#################################################################################### #################################################################################### #################################################################################### ### ### ### ### ### TRIANGULATIONS OF AKBULUT-KIRBY SPHERES ### ### by Frank H. Lutz and Mimi Tsuruga ### ### http://matsguru.com/research/AKspheres.html ### ### ### ### name of example: ### ### ### ### AK_II_008 ### ### ### ### ### ### description: ### ### ### ### Type 2 construction of triangulation of Akbulut-Kirby sphere ### ### ### ### ### ### properties: ### ### ### ### f=(560,8454,28236,33900,13560) ### ### H_*=(Z,0,0,0,Z) ### ### PL 4-sphere ### ### ### ### ### ### references: ### ### ### ### Frank H. Lutz and Mimi Tsuruga. ### ### Constructing complicated spheres. ### ### arXiv:1302.6856[math.CO], 2013. ### ### ### ### ### #################################################################################### #################################################################################### #################################################################################### facets:=[ [0,1,2,5,281],[0,1,2,35,311],[0,1,2,278,281],[0,1,2,278,311], [0,1,4,5,281],[0,1,4,142,418],[0,1,4,280,281],[0,1,4,280,418], [0,1,34,35,311],[0,1,34,130,268],[0,1,34,130,406],[0,1,34,172,268], [0,1,34,310,311],[0,1,34,310,406],[0,1,129,130,268],[0,1,129,130,406], [0,1,129,267,268],[0,1,129,267,543],[0,1,129,405,406],[0,1,129,405,543], [0,1,139,142,418],[0,1,139,172,268],[0,1,139,267,268],[0,1,139,267,543], [0,1,139,415,418],[0,1,139,415,543],[0,1,277,278,281],[0,1,277,278,311], [0,1,277,280,281],[0,1,277,280,418],[0,1,277,310,311],[0,1,277,310,406], [0,1,277,405,406],[0,1,277,405,543],[0,1,277,415,418],[0,1,277,415,543], [0,2,5,26,164],[2,5,26,164,440],[0,2,5,26,302],[2,5,26,302,440], [0,2,5,143,164],[2,5,143,164,440],[2,5,143,419,440],[0,2,5,281,302], [2,5,281,302,440],[2,5,281,419,440],[0,2,26,29,167],[2,26,29,167,443], [0,2,26,29,305],[2,26,29,305,443],[0,2,26,164,167],[2,26,164,167,443], [2,26,164,440,443],[0,2,26,302,305],[2,26,302,305,443],[2,26,302,440,443], [0,2,29,133,271],[0,2,29,133,409],[0,2,29,167,271],[0,2,29,305,409], [0,2,35,133,271],[0,2,35,133,409],[0,2,35,173,271],[0,2,35,311,409], [0,2,140,143,164],[2,140,143,164,440],[2,140,143,419,440],[0,2,140,164,167], [2,140,164,167,443],[2,140,164,440,443],[0,2,140,167,271],[0,2,140,173,271], [2,140,416,419,440],[2,140,416,440,443],[0,2,278,281,302],[2,278,281,302,440], [2,278,281,419,440],[0,2,278,302,305],[2,278,302,305,443],[2,278,302,440,443], [0,2,278,305,409],[0,2,278,311,409],[2,278,416,419,440],[2,278,416,440,443], [0,3,4,5,281],[0,3,4,142,418],[0,3,4,280,281],[0,3,4,280,418], [0,3,5,26,164],[0,3,5,26,302],[0,3,5,143,164],[0,3,5,281,302], [0,3,24,26,164],[0,3,24,26,302],[0,3,24,162,164],[0,3,24,162,438], [0,3,24,300,302],[0,3,24,300,438],[0,3,141,142,418],[0,3,141,143,164], [0,3,141,162,164],[0,3,141,162,438],[0,3,141,417,418],[0,3,141,417,438], [0,3,279,280,281],[0,3,279,280,418],[0,3,279,281,302],[0,3,279,300,302], [0,3,279,300,438],[0,3,279,417,418],[0,3,279,417,438],[0,24,26,29,167], [0,24,26,29,305],[0,24,26,164,167],[0,24,26,302,305],[