators under unknown group | dim = 2 | n \ = 7> gap> SCLib.DetermineTopologicalType(c); <SimplicialComplex: complex from generators under unknown group | dim = 2 | n \ = 7> </pre></div> <p><a id="X867E1FF580230E20" name="X867E1FF580230E20"></a></p> <h4>6.2 <span class="Heading">Isomorphism signatures</span></h4> <p>This section contains functions to construct simplicial complexes from isomorphism signat <p>The isomorphism signature of a closed and strongly connected weak pseudomanifold is a repres <p>To compute the isomorphism signature of a closed and strongly connected weak pseudomanifold < <p>A canonical labeling of <span class="SimpleMath">\(P\)</span> is determined by chosing a facet < <p>Note that there are precisely <span class="SimpleMath">\((d+1)! m\)</span> canonical labelin <p>Note: Another way of storing and loading complexes is provided by simpcomp's library functio <p><a id="X7E95E36680C188C4" name="X7E95E36680C188C4"></a></p> <h5>6.2-1 SCExportToString</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: string upon success, <code class="keyw">fail</code> otherwise.</p> <p>Computes one string representation of a closed and strongly connected weak pseudomanifold. <div class="example"><pre> gap> c:=SCSeriesBdHandleBody(3,9);; gap> s:=SCExportToString(c); time; "deffg.h.f.fahaiciai.i.hai.fbgeiagihbhceceba.g.gag" 3 gap> s:=SCExportIsoSig(c); time; "deefgaf.hbi.gbh.eaiaeaicg.g.ibf.heg.iff.hggcfffgg" 11 </pre></div> <p><a id="X80098C5F7B80A621" name="X80098C5F7B80A621"></a></p> <h5>6.2-2 SCExportIsoSig</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: string upon success, <code class="keyw">fail</code> otherwise.</p> <p>Computes the isomorphism signature of a closed, strongly connected weak pseudomanifold. Th <div class="example"><pre> gap> c:=SCSeriesBdHandleBody(3,9);; gap> s:=SCExportIsoSig(c); "deefgaf.hbi.gbh.eaiaeaicg.g.ibf.heg.iff.hggcfffgg" </pre></div> <p><a id="X7E915DA7821DD513" name="X7E915DA7821DD513"></a></p> <h5>6.2-3 SCFromIsoSig</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a SCSimplicialComplex object upon success, <code class="keyw">fail</code> otherwis <p>Computes a simplicial complex from its isomorphism signature. If a file with isomorphism sig <div class="example"><pre> gap> s:="deeee";; gap> c:=SCFromIsoSig(s);; gap> SCIsIsomorphic(c,SCBdSimplex(4)); true </pre></div> <div class="example"><pre> gap> s:="deeee";; gap> PrintTo("tmp.txt",s,"\n");; gap> cc:=SCFromIsoSig("tmp.txt"); [ <SimplicialComplex: unnamed complex 9 | dim = 3 | n = 5> ] gap> cc[1].F; [ 5, 10, 10, 5 ] </pre></div> <p><a id="X79072405786FEA0B" name="X79072405786FEA0B"></a></p> <h4>6.3 <span class="Heading">Generating some standard triangulations</span></h4> <p><a id="X7E04DD807AF33B78" name="X7E04DD807AF33B78"></a></p> <h5>6.3-1 SCBdCyclicPolytope</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>Generates the boundary complex of the <var class="Arg">d</var>-dimensional cyclic polytope (a <div class="example"><pre> gap> SCBdCyclicPolytope(3,8); <SimplicialComplex: Bd(C_3(8)) | dim = 2 | n = 8> </pre></div> <p><a id="X839F3BD37DBA3F3C" name="X839F3BD37DBA3F3C"></a></p> <h5>6.3-2 SCBdSimplex</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>Generates the boundary of the <span class="SimpleMath">\(d\)</span>-simplex <span class="Sim <div class="example"><pre> gap> SCBdSimplex(5); <SimplicialComplex: S^4_6 | dim = 4 | n = 6> </pre></div> <p><a id="X856E48967BBFCF0E" name="X856E48967BBFCF0E"></a></p> <h5>6.3-3 SCEmpty</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>Generates an empty complex (of dimension <span class="SimpleMath">\(-1\)</span>), i. e. a <code <div class="example"><pre> gap> SCEmpty(); <SimplicialComplex: empty complex | dim = -1 | n = 0> </pre></div> <p><a id="X7A23532F7A8A3988" name="X7A23532F7A8A3988"></a></p> <h5>6.3-4 SCSimplex</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>Generates the <var class="Arg">d</var>-simplex.