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{{Rolfsen Knot Page| |
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n = 10 | |
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k = 142 | |
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KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,-4,5,-6,7,10,-2,-3,9,-8,4,-7,6,-5,3,-9,8/goTop.html | |
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braid_table = <table cellspacing=0 cellpadding=0 border=0> |
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<tr><td>[[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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{{Knot Navigation Links|ext=gif}} |
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<tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr> |
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{{Rolfsen Knot Page Header|n=10|k=142|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,-4,5,-6,7,10,-2,-3,9,-8,4,-7,6,-5,3,-9,8/goTop.html}} |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr> |
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</table> | |
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<br style="clear:both" /> |
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braid_crossings = 11 | |
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braid_width = 4 | |
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{{:{{PAGENAME}} Further Notes and Views}} |
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braid_index = 4 | |
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same_alexander = | |
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{{Knot Presentations}} |
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same_jones = | |
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{{3D Invariants}} |
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khovanov_table = <table border=1> |
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{{4D Invariants}} |
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{{Polynomial Invariants}} |
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{{Vassiliev Invariants}} |
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{{Khovanov Homology|table=<table border=1> |
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<tr align=center> |
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<td width=16.6667%><table cellpadding=0 cellspacing=0> |
<td width=16.6667%><table cellpadding=0 cellspacing=0> |
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<tr><td>\</td><td> </td><td>r</td></tr> |
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<tr><td> </td><td> \ </td><td> </td></tr> |
<tr><td> </td><td> \ </td><td> </td></tr> |
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<tr><td>j</td><td> </td><td>\</td></tr> |
<tr><td>j</td><td> </td><td>\</td></tr> |
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<td width=8.33333%>0</td ><td width=8.33333%>1</td ><td width=8.33333%>2</td ><td width=8.33333%>3</td ><td width=8.33333%>4</td ><td width=8.33333%>5</td ><td width=8.33333%>6</td ><td width=8.33333%>7</td ><td width=16.6667%>χ</td></tr> |
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<tr align=center><td>21</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td>-2</td></tr> |
<tr align=center><td>21</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td>-2</td></tr> |
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<tr align=center><td>19</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>0</td></tr> |
<tr align=center><td>19</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>0</td></tr> |
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<tr align=center><td>7</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
<tr align=center><td>7</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>5</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
<tr align=center><td>5</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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coloured_jones_2 = <math>q^{29}-q^{26}+q^{25}-q^{24}-3 q^{23}+3 q^{22}-3 q^{20}+2 q^{19}+3 q^{18}-4 q^{17}+4 q^{15}-4 q^{14}-q^{13}+5 q^{12}-2 q^{11}-2 q^{10}+3 q^9-q^7+q^6</math> | |
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{{Computer Talk Header}} |
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coloured_jones_3 = <math>-2 q^{55}+2 q^{53}+4 q^{52}-5 q^{51}-3 q^{50}+2 q^{49}+10 q^{48}-2 q^{47}-10 q^{46}-2 q^{45}+12 q^{44}+3 q^{43}-13 q^{42}-4 q^{41}+11 q^{40}+5 q^{39}-12 q^{38}-3 q^{37}+9 q^{36}+4 q^{35}-9 q^{34}-q^{33}+4 q^{32}+3 q^{31}-4 q^{30}-q^{29}-q^{28}+2 q^{27}+q^{26}+2 q^{25}-4 q^{24}-2 q^{23}+2 q^{22}+5 q^{21}-2 q^{20}-4 q^{19}-q^{18}+5 q^{17}+q^{16}-2 q^{15}-2 q^{14}+2 q^{13}+q^{12}-q^{10}+q^9</math> | |