0,24,27,29,167], [0,24,27,29,305],[0,24,27,165,167],[0,24,27,165,441],[0,24,27,303,305], [0,24,27,303,441],[0,24,162,164,167],[0,24,162,165,167],[0,24,162,165,441], [0,24,162,438,441],[0,24,300,302,305],[0,24,300,303,305],[0,24,300,303,441], [0,24,300,438,441],[0,27,29,133,271],[0,27,29,133,409],[0,27,29,167,271], [0,27,29,305,409],[0,27,126,130,268],[0,27,126,130,406],[0,27,126,133,271], [0,27,126,133,409],[0,27,126,264,268],[0,27,126,264,271],[0,27,126,402,406], [0,27,126,402,409],[0,27,129,130,268],[0,27,129,130,406],[0,27,129,267,268], [0,27,129,267,543],[0,27,129,405,406],[0,27,129,405,543],[0,27,165,167,271], [0,27,165,264,268],[0,27,165,264,271],[0,27,165,267,268],[0,27,165,267,543], [0,27,165,441,543],[0,27,303,305,409],[0,27,303,402,406],[0,27,303,402,409], [0,27,303,405,406],[0,27,303,405,543],[0,27,303,441,543],[0,33,34,35,311], [0,33,34,130,268],[0,33,34,130,406],[0,33,34,172,268],[0,33,34,310,311], [0,33,34,310,406],[0,33,35,133,271],[0,33,35,133,409],[0,33,35,173,271], [0,33,35,311,409],[0,33,126,130,268],[0,33,126,130,406],[0,33,126,133,271], [0,33,126,133,409],[0,33,126,264,268],[0,33,126,264,271],[0,33,126,402,406], [0,33,126,402,409],[0,33,171,172,268],[0,33,171,173,271],[0,33,171,264,268], [0,33,171,264,271],[0,33,309,310,311],[0,33,309,310,406],[0,33,309,311,409], [0,33,309,402,406],[0,33,309,402,409],[138,139,140,143,419],[138,139,140,173,449], [138,139,140,416,419],[138,139,140,416,449],[138,139,142,143,419],[0,138,139,142,418], [138,139,142,418,419],[138,139,172,173,449],[0,138,139,172,268],[138,139,172,268,544], [138,139,172,448,449],[138,139,172,448,544],[0,138,139,267,268],[138,139,267,268,544], [0,138,139,267,543],[138,139,267,543,544],[138,139,415,416,419],[138,139,415,416,449], [0,138,139,415,418],[138,139,415,418,419],[138,139,415,448,449],[138,139,415,448,544], [0,138,139,415,543],[138,139,415,543,544],[0,138,140,143,164],[138,140,143,164,440], [138,140,143,419,440],[0,138,140,164,167],[138,140,164,167,443],[138,140,164,440,443], [0,138,140,167,271],[138,140,167,271,547],[138,140,167,443,547],[0,138,140,173,271], [138,140,173,271,547],[138,140,173,449,547],[138,140,416,419,440],[138,140,416,440,443], [138,140,416,443,547],[138,140,416,449,547],[138,141,142,143,419],[0,138,141,142,418], [138,141,142,418,419],[0,138,141,143,164],[138,141,143,164,440],[138,141,143,419,440], [0,138,141,162,164],[138,141,162,164,440],[0,138,141,162,438],[138,141,162,438,440], [0,138,141,417,418],[138,141,417,418,419],[138,141,417,419,440],[0,138,141,417,438], [138,141,417,438,440],[0,138,162,164,167],[138,162,164,167,443],[138,162,164,440,443], [0,138,162,165,167],[138,162,165,167,443],[0,138,162,165,441],[138,162,165,441,443], [138,162,438,440,443],[0,138,162,438,441],[138,162,438,441,443],[0,138,165,167,271], [138,165,167,271,547],[138,165,167,443,547],[0,138,165,264,268],[138,165,264,268,544], [0,138,165,264,271],[138,165,264,271,547],[138,165,264,540,544],[138,165,264,540,547], [0,138,165,267,268],[138,165,267,268,544],[0,138,165,267,543],[138,165,267,543,544], [138,165,441,443,547],[138,165,441,540,544],[138,165,441,540,547],[0,138,165,441,543], [138,165,441,543,544],[138,171,172,173,449],[0,138,171,172,268],[138,171,172,268,544], [138,171,172,448,449],[138,171,172,448,544],[0,138,171,173,271],[138,171,173,271,547], [138,171,173,449,547],[0,138,171,264,268],[138,171,264,268,544],[0,138,171,264,271], [138,171,264,271,547],[138,171,264,540,544],[138,171,264,540,547],[138,171,447,448,449], [138,171,447,448,544],[138,171,447,449,547],[138,171,447,540,544],[138,171,447,540,547], [138,414,415,416,419],[138,414,415,416,449],[0,138,414,415,418],[138,414,415,418,419], [138,414,415,448,449],[138,414,415,448,544],[0,138,414,415,543],[138,414,415,543,544], [138,414,416,419,440],[138,414,416,440,443],[138,414,416,443,547],[138,414,416,449,547], [0,138,414,417,418],[138,414,417,418,419],[138,414,417,419,440],[0,138,414,417,438], [138,414,417,438,440],[138,414,438,440,443],[0,138,414,438,441],[138,414,438,441,443], [138,414,441,443,547],[138,414,441,540,544],[138,414,441,540,547],[0,138,414,441,543], [138,414,441,543,544],[138,414,447,448,449],[138,414,447,448,544],[138,414,447,449,547], [138,414,447,540,544],[138,414,447,540,547],[0,276,277,278,281],[276,277,278,281,419], [0,276,277,278,311],[276,277,278,311,449],[276,277,278,416,419],[276,277,278,416,449], [0,276,277,280,281],[276,277,280,281,419],[0,276,277,280,418],[276,277,280,418,419], [0,276,277,310,311],[276,277,310,311,449],[0,276,277,310,406],[276,277,310,406,544], [276,277,310,448,449],[276,277,310,448,544],[0,276,277,405,406],[276,277,405,406,544], [0,276,277,405,543],[276,277,405,543,544],[276,277,415,416,419],[276,277,415,416,449], [0,276,277,415,418],[276,277,415,418,419],[276,277,415,448,449],[276,277,415,448,544], [0,276,277,415,543],[276,277,415,543,544],[0,276,278,281,302],[276,278,281,302,440], [276,278,281,419,440],[0,276,278,302,305],[276,278,302,305,443],[276,278,302,440,443], [0,276,278,305,409],[276,278,305,409,547],[276,278,305,443,547],[0,276,278,311,409], [276,278,311,409,547],[276,278,311,449,547],[276,278,416,419,440],[276,278,416,440,443], [276,278,416,443,547],[276,278,416,449,547],[0,276,279,280,281],[276,279,280,281,419], [0,276,279,280,418],[276,279,280,418,419],[0,276,279,281,302],[276,279,281,302,440], [276,279,281,419,440],[0,276,279,300,302],[276,279,300,302,440],[0,276,279,300,438], [276,279,300,438,440],[0,276,279,417,418],[276,279,417,418,419],[276,279,417,419,440], [0,276,279,417,438],[276,279,417,438,440],[0,276,300,302,305],[276,300,302,305,443], [276,300,302,440,443],[0,276,300,303,305],[276,300,303,305,443],[0,276,300,303,441], [276,300,303,441,443],[276,300,438,440,443],[0,276,300,438,441],[276,300,438,441,443], [0,276,303,305,409],[276,303,305,409,547],[276,303,305,443,547],[0,276,303,402,406], [276,303,402,406,544],[0,276,303,402,409],[276,303,402,409,547],[276,303,402,540,544], [276,303,402,540,547],[0,276,303,405,406],[276,303,405,406,544],[0,276,303,405,543], [276,303,405,543,544],[276,303,441,443,547],[276,303,441,540,544],[276,303,441,540,547], [0,276,303,441,543],[276,303,441,543,544],[0,276,309,310,311],[276,309,310,311,449], [0,276,309,310,406],[276,309,310,406,544],[276,309,310,448,449],[276,309,310,448,544], [0,276,309,311,409],[276,309,311,409,547],[276,309,311,449,547],[0,276,309,402,406], [276,309,402,406,544],[0,276,309,402,409],[276,309,402,409,547],[276,309,402,540,544], [276,309,402,540,547],[276,309,447,448,449],[276,309,447,448,544],[276,309,447,449,547], [276,309,447,540,544],[276,309,447,540,547],[276,414,415,416,419],[276,414,415,416,449], [0,276,414,415,418],[276,414,415,418,419],[276,414,415,448,449],[276,414,415,448,544], [0,276,414,415,543],[276,414,415,543,544],[276,414,416,419,440],[276,414,416,440,443], [276,414,416,443,547],[276,414,416,449,547],[0,276,414,417,418],[276,414,417,418,419], [276,414,417,419,440],[0,276,414,417,438],[276,414,417,438,440],[276,414,438,440,443], [0,276,414,438,441],[276,414,438,441,443],[276,414,441,443,547],[276,414,441,540,544], [276,414,441,540,547],[0,276,414,441,543],[276,414,441,543,544],[276,414,447,448,449], [276,414,447,448,544],[276,414,447,449,547],[276,414,447,540,544],[276,414,447,540,547], [1,2,5,41,179],[2,5,41,179,455],[1,2,5,41,317],[2,5,41,317,455], [1,2,5,143,179],[2,5,143,179,455],[2,5,143,419,455],[1,2,5,281,317], [2,5,281,317,455],[2,5,281,419,455],[1,2,35,133,271],[1,2,35,133,409], [1,2,35,173,271],[1,2,35,311,409],[1,2,38,41,179],[2,38,41,179,455], [1,2,38,41,317],[2,38,41,317,455],[1,2,38,136,274],[1,2,38,136,412], [1,2,38,176,179],[2,38,176,179,455],[1,2,38,176,274],[2,38,176,452,455], [1,2,38,314,317],[2,38,314,317,455],[1,2,38,314,412],[2,38,314,452,455], [1,2,132,133,271],[1,2,132,133,409],[1,2,132,136,274],[1,2,132,136,412], [1,2,132,270,271],[1,2,132,270,274],[1,2,132,408,409],[1,2,132,408,412], [1,2,140,143,179],[2,140,143,179,455],[2,140,143,419,455],[1,2,140,173,271], [1,2,140,176,179],[2,140,176,179,455],[1,2,140,176,274],[2,140,176,452,455], [1,2,140,270,271],[1,2,140,270,274],[2,140,416,419,455],[2,140,416,452,455], [1,2,278,281,317],[2,278,281,317,455],[2,278,281,419,455],[1,2,278,311,409], [1,2,278,314,317],[2,278,314,317,455],[1,2,278,314,412],[2,278,314,452,455], [1,2,278,408,409],[1,2,278,408,412],[2,278,416,419,455],[2,278,416,452,455], [1,4,5,41,179],[1,4,5,41,317],[1,4,5,143,179],[1,4,5,281,317], [1,4,39,41,179],[1,4,39,41,317],[1,4,39,177,179],[1,4,39,177,453], [1,4,39,315,317],[1,4,39,315,453],[1,4,142,143,179],[1,4,142,177,179], [1,4,142,177,453],[1,4,142,418,453],[1,4,280,281,317],[1,4,280,315,317], [1,4,280,315,453],[1,4,280,418,453],[1,34,35,133,271],[1,34,35,133,409], [1,34,35,173,271],[1,34,35,311,409],[1,34,127,130,268],[1,34,127,130,406], [1,34,127,133,271],[1,34,127,133,409],[1,34,127,265,268],[1,34,127,265,271], [1,34,127,403,406],[1,34,127,403,409],[1,34,172,173,271],[1,34,172,265,268], [1,34,172,265,271],[1,34,310,311,409],[1,34,310,403,406],[1,34,310,403,409], [1,36,38,41,179],[1,36,38,41,317],[1,36,38,136,274],[1,36,38,136,412], [1,36,38,176,179],[1,36,38,176,274],[1,36,38,314,317],[1,36,38,314,412], [1,36,39,41,179],[1,36,39,41,317],[1,36,39,177,179],[1,36,39,177,453], [1,36,39,315,317],[1,36,39,315,453],[1,36,127,130,268],[1,36,127,130,406], [1,36,127,133,271],[1,36,127,133,409],[1,36,127,265,268],[1,36,127,265,271], [1,36,127,403,406],[1,36,127,403,409],[1,36,129,130,268],[1,36,129,130,406], [1,36,129,267,268],[1,36,129,267,543],[1,36,129,405,406],[1,36,129,405,543], [1,36,132,133,271],[1,36,132,133,409],[1,36,132,136,274],[1,36,132,136,412], [1,36,132,270,271],[1,36,132,270,274],[1,36,132,408,409],[1,36,132