</p> <div class="example"><pre> gap> SCSimplex(3); <SimplicialComplex: B^3_4 | dim = 3 | n = 4> </pre></div> <p><a id="X8664A90879248282" name="X8664A90879248282"></a></p> <h5>6.3-5 SCSeriesTorus</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>Generates the <span class="SimpleMath">\(d\)</span>-torus described in <a href="chapBib_mj. <div class="example"><pre> gap> t4:=SCSeriesTorus(4); <SimplicialComplex: 4-torus T^4 | dim = 4 | n = 31> gap> t4.Homology; [ [ 0, [ ] ], [ 4, [ ] ], [ 6, [ ] ], [ 4, [ ] ], [ 1, [ ] ] ] </pre></div> <p><a id="X87C67A0087F645C1" name="X87C67A0087F645C1"></a></p> <h5>6.3-6 SCSurface</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>Generates the surface of genus <var class="Arg">g</var> where the boolean argument <var class="A <div class="example"><pre> gap> c:=SCSurface(23,true); <SimplicialComplex: S_23^or | dim = 2 | n = 88> gap> c.Homology; [ [ 0, [ ] ], [ 46, [ ] ], [ 1, [ ] ] ] gap> c.TopologicalType; "(T^2)^#23" gap> c:=SCSurface(23,false); <SimplicialComplex: S_23^non | dim = 2 | n = 21> gap> c.Homology; [ [ 0, [ ] ], [ 22, [ 2 ] ], [ 0, [ ] ] ] gap> c.TopologicalType; "(RP^2)^#23" </pre></div> <div class="example"><pre> gap> dc:=SCSurface(345,true); [ [ 1, 1, 1374 ], [ 2, 343, 1031 ], [ 343, 345, 688 ] ] gap> c:=SCFromDifferenceCycles(dc); <SimplicialComplex: complex from diffcycles [ [ 1, 1, 1374 ], [ 2, 343, 1031 ]\ , [ 343, 345, 688 ] ] | dim = 2 | n = 1376> gap> c.Chi; -688 gap> dc:=SCSurface(12345678910,true); time; [ [ 1, 1, 24691357816 ], [ 2, 4, 24691357812 ], [ 3, 3, 24691357812 ], [ 4, 12345678907, 12345678907 ] ] 0 </pre></div> <p><a id="X831315CD80BA3654" name="X831315CD80BA3654"></a></p> <h5>6.3-7 SCFVectorBdCrossPolytope</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a list of integers of size <code class="code">d + 1</code> upon success, <code class="key <p>Computes the <span class="SimpleMath">\(f\)</span>-vector of the <span class="SimpleMath">\( <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">SCFVectorBdCrossPolytope(50);</s [ 100, 4900, 156800, 3684800, 67800320, 1017004800, 12785203200, 137440934400, 1282782054400, 10518812846080, 76500457062400, 497252970905600, 2907017368371200, 15365663232819200, 73755183517532160, 322678927889203200, 1290715711556812800, 4732624275708313600, 15941471244491161600, 49418560857922600960, 141195888165493145600, 372243705163572838400, 906332499528699084800, 2039248123939572940800, 4241636097794311716864, 8156992495758291763200, 14501319992459185356800, 23823597130468661657600, 36146147370366245273600, 50604606318512743383040, 65296266217435797913600, 77539316133205010022400, 84588344872587283660800, 84588344872587283660800, 77337915312079802204160, 64448262760066501836800, 48771658304915190579200, 33370081998099867238400, 20535435075753764454400, 11294489291664570449920, 5509506971543692902400, 2361217273518725529600, 878592473867432755200, 279552150776001331200, 74547240206933688320, 16205921784116019200, 2758454771764428800, 344806846470553600, 28147497671065600, 1125899906842624 ] </pre></div> <p><a id="X7C010361858F0214" name="X7C010361858F0214"></a></p> <h5>6.3-8 SCFVectorBdCyclicPolytope</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a list of integers of size <code class="code">d+1</code> upon success, <code class="key <p>Computes the <span class="SimpleMath">\(f\)</span>-vector of the <var class="Arg">d</var>-dime <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">SCFVectorBdCyclicPolytope(25,19 [ 198, 19503, 1274196, 62117055, 2410141734, 77526225777, 2126433621312, 50768602708824, 1071781612741840, 20256672480820776, 346204947854027808, 5395027104058600008, 48354596155522298656, 262068846498922699590, 940938105142239825104, 2379003007642628680027, 4396097923113038784642, 6062663500381642763609, 6294919173643129209180, 4911378208855785427761, 2840750019404460890298, 1183225500922302444568, 335951678686835900832, 58265626173398052500, 4661250093871844200 ] </pre></div> <p><a id="X7C3E0F7687AC966E" name="X7C3E0F7687AC966E"></a></p> <h5>6.3-9 SCFVectorBdSimplex</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a list of integers of size <code class="code">d + 1</code> upon success, <code class="key <p>Computes the <span class="SimpleMath">\(f\)</span>-vector of the <span class="SimpleMath">\( <div class="example"><pre> gap> SCFVectorBdSimplex(100); [ 101, 5050, 166650, 4082925, 79208745, 1267339920, 17199613200, 202095455100, 2088319702700, 19212541264840, 158940114100040, 1192050855750300, 8160963550905900, 51297485177122800, 297525414027312240, 1599199100396803290, 7995995501984016450, 37314645675925410100, 163006083742200475700, 668324943343021950370, 2577824781465941808570, 9373908296239788394800, 32197337191432316660400, 104641345872155029146300, 322295345286237489770604, 942094086221309585483304, 2616928017281415515231400, 6916166902815169575968700, 17409661513983013070541900, 41783187633559231369300560, 95696978128474368620010960, 209337139656037681356273975, 437704928371715151926754675, 875409856743430303853509350, 1675784582908852295948146470, 3072271735332895875904935195, 5397234129638871133346507775, 9090078534128625066688855200, 14683973016669317415420458400, 22760158175837441993901710520, 33862674359172779551902544920, 48375249084532542217003635600, 66375341767149302111702662800, 87494768693060443692698964600, 110826707011209895344085355160, 134919469404951176940625649760, 157884485473879036845412994400, 177620046158113916451089618700, 192119641762857909630770403900, 199804427433372226016001220056, 199804427433372226016001220056, 192119641762857909630770403900, 177620046158113916451089618700, 157884485473879036845412994400, 134919469404951176940625649760, 110826707011209895344085355160, 87494768693060443692698964600, 66375341767149302111702662800, 48375249084532542217003635600, 33862674359172779551902544920, 22760158175837441993901710520, 14683973016669317415420458400, 9090078534128625066688855200, 5397234129638871133346507775, 3072271735332895875904935195, 1675784582908852295948146470, 875409856743430303853509350, 437704928371715151926754675, 209337139656037681356273975, 95696978128474368620010960, 41783187633559231369300560, 17409661513983013070541900, 6916166902815169575968700, 2616928017281415515231400, 942094086221309585483304, 322295345286237489770604, 104641345872155029146300, 32197337191432316660400, 9373908296239788394800, 2577824781465941808570, 668324943343021950370, 163006083742200475700, 37314645675925410100, 7995995501984016450, 1599199100396803290, 297525414027312240, 51297485177122800, 8160963550905900, 1192050855750300, 158940114100040, 19212541264840, 2088319702700, 202095455100, 17199613200, 1267339920, 79208745, 4082925, 166650, 5050, 101 ] </pre></div> <p><a id="X814FE0267D7C54A9" name="X814FE0267D7C54A9"></a></p> <h4>6.4 <span class="Heading">Generating infinite series of transitive triangulations</span></ <p><a id="X7EA6421A8156EBDF" name="X7EA6421A8156EBDF"></a></p> <h5>6.4-1 SCSeriesAGL</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a permutation group and a list of <span class="SimpleMath">\(5\)</span>-tuples of int <p>For a given