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coloured_jones_4 = <math>q^{90}+2 q^{88}-2 q^{87}-4 q^{86}-q^{85}-q^{84}+10 q^{83}+5 q^{82}-9 q^{81}-11 q^{80}-11 q^{79}+19 q^{78}+23 q^{77}-4 q^{76}-20 q^{75}-30 q^{74}+19 q^{73}+36 q^{72}+8 q^{71}-22 q^{70}-42 q^{69}+12 q^{68}+39 q^{67}+15 q^{66}-19 q^{65}-45 q^{64}+9 q^{63}+37 q^{62}+13 q^{61}-14 q^{60}-41 q^{59}+6 q^{58}+34 q^{57}+12 q^{56}-9 q^{55}-36 q^{54}-q^{53}+29 q^{52}+11 q^{51}-q^{50}-28 q^{49}-10 q^{48}+18 q^{47}+9 q^{46}+9 q^{45}-14 q^{44}-13 q^{43}+6 q^{42}+q^{41}+12 q^{40}-q^{39}-8 q^{38}+2 q^{37}-8 q^{36}+5 q^{35}+3 q^{34}+7 q^{32}-9 q^{31}-q^{30}-2 q^{29}-q^{28}+10 q^{27}-2 q^{26}-4 q^{24}-4 q^{23}+6 q^{22}+2 q^{20}-q^{19}-3 q^{18}+2 q^{17}+q^{15}-q^{13}+q^{12}</math> | |
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<table> |
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coloured_jones_5 = <math>-2 q^{132}+2 q^{129}+2 q^{128}+4 q^{127}-q^{126}-3 q^{125}-7 q^{124}-5 q^{123}+2 q^{122}+10 q^{121}+14 q^{120}+10 q^{119}-13 q^{118}-27 q^{117}-23 q^{116}+37 q^{114}+48 q^{113}+11 q^{112}-45 q^{111}-63 q^{110}-33 q^{109}+41 q^{108}+83 q^{107}+51 q^{106}-37 q^{105}-89 q^{104}-64 q^{103}+26 q^{102}+92 q^{101}+74 q^{100}-19 q^{99}-90 q^{98}-76 q^{97}+15 q^{96}+86 q^{95}+75 q^{94}-12 q^{93}-85 q^{92}-71 q^{91}+11 q^{90}+81 q^{89}+71 q^{88}-12 q^{87}-79 q^{86}-65 q^{85}+8 q^{84}+72 q^{83}+66 q^{82}-2 q^{81}-66 q^{80}-63 q^{79}-5 q^{78}+53 q^{77}+63 q^{76}+17 q^{75}-44 q^{74}-59 q^{73}-24 q^{72}+24 q^{71}+53 q^{70}+36 q^{69}-13 q^{68}-42 q^{67}-36 q^{66}-6 q^{65}+27 q^{64}+38 q^{63}+16 q^{62}-13 q^{61}-27 q^{60}-24 q^{59}-3 q^{58}+17 q^{57}+20 q^{56}+14 q^{55}-4 q^{54}-15 q^{53}-14 q^{52}-5 q^{51}+2 q^{50}+12 q^{49}+9 q^{48}+3 q^{47}-3 q^{46}-5 q^{45}-8 q^{44}-3 q^{43}+q^{42}+5 q^{41}+4 q^{40}+5 q^{39}-q^{38}-3 q^{37}-5 q^{36}-4 q^{35}+5 q^{33}+3 q^{32}+3 q^{31}-q^{30}-4 q^{29}-3 q^{28}+2 q^{27}+2 q^{25}+2 q^{24}-q^{23}-2 q^{22}+q^{21}+q^{18}-q^{16}+q^{15}</math> | |
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coloured_jones_6 = <math>q^{183}+2 q^{181}-2 q^{179}-4 q^{178}-4 q^{177}-3 q^{176}+q^{175}+10 q^{174}+9 q^{173}+8 q^{172}-q^{171}-7 q^{170}-25 q^{169}-23 q^{168}-q^{167}+14 q^{166}+37 q^{165}+38 q^{164}+33 q^{163}-28 q^{162}-65 q^{161}-63 q^{160}-45 q^{159}+27 q^{158}+88 q^{157}+136 q^{156}+42 q^{155}-58 q^{154}-131 q^{153}-156 q^{152}-60 q^{151}+83 q^{150}+227 q^{149}+151 q^{148}+16 q^{147}-138 q^{146}-234 q^{145}-161 q^{144}+27 q^{143}+247 q^{142}+211 q^{141}+87 q^{140}-101 q^{139}-245 q^{138}-204 q^{137}-20 q^{136}+228 q^{135}+215 q^{134}+111 q^{133}-76 q^{132}-230 q^{131}-205 q^{130}-29 q^{129}+216 q^{128}+203 q^{127}+106 q^{126}-73 q^{125}-220 q^{124}-197 q^{123}-25 q^{122}+211 q^{121}+192 q^{120}+103 q^{119}-68 q^{118}-208 q^{117}-190 q^{116}-33 q^{115}+193 q^{114}+178 q^{113}+110 q^{112}-43 q^{111}-180 q^{110}-182 q^{109}-60 q^{108}+145 q^{107}+154 q^{106}+126 q^{105}+5 q^{104}-129 q^{103}-167 q^{102}-100 q^{101}+70 q^{100}+110 q^{99}+135 q^{98}+65 q^{97}-52 q^{96}-127 q^{95}-126 q^{94}-12 q^{93}+37 q^{92}+110 q^{91}+102 q^{90}+30 q^{89}-50 q^{88}-104 q^{87}-62 q^{86}-41 q^{85}+38 q^{84}+80 q^{83}+69 q^{82}+28 q^{81}-32 q^{80}-43 q^{79}-68 q^{78}-29 q^{77}+12 q^{76}+37 q^{75}+49 q^{74}+23 q^{73}+14 q^{72}-29 q^{71}-33 q^{70}-28 q^{69}-16 q^{68}+14 q^{67}+14 q^{66}+32 q^{65}+13 q^{64}+4 q^{63}-11 q^{62}-22 q^{61}-8 q^{60}-17 q^{59}+7 q^{58}+8 q^{57}+15 q^{56}+8 q^{55}+6 q^{53}-16 q^{52}-4 q^{51}-7 q^{50}+q^{49}-q^{48}+2 q^{47}+14 q^{46}-3 q^{45}+4 q^{44}-3 q^{43}-q^{42}-8 q^{41}-5 q^{40}+7 q^{39}-2 q^{38}+5 q^{37}+2 q^{36}+3 q^{35}-4 q^{34}-4 q^{33}+3 q^{32}-3 q^{31}+q^{30}+q^{29}+3 q^{28}-q^{27}-2 q^{26}+2 q^{25}-q^{24}+q^{21}-q^{19}+q^{18}</math> | |
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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coloured_jones_7 = | |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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computer_talk = |
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<table> |
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<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 17, 2005, 14:44:34)...</pre></td></tr> |
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<tr valign=top> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Knot[10, 142]]</nowiki></pre></td></tr> |
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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</tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[11, 19, 12, 18], X[5, 15, 6, 14], |
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<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[10, 142]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[11, 19, 12, 18], X[5, 15, 6, 14], |