,408,412], [1,36,174,176,179],[1,36,174,176,274],[1,36,174,177,179],[1,36,174,177,453], [1,36,174,265,268],[1,36,174,265,271],[1,36,174,267,268],[1,36,174,267,543], [1,36,174,270,271],[1,36,174,270,274],[1,36,174,450,453],[1,36,174,450,543], [1,36,312,314,317],[1,36,312,314,412],[1,36,312,315,317],[1,36,312,315,453], [1,36,312,403,406],[1,36,312,403,409],[1,36,312,405,406],[1,36,312,405,543], [1,36,312,408,409],[1,36,312,408,412],[1,36,312,450,453],[1,36,312,450,543], [1,139,140,143,179],[139,140,143,179,455],[139,140,143,419,455],[1,139,140,173,271], [139,140,173,271,547],[139,140,173,449,547],[1,139,140,176,179],[139,140,176,179,455], [1,139,140,176,274],[139,140,176,274,550],[139,140,176,452,455],[139,140,176,452,550], [1,139,140,270,271],[139,140,270,271,547],[1,139,140,270,274],[139,140,270,274,550], [139,140,270,546,547],[139,140,270,546,550],[139,140,416,419,455],[139,140,416,449,547], [139,140,416,452,455],[139,140,416,452,550],[139,140,416,546,547],[139,140,416,546,550], [1,139,142,143,179],[139,142,143,179,455],[139,142,143,419,455],[1,139,142,177,179], [139,142,177,179,455],[1,139,142,177,453],[139,142,177,453,455],[139,142,418,419,455], [1,139,142,418,453],[139,142,418,453,455],[1,139,172,173,271],[139,172,173,271,547], [139,172,173,449,547],[1,139,172,265,268],[139,172,265,268,544],[1,139,172,265,271], 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[99,102,378,385,523],[99,102,378,516,517],[99,102,378,516,523],[99,106,107,110,248], [99,106,107,110,386],[99,106,107,245,248],[99,106,107,383,386],[99,106,109,110,248], [99,106,109,110,386],[99,106,109,247,248],[99,106,109,247,523],[99,106,109,385,386], [99,106,109,385,523],[99,106,244,245,248],[99,106,244,247,248],[99,106,244,247,523], [99,106,244,520,523],[99,106,382,383,386],[99,106,382,385,386],[99,106,382,385,523], [99,106,382,520,523],[237,238,239,242,518],[237,238,239,515,518],[237,238,241,242,518], [99,237,238,241,517],[237,238,241,517,518],[237,238,514,515,518],[99,237,238,514,517], [237,238,514,517,518],[99,237,239,242,248],[237,239,242,248,524],[237,239,242,518,524], [99,237,239,245,248],[237,239,245,248,524],[237,239,245,521,524],[237,239,515,518,524], [237,239,515,521,524],[237,240,241,242,518],[99,237,240,241,517],[237,240,241,517,518], [99,237,240,242,248],[237,240,242,248,524],[237,240,242,518,524],[99,237,240,247,248], [237,240,247,248,524],[99,237,240,247,523],[237,240,247,523,524],[99,237,240,516,517], [237,240,516,517,518],[237,240,516,518,524],[99,237,240,516,523],[237,240,516,523,524], [99,237,244,245,248],[237,244,245,248,524],[237,244,245,521,524],[99,237,244,247,248], [237,244,247,248,524],[99,237,244,247,523],[237,244,247,523,524],[237,244,520,521,524], [99,237,244,520,523],[237,244,520,523,524],[237,513,514,515,518],[99,237,513,514,517], [237,513,514,517,518],[237,513,515,518,524],[237,513,515,521,524],[99,237,513,516,517], [237,513,516,517,518],[237,513,516,518,524],[99,237,513,516,523],[237,513,516,523,524], [237,513,520,521,524],[99,237,513,520,523],[237,513,520,523,524],[99,375,376,377,380], [375,376,377,380,518],[375,376,377,515,518],[99,375,376,379,380],[375,376,379,380,518], [99,375,376,379,517],[375,376,379,517,518],[375,376,514,515,518],[99,375