prime <var class="Arg">p</var> the automorphism group (AGL<span class="SimpleMath"> <div class="example"><pre> gap> gens:=SCSeriesAGL(17); [ AGL(1,17), [ [ 1, 2, 4, 8, 16 ] ] ] gap> c:=SCFromGenerators(gens[1],gens[2]);; gap> SCIsManifold(SCLink(c,1)); true </pre></div> <div class="example"><pre> gap> List([19..23],x->SCSeriesAGL(x)); #I SCSeriesAGL: argument must be a prime > 13. #I SCSeriesAGL: argument must be a prime > 13. #I SCSeriesAGL: argument must be a prime > 13. [ [ AGL(1,19), [ [ 1, 2, 10, 12, 17 ] ] ], fail, fail, fail, [ AGL(1,23), [ [ 1, 2, 7, 9, 19 ], [ 1, 2, 4, 8, 22 ] ] ] ] gap> for i in [80000..80100] do if IsPrime(i) then Print(i,"\n"); fi; od; 80021 80039 80051 80071 80077 gap> SCSeriesAGL(80021); AGL(1,80021) gap> SCSeriesAGL(80039); [ AGL(1,80039), [ [ 1, 2, 6496, 73546, 78018 ] ] ] gap> SCSeriesAGL(80051); [ AGL(1,80051), [ [ 1, 2, 31498, 37522, 48556 ] ] ] gap> SCSeriesAGL(80071); AGL(1,80071) gap> SCSeriesAGL(80077); [ AGL(1,80077), [ [ 1, 2, 4126, 39302, 40778 ] ] ] </pre></div> <p><a id="X85E6FD6D84FF762B" name="X85E6FD6D84FF762B"></a></p> <h5>6.4-2 SCSeriesBrehmKuehnelTorus</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>Generates a neighborly 3-torus with <var class="Arg">n</var> vertices if <var class="Arg">n</var> <div class="example"><pre> gap> T3:=SCSeriesBrehmKuehnelTorus(15); <SimplicialComplex: Neighborly 3-Torus NT3(15) | dim = 3 | n = 15> gap> T3.Homology; [ [ 0, [ ] ], [ 3, [ ] ], [ 3, [ ] ], [ 1, [ ] ] ] gap> T3.Neighborliness; 2 gap> T3:=SCSeriesBrehmKuehnelTorus(16); <SimplicialComplex: Centrally symmetric 3-Torus SCT3(16) | dim = 3 | n = 16> gap> T3.Homology; [ [ 0, [ ] ], [ 3, [ ] ], [ 3, [ ] ], [ 1, [ ] ] ] gap> T3.IsCentrallySymmetric; true </pre></div> <p><a id="X786AD599875BD006" name="X786AD599875BD006"></a></p> <h5>6.4-3 SCSeriesBdHandleBody</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><code class="code">SCSeriesBdHandleBody(d,n)</code> generates a transitive <span class="Si <div class="example"><pre> gap> c:=SCSeriesBdHandleBody(2,7); <SimplicialComplex: Sphere bundle S^1 x S^1 | dim = 2 | n = 7> gap> SCLib.DetermineTopologicalType(c); <SimplicialComplex: Sphere bundle S^1 x S^1 | dim = 2 | n = 7> gap> SCIsIsomorphic(c,SCSeriesHandleBody(3,7).Boundary); true </pre></div> <p><a id="X8787A3A4788E950C" name="X8787A3A4788E950C"></a></p> <h5>6.4-4 SCSeriesBid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a simplicial complex upon success, <code class="keyw">fail</code> otherwise.</p> <p>Constructs the complex <span class="SimpleMath">\(B(i,d)\)</span> as described in <a href="ch <div class="example"><pre> gap> b26:=SCSeriesBid(2,6); <SimplicialComplex: B(2,6) | dim = 5 | n = 12> gap> s2s2:=SCBoundary(b26); <SimplicialComplex: Bd(B(2,6)) | dim = 4 | n = 12> gap> SCFVector(s2s2); [ 12, 60, 160, 180, 72 ] gap> SCAutomorphismGroup(s2s2); Group([ (1,3)(4,6)(7,9)(10,12), (1,5)(2,10)(4,8)(6,12)(7,11), (1,10,7,4) (2,3,8,9)(5,12,11,6) ]) gap> SCIsManifold(s2s2); true gap> SCHomology(s2s2); [ [ 0, [ ] ], [ 0, [ ] ], [ 2, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ] </pre></div> <p><a id="X7C0223DF83CC961B" name="X7C0223DF83CC961B"></a></p> <h5>6.4-5 SCSeriesC2n</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>Generates the combinatorial <span class="SimpleMath">\(3\)</span>-manifold <span class="Si <div class="example"><pre> gap> c:=SCSeriesC2n(8); <SimplicialComplex: C_16 = { (1:1:3:11),(1:1:11:3),(1:3:1:11),(2:3:2:9),(2:5:2\ :7) } | dim = 3 | n = 16> gap> SCGenerators(c); [ [ [ 1, 2, 3, 6 ], 32 ], [ [ 1, 2, 5, 6 ], 16 ], [ [ 1, 3, 6, 8 ], 16 ], [ [ 1, 3, 8, 10 ], 16 ] ] </pre></div> <div class="example"><pre> gap> c:=SCSeriesC2n(8);; gap> d:=SCSeriesD2n(8); <SimplicialComplex: D_16 = { (1:1:1:13),(1:2:11:2),(3:4:5:4),(2:3:4:7),(2:7:4:\ 3) } | dim = 3 | n = 16> gap> SCIsIsomorphic(c,d); true gap> c:=SCSeriesC2n(11);; gap> d:=SCSeriesD2n(11);; gap> c.Homology; [ [ 0, [ ] ], [ 1, [ ] ], [ 1, [ ] ], [ 1, [ ] ] ] gap> d.Homology; [ [ 0, [ ] ], [ 1, [ ] ], [ 0, [ 2 ] ], [ 0, [ ] ] ] </pre></div> <p><a id="X7E2927DA7F60D957" name="X7E2927DA7F60D957"></a></p> <h5>6.4-6 SCSeriesConnectedSum</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>Generates a combinatorial manifold of type <span class="SimpleMath">\((S^2 x S^1)^{\#k}\)</ <div class="example"><pre> gap> c:=SCSeriesConnectedSum(12); <SimplicialComplex: (S^2xS^1)^#12) | dim = 3 | n = 52> gap> c.Homology; [ [ 0, [ ] ], [ 12, [ ] ], [ 12, [ ] ], [ 1, [ ] ] ] gap> g:=SimplifiedFpGroup(SCFundamentalGroup(c)); <fp group of size infinity on the generators [ [2,3], [2,14], [3,4], [6,7], [9,10], [10,11], [11,12], [12,13], [26,32], [26,34], [29,31], [33,35] ]> gap> RelatorsOfFpGroup(g); [ ] </pre></div> <p><a id="X7D1CEF9F86D3AE66" name="X7D1CEF9F86D3AE66"></a></p> <h5>6.4-7 SCSeriesCSTSurface</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><code class="code">SCSeriesCSTSurface(l,j,2k)</code> generates the centrally symmetric tr <div class="example"><pre> gap> SCSeriesCSTSurface(2,4,14); <SimplicialComplex: cst surface S_{(2,4,14)} = { (2:4:8),(2:8:4) } | dim = 2 |\ n = 14> gap> last.Homology; [ [ 1, [ ] ], [ 4, [ ] ], [ 2, [ ] ] ] gap> SCSeriesCSTSurface(2,10); <SimplicialComplex: cst surface S_{(2,10)} = { (2:2:6),(3:3:4) } | dim = 2 | n\ = 10> gap> last.Homology; [ [ 0, [ ] ], [ 1, [ 2 ] ], [ 0, [ ] ] ] </pre></div> <p><a id="X7C56D2B7858A80C7" name="X7C56D2B7858A80C7"></a></p> <h5>6.4-8 SCSeriesD2n</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>Generates the combinatorial <span class="SimpleMath">\(3\)</span>-manifold <span class="Si <div class="example"><pre> gap> d:=SCSeriesD2n(15); <SimplicialComplex: D_30 = { (1:1:1:27),(1:2:25:2),(3:11:5:11),(2:3:11:14),(2:\ 14:11:3) } | dim = 3 | n = 30> gap> SCAutomorphismGroup(d); Group([ (1,3)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21) (14,20)(15,19)(16,18), (1,4)(2,3)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25) (11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18) ]) gap> StructureDescription(last); "D60" </pre></div> <div class="example"><pre> gap> c:=SCSeriesC2n(8);; gap> d:=SCSeriesD2n(8); <SimplicialComplex: D_16 = { (1:1:1:13),(1:2:11:2),(3:4:5:4),(2:3:4:7),(2:7:4:\ 3) } | dim = 3 | n = 16> gap> SCIsIsomorphic(c,d); true gap> c:=SCSeriesC2n(11);; gap> d:=SCSeriesD2n(11);; gap> c.Homology; [ [ 0, [ ] ], [ 1, [ ] ], [ 1, [ ] ], [ 1, [ ] ] ] gap> d.Homology; [ [ 0, [ ] ], [ 1, [ ] ], [ 0, [ 2 ] ], [ 0, [ ] ] ] </pre></div> <p><a id="X7CCBF8F487036415" name="X7CCBF8F487036415"></a></p> <h5>6.4-9 SCSeriesHandleBody</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><code class="code">SCSeriesHandleBody(d,n)</code> generates a transitive <span class="Simp <div class="example"><pre> gap> c:=SCSeriesHandleBody(3,7); <SimplicialComplex: Handle body B^2 x S^1 | dim = 3 | n = 7> gap> SCAutomorphismGroup(c); Group([ (1,3)(4,7)(5,6), (1,4)(2,3)(5,7) ]) gap> bd:=SCBoundary(c);; gap> SCAutomorphismGroup(bd); Group([ (1,2)(3,7)(4,6), (1,4,2)(3,5,6) ]) gap> SCIsIsomorphic(bd,SCSeriesBdHandleBody(2,7)); true </pre></div> <p><a id="X8519C1B678C101BF" name="X8519C1B678C101BF"></a></p> <h5>6.4-10 SCSeriesHomologySphere</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>Generates a combinatorial Brieskorn homology sphere of type <span class="SimpleMath">\(\Si <div class="example"><pre> gap> c:=SCSeriesHomologySphere(2,3,5); <SimplicialComplex: Homology sphere