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X[17, 7, 18, 6], X[7, 17, 8, 16], X[15, 9, 16, 8], X[13, 1, 14, 20], |
X[17, 7, 18, 6], X[7, 17, 8, 16], X[15, 9, 16, 8], X[13, 1, 14, 20], |
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X[19, 13, 20, 12], X[2, 10, 3, 9]]</nowiki></ |
X[19, 13, 20, 12], X[2, 10, 3, 9]]</nowiki></code></td></tr> |
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</table> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 142]]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[1, -10, 2, -1, -4, 5, -6, 7, 10, -2, -3, 9, -8, 4, -7, 6, -5, |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[10, 142]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[1, -10, 2, -1, -4, 5, -6, 7, 10, -2, -3, 9, -8, 4, -7, 6, -5, |
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3, -9, 8]</nowiki></ |
3, -9, 8]</nowiki></code></td></tr> |
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</table> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[Knot[10, 142]]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {1, 1, 1, 2, 1, 1, 1, 2, 3, -2, 3}]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[10, 142]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 10, -14, -16, 2, -18, -20, -8, -6, -12]</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[10, 142]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[4, {1, 1, 1, 2, 1, 1, 1, 2, 3, -2, 3}]</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{4, 11}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[10, 142]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>4</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[10, 142]]]</nowiki></code></td></tr> |
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<tr align=left><td></td><td>[[Image:10_142_ML.gif]]</td></tr><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[10, 142]]&) /@ { |
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SymmetryType, UnknottingNumber, ThreeGenus, |
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BridgeIndex, SuperBridgeIndex, NakanishiIndex |
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}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 3, 3, 3, NotAvailable, 2}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[10, 142]][t]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 3 2 2 3 |
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-1 + -- - -- + - + 2 t - 3 t + 2 t |
-1 + -- - -- + - + 2 t - 3 t + 2 t |
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3 2 t |
3 2 t |
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t t</nowiki></ |
t t</nowiki></code></td></tr> |
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</table> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[10, 142]][z]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td> |
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1 + 8 z + 9 z + 2 z</nowiki></pre></td></tr> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[10, 142]][z]</nowiki></code></td></tr> |
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<tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 142]}</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6 |
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1 + 8 z + 9 z + 2 z</nowiki></code></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[Knot[10, 142]][q]</nowiki></pre></td></tr> |
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</table> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 4 5 6 7 8 9 10 |
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<table><tr align=left> |
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q - q + 2 q - 2 q + 3 q - 2 q + 2 q - 2 q</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr> |
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<tr align=left> |
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<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 142]}</nowiki></code></td></tr> |
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</table> |
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q + q + q + q + 2 q + 3 q - q - 3 q - 2 q - q + q</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 142]][a, z]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[10, 142]], KnotSignature[Knot[10, 142]]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{15, 6}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[10, 142]][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 3 4 5 6 7 8 9 10 |
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q - q + 2 q - 2 q + 3 q - 2 q + 2 q - 2 q</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 142]}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[10, 142]][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 10 14 18 20 22 24 28 30 32 34 38 |