,376,514,517], [375,376,514,517,518],[99,375,377,380,386],[375,377,380,386,524],[375,377,380,518,524], [99,375,377,383,386],[375,377,383,386,524],[375,377,383,521,524],[375,377,515,518,524], [375,377,515,521,524],[99,375,378,379,380],[375,378,379,380,518],[99,375,378,379,517], [375,378,379,517,518],[99,375,378,380,386],[375,378,380,386,524],[375,378,380,518,524], [99,375,378,385,386],[375,378,385,386,524],[99,375,378,385,523],[375,378,385,523,524], [99,375,378,516,517],[375,378,516,517,518],[375,378,516,518,524],[99,375,378,516,523], [375,378,516,523,524],[99,375,382,383,386],[375,382,383,386,524],[375,382,383,521,524], [99,375,382,385,386],[375,382,385,386,524],[99,375,382,385,523],[375,382,385,523,524], [375,382,520,521,524],[99,375,382,520,523],[375,382,520,523,524],[375,513,514,515,518], [99,375,513,514,517],[375,513,514,517,518],[375,513,515,518,524],[375,513,515,521,524], [99,375,513,516,517],[375,513,516,517,518],[375,513,516,518,524],[99,375,513,516,523], [375,513,516,523,524],[375,513,520,521,524],[99,375,513,520,523],[375,513,520,523,524], 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[75,213,216,217,556],[75,213,216,218,556],[213,216,217,218,556],[78,80,83,221,556], [79,80,83,221,556],[78,80,218,221,556],[79,80,218,221,556],[79,82,83,221,556], [78,79,82,220,556],[79,82,220,221,556],[79,217,218,221,556],[78,79,217,220,556], [79,217,220,221,556],[78,81,83,221,556],[78,81,82,220,556],[78,81,219,220,556], [78,81,219,221,556],[78,216,218,221,556],[216,217,218,221,556],[78,216,217,220,556], [216,217,220,221,556],[78,216,219,220,556],[78,216,219,221,556],[216,219,220,221,556], [81,83,86,224,556],[82,83,86,224,556],[81,83,221,224,556],[82,83,221,224,556], [82,85,86,224,556],[81,82,85,223,556],[82,85,223,224,556],[82,220,221,224,556], [81,82,220,223,556],[82,220,223,224,556],[81,84,86,224,556],[81,84,85,223,556], [81,84,222,223,556],[81,84,222,224,556],[81,219,221,224,556],[219,220,221,224,556], [81,219,220,223,556],[219,220,223,224,556],[81,219,222,223,556],[81,219,222,224,556], [219,222,223,224,556],[84,86,89,227,556],[85,86,89,227,556],[84,86,224,227,556], 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[91,229,232,233,556],[90,93,95,233,556],[90,93,94,232,556],[90,93,231,232,556], [90,93,231,233,556],[90,228,230,233,556],[228,229,230,233,556],[90,228,229,232,556], [228,229,232,233,556],[90,228,231,232,556],[90,228,231,233,556],[228,231,232,233,556], [93,95,98,236,556],[94,95,98,236,556],[93,95,233,236,556],[94,95,233,236,556], [94,97,98,236,556],[93,94,97,235,556],[94,97,235,236,556],[94,232,233,236,556], [93,94,232,235,556],[94,232,235,236,556],[93,96,98,236,556],[93,96,97,235,556], [93,96,234,235,556],[93,96,234,236,556],[93,231,233,236,556],[231,232,233,236,556], [93,231,232,235,556],[231,232,235,236,556],[93,231,234,235,556],[93,231,234,236,556], [231,234,235,236,556],[96,98,101,239,556],[97,98,101,239,556],[96,98,236,239,556], [97,98,236,239,556],[97,100,101,239,556],[96,97,100,238,556],[97,100,238,239,556], [97,235,236,239,556],[96,97,235,238,556],[97,235,238,239,556],[96,99,101,239,556], [96,99,100,238,556],[96,99,237,238,556],[96,99,237,239,556],[96,234,236,239,556], 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