Sigma(2,3,5) | dim = 3 | n = 17> gap> c.Homology; [ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ] gap> c:=SCSeriesHomologySphere(3,4,13); <SimplicialComplex: Homology sphere Sigma(3,4,13) | dim = 3 | n = 37> gap> c.Homology; [ [ 0, [ ] ], [ 0, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ] </pre></div> <p><a id="X78DA125479E1D77F" name="X78DA125479E1D77F"></a></p> <h5>6.4-11 SCSeriesK</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>Generates the <var class="Arg">k</var>-th member (<span class="SimpleMath">\(k \geq 0\)</span>) <div class="example"><pre> gap> cc:=List([1..10],x->SCSeriesK(x,0));; gap> Set(List(cc,x->x.F)); [ [ 9, 36, 54, 27 ], [ 11, 55, 88, 44 ], [ 13, 65, 104, 52 ], [ 13, 78, 130, 65 ], [ 15, 90, 150, 75 ], [ 15, 105, 180, 90 ] ] gap> cc:=List([1..10],x->SCSeriesK(x,10));; gap> gap> cc:=List([1..10],x->SCSeriesK(x,10));; gap> Set(List(cc,x->x.Homology)); [ [ [ 0, [ ] ], [ 1, [ ] ], [ 0, [ 2 ] ], [ 0, [ ] ] ] ] gap> Set(List(cc,x->x.IsManifold)); [ true ] </pre></div> <p><a id="X7B8300428516DAD8" name="X7B8300428516DAD8"></a></p> <h5>6.4-12 SCSeriesKu</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 the symmetric orientable sphere bundle Ku<span class="SimpleMath">\((n)\)</span> w <div class="example"><pre> gap> c:=SCSeriesKu(4); <SimplicialComplex: Sl_16 = G{ [1,2,5,9],[1,2,9,10],[1,5,9,16] } | dim = 3 | n\ = 16> gap> SCSlicing(c,[[1,2,3,4,9,10,11,12],[5,6,7,8,13,14,15,16]]); <NormalSurface: slicing [ [ 1, 2, 3, 4, 9, 10, 11, 12 ], [ 5, 6, 7, 8, 13, 14,\ 15, 16 ] ] of Sl_16 = G{ [1,2,5,9],[1,2,9,10],[1,5,9,16] } | dim = 2> gap> Mminus:=SCSpan(c,[1,2,3,4,9,10,11,12]);; gap> Mplus:=SCSpan(c,[5,6,7,8,13,14,15,16]);; gap> SCCollapseGreedy(Mminus).Facets; [ [ 3, 4 ], [ 3, 10 ], [ 4, 12 ], [ 9, 10 ], [ 9, 12 ] ] gap> SCCollapseGreedy(Mplus).Facets; [ [ 5, 6 ], [ 5, 8 ], [ 6, 14 ], [ 7, 8 ], [ 7, 15 ], [ 14, 15 ] ] </pre></div> <p><a id="X813C5B0E7FA7C1A3" name="X813C5B0E7FA7C1A3"></a></p> <h5>6.4-13 SCSeriesL</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>Generates the <var class="Arg">k</var>-th member (<span class="SimpleMath">\(k \geq 0\)</span>) <div class="example"><pre> gap> cc:=List([1..18],x->SCSeriesL(x,0));; gap> Set(List(cc,x->x.F)); [ [ 10, 45, 70, 35 ], [ 12, 60, 96, 48 ], [ 12, 66, 108, 54 ], [ 14, 77, 126, 63 ], [ 14, 84, 140, 70 ], [ 14, 91, 154, 77 ] ] gap> cc:=List([1..18],x->SCSeriesL(x,10));; gap> Set(List(cc,x->x.IsManifold)); [ true ] </pre></div> <p><a id="X7EAC6828812A241A" name="X7EAC6828812A241A"></a></p> <h5>6.4-14 SCSeriesLe</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>Generates the <var class="Arg">k</var>-th member (<span class="SimpleMath">\(k \geq 7\)</span>) <div class="example"><pre> gap> c:=SCSeriesLe(7); <SimplicialComplex: Le_14 = { (1:1:1:11),(1:2:4:7),(1:4:2:7),(2:1:4:7),(2:5:2:\ 5),(2:4:2:6) } | dim = 3 | n = 14> gap> d:=SCLib.DetermineTopologicalType(c);; gap> SCReference(d); "manifold_3_14_1_5 in F.H.Lutz: 'The Manifold Page', http://www.math.tu-berlin\ .de/diskregeom/stellar/,\r\nF.H.Lutz: 'Triangulated manifolds with few vertice\ s and vertex-transitive group actions', Doctoral Thesis TU Berlin 1999, Shaker\ -Verlag, Aachen 1999" </pre></div> <p><a id="X8280ED8280FF9218" name="X8280ED8280FF9218"></a></p> <h5>6.4-15 SCSeriesLensSpace</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>Generates the lens space <span class="SimpleMath">\(L(p,q)\)</span> whenever <span class="Si <div class="example"><pre> gap> l154:=SCSeriesLensSpace(15,4); <SimplicialComplex: Lens space L(15,4) | dim = 3 | n = 22> gap> l154.Homology; [ [ 0, [ ] ], [ 0, [ 15 ] ], [ 0, [ ] ], [ 1, [ ] ] ] gap> g:=SimplifiedFpGroup(SCFundamentalGroup(l154)); <fp group on the generators [ [2,5] ]> gap> StructureDescription(g); "C15" </pre></div> <div class="example"><pre> gap> l151:=SCSeriesLensSpace(15,1); <SimplicialComplex: Lens space L(15,1) | dim = 3 | n = 62> gap> l151.Homology; [ [ 0, [ ] ], [ 0, [ 15 ] ], [ 0, [ ] ], [ 1, [ ] ] ] gap> g:=SimplifiedFpGroup(SCFundamentalGroup(l151)); <fp group on the generators [ [2,3] ]> gap> StructureDescription(g); "C15" </pre></div> <p><a id="X7DDC1B127F21CFA4" name="X7DDC1B127F21CFA4"></a></p> <h5>6.4-16 SCSeriesPrimeTorus</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>Generates the well known triangulated torus <span class="SimpleMath">\(\{ (l:j:p-l-j),(l: <div class="example"><pre> gap> l:=List([2..19],x->SCSeriesPrimeTorus(1,x,41));; gap> Set(List(l,x->SCHomology(x))); [ [ [ 0, [ ] ], [ 2, [ ] ], [ 1, [ ] ] ] ] </pre></div> <p><a id="X7CC3944D7E2F6458" name="X7CC3944D7E2F6458"></a></p> <h5>6.4-17 SCSeriesSeifertFibredSpace</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>Generates a combinatorial Seifert fibred space of type</p> <p class="center">\[SFS [ (\mathbb{T}^2)^{(a-1)(b-1)} : (p/a,b_1)^b , (q/b,b_2)^a, (r/ab,b <p>where <span class="SimpleMath">\(p\)</span> and <span class="SimpleMath">\(q\)</span> are co-p <p class="center">\[\frac{b_1}{p} + \frac{b_2}{q} + \frac{b_3}{r} = \frac{\pm ab}{pqr}.\]</ <p>This <span class="SimpleMath">\(3\)</span>-parameter family of combinatorial <span class="S <div class="example"><pre> gap> c:=SCSeriesSeifertFibredSpace(2,3,15); <SimplicialComplex: SFS [ S^2 : (2,b1)^3, (5,b3) ] | dim = 3 | n = 27> gap> c.Homology; [ [ 0, [ ] ], [ 0, [ 2, 2 ] ], [ 0, [ ] ], [ 1, [ ] ] ] </pre></div> <p><a id="X7B71BC8B7D74AFD5" name="X7B71BC8B7D74AFD5"></a></p> <h5>6.4-18 SCSeriesS2xS2</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>Generates a combinatorial version of <span class="SimpleMath">\((S^2 \times S^2)^{\# k}\)</ <div class="example"><pre> gap> c:=SCSeriesS2xS2(3); <SimplicialComplex: (S^2 x S^2)^(# 3) | dim = 4 | n = 24> gap> c.Homology; [ [ 0, [ ] ], [ 0, [ ] ], [ 6, [ ] ], [ 0, [ ] ], [ 1, [ ] ] ] </pre></div> <p><a id="X7899878881EA47F8" name="X7899878881EA47F8"></a></p> <h4>6.5 <span class="Heading">A census of regular and chiral maps</span></h4> <p><a id="X867D2AFC79B11405" name="X867D2AFC79B11405"></a></p> <h5>6.5-1 SCChiralMap</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a <code class="keyw">SCSimplicialComplex</code> object upon success, <code class="k <p>Returns the (hyperbolic) chiral map of vertex valence <var class="Arg">m</var> and genus <var cla <div class="example"><pre> gap> SCChiralMaps(); [ [ 7, 17 ], [ 8, 10 ], [ 8, 28 ], [ 8, 37 ], [ 8, 46 ], [ 8, 82 ], [ 9, 43 ], [ 10, 73 ], [ 12, 22 ], [ 12, 33 ], [ 12, 40 ], [ 12, 51 ], [ 12, 58 ], [ 12, 64 ], [ 12, 85 ], [ 12, 94 ], [ 12, 97 ], [ 18, 28 ] ] gap> c:=SCChiralMap(8,10); <SimplicialComplex: Chiral map {8,10} | dim = 2 | n = 54> gap> c.Homology; [ [ 0, [ ] ], [ 20, [ ] ], [ 1, [ ] ] ] </pre></div> <p><a id="X85BB97CB8240E59B" name="X85BB97CB8240E59B"></a></p> <h5>6.5-2 SCChiralMaps</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a list of lists upon success, <code class="keyw">fail</code> otherwise.</p> <p>Returns a list of all simplicial (hyperbolic) chiral maps of orientable genus up to <span class <div class="example"><pre> gap> ll:=SCChiralMaps(); [ [ 7, 17 ], [ 8, 10 ], [ 8, 28 ], [ 8, 37 ], [ 8, 46 ], [ 8, 82 ], [ 9, 43 ], [ 10, 73 ], [ 12, 22 ], [ 12, 33 ], [ 12, 40 ], [ 12, 51 ], [ 12, 58 ], [ 12, 64 ], [ 12, 85 ], [ 12, 94 ], [ 12, 97 ], [ 18, 28 ] ] gap> c:=SCChiralMap(ll[18][1],ll[18][2]); <SimplicialComplex: Chiral map {18,28} | dim = 2 | n = 27> gap> SCHomology(c); [ [ 0, [ ] ], [ 56, [ ] ], [ 1, [ ] ] ] </pre></div> <p><a id="X7B362D25784E7217" name="X7B362D25784E7217"></a></p> <h5>6.5-3 SCChiralTori</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a <code class="keyw">SCSimplicialComplex</code> object upon success, <code class="k <p>Returns a list of chiral triangulations of the torus with <span class="SimpleMath">\(n\)</spa <div class="example"><pre> gap> cc:=SCChiralTori(91); [ <SimplicialComplex: {3,6}_(9,1) | dim = 2 | n = 91>, <SimplicialComplex: {3,6}_(6,5) | dim = 2 | n = 91> ] gap> SCIsIsomorphic(cc[1],cc[2]); false </pre></div> <p><a id="X7AB15983833FCA6B" name="X7AB15983833FCA6B"></a></p> <h5>6.5-4 SCNrChiralTori</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>Returns the number of simplicial chiral maps on the torus with <span class="SimpleMath">\(n\)< <div class="example"><pre> gap> SCNrChiralTori(7); 1 gap> SCNrChiralTori(343); 2 </pre></div> <p><a id="X87180AD07F799C5A" name="X87180AD07F799C5A"></a></p> <h5>6.5-5 SCNrRegularTorus</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>Returns the number of simplicial regular maps on the torus with <span class="SimpleMath">\(n\ <div class="example"><pre> gap> SCNrRegularTorus(9); 1 gap> SCNrRegularTorus(10); 0 </pre></div> <p><a id="X83D0946E7E2C4163" name="X83D0946E7E2C4163"></a></p> <h5>6.5-6 SCRegularMap</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a <code class="keyw">SCSimplicialComplex</code> object upon success, <code class="k <p>Returns the (hyperbolic) regular map of vertex valence <var class="Arg">m</var>, genus <var clas <div class="example"><pre> gap> SCRegularMaps(){[1..10]}; [ [ 7, 3, true ], [ 7, 7, true ], [ 7, 8, false ], [ 7, 14, true ], [ 7, 15, false ], [ 7, 147, false ], [ 8, 3, true ], [ 8, 5, true ], [ 8, 8, true ], [ 8, 9, false ] ] gap> c:=SCRegularMap(7,7,true); <SimplicialComplex: Orientable regular map {7,7} | dim = 2 | n = 72> gap> g:=SCAutomorphismGroup(c); #I group not listed C2 x PSL(2,8) gap> Size(g); 1008 </pre></div> <p><a id="X7F75F5E183CC097E" name="X7F75F5E183CC097E"></a></p> <h5>6.5-7 SCRegularMaps</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a list of lists upon success, <code class="keyw">fail</code> otherwise.</p> <p>Returns a list of all simplicial (hyperbolic) regular maps of orientable genus up to <span clas <div class="example"><pre> gap> ll:=SCRegularMaps(){[1..10]}; [ [ 7, 3, true ], [ 7, 7, true ], [ 7, 8, false ], [ 7, 14, true ], [ 7, 15, false ], [ 7, 147, false ], [ 8, 3, true ], [ 8, 5, true ], [ 8, 8, true ], [ 8, 9, false ] ] gap> c:=SCRegularMap(ll[5][1],ll[5][2],ll[5][3]); <SimplicialComplex: Non-orientable regular map {7,15} | dim = 2 | n = 78> gap> SCHomology(c); [ [ 0, [ ] ], [ 14, [ 2 ] ], [ 0, [ ] ] ] gap> SCGenerators(c); [ [ [ 1, 4, 7 ], 182 ] ] </pre></div> <p><a id="X79B6F47187668CDF" name="X79B6F47187668CDF"></a></p> <h5>6.5-8 SCRegularTorus</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a <code class="keyw">SCSimplicialComplex</code> object upon success, <code class="k <p>Returns a list of regular triangulations of the torus with <span class="SimpleMath">\(n\)</sp <div class="example"><pre> gap> cc:=SCRegularTorus(9); [ <SimplicialComplex: {3,6}_(3,0) | dim = 2 | n = 9> ] gap> g:=SCAutomorphismGroup(cc[1]); Group([ (2,7)(3,4)(5,9), (1,4,2)(3,7,9)(5,8,6), (2,8,7,3,6,4)(5,9) ]) gap> SCNumFaces(cc[1],0)*12 = Size(g); true </pre></div> <p><a id="X87CE08247BE77E44" name="X87CE08247BE77E44"></a></p> <h5>6.5-9 SCSeriesSymmetricTorus</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: a <code class="keyw">SCSimplicialComplex</code> object upon success, <code class="k <p>Returns the equivarient triangulation of the torus <span class="SimpleMath">\(\{ 3,6 \}_{(p |