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q + q + q + q + 2 q + 3 q - q - 3 q - 2 q - q + q</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[10, 142]][a, z]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 2 2 4 4 4 6 6 |
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-12 5 4 -6 5 z 7 z 6 z z 5 z 5 z z z |
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a - --- + -- + a - ---- + ---- + ---- - --- + ---- + ---- + -- + -- |
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10 8 10 8 6 10 8 6 8 6 |
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a a a a a a a a a a</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[10, 142]][a, z]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 2 2 2 |
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-12 5 4 -6 2 z 4 z 6 z z 17 z 10 z 6 z |
-12 5 4 -6 2 z 4 z 6 z z 17 z 10 z 6 z |
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a + --- + -- - a + --- - --- - --- - --- - ----- - ----- + ---- + |
a + --- + -- - a + --- - --- - --- - --- - ----- - ----- + ---- + |
||
Line 99: | Line 204: | ||
---- - ---- - ---- + -- + --- + ---- + -- + --- + -- |
---- - ---- - ---- + -- + --- + ---- + -- + --- + -- |
||
7 10 8 6 11 9 7 10 8 |
7 10 8 6 11 9 7 10 8 |
||
a a a a a a a a a</nowiki></ |
a a a a a a a a a</nowiki></code></td></tr> |
||
</table> |
|||
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[10, 142]], Vassiliev[3][Knot[10, 142]]}</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 21}</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td> |
||
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 142]], Vassiliev[3][Knot[10, 142]]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{8, 21}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[10, 142]][q, t]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 5 7 7 9 2 11 2 11 3 13 3 13 4 15 4 |
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q + q + q t + q t + q t + q t + q t + 2 q t + q t + |
q + q + q t + q t + q t + q t + q t + 2 q t + q t + |
||
17 5 17 6 21 7 |
17 5 17 6 21 7 |
||
2 q t + 2 q t + 2 q t</nowiki></ |
2 q t + 2 q t + 2 q t</nowiki></code></td></tr> |
||
</table> |
</table> |
||
<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td> |
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[[Category:Knot Page]] |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[10, 142], 2][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 6 7 9 10 11 12 13 14 15 17 |
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q - q + 3 q - 2 q - 2 q + 5 q - q - 4 q + 4 q - 4 q + |
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18 19 20 22 23 24 25 26 29 |
|||
3 q + 2 q - 3 q + 3 q - 3 q - q + q - q + q</nowiki></code></td></tr> |
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</table> }} |
Latest revision as of 17:02, 1 September 2005
|
|
(KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 142's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
10_142 is also known as the pretzel knot P(-4,3,3). |
Knot presentations
Planar diagram presentation | X4251 X10,4,11,3 X11,19,12,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X13,1,14,20 X19,13,20,12 X2,10,3,9 |
Gauss code | 1, -10, 2, -1, -4, 5, -6, 7, 10, -2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8 |
Dowker-Thistlethwaite code | 4 10 -14 -16 2 -18 -20 -8 -6 -12 |
Conway Notation | [31,3,3-] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
[{4, 10}, {3, 5}, {1, 4}, {6, 9}, {5, 8}, {9, 7}, {11, 6}, {10, 12}, {2, 11}, {12, 3}, {8, 2}, {7, 1}] |
[edit Notes on presentations of 10 142]
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["10 142"];
|
In[4]:=
|
PD[K]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
X4251 X10,4,11,3 X11,19,12,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X13,1,14,20 X19,13,20,12 X2,10,3,9 |
In[5]:=
|
GaussCode[K]
|
Out[5]=
|
1, -10, 2, -1, -4, 5, -6, 7, 10, -2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8 |
In[6]:=
|
DTCode[K]
|
Out[6]=
|
4 10 -14 -16 2 -18 -20 -8 -6 -12 |
(The path below may be different on your system)
In[7]:=
|
AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
|
ConwayNotation[K]
|
Out[8]=
|
[31,3,3-] |
In[9]:=
|
br = BR[K]
|
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
|
Out[9]=
|
In[10]:=
|
{First[br], Crossings[br], BraidIndex[K]}
|
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
|
KnotTheory::loading: Loading precomputed data in IndianaData`.
|
Out[10]=
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{ 4, 11, 4 } |
In[11]:=
|
Show[BraidPlot[br]]
|
Out[11]=
|
-Graphics- |
In[12]:=
|
Show[DrawMorseLink[K]]
|
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
Out[12]=
|
-Graphics- |
In[13]:=
|
ap = ArcPresentation[K]
|
Out[13]=
|
ArcPresentation[{4, 10}, {3, 5}, {1, 4}, {6, 9}, {5, 8}, {9, 7}, {11, 6}, {10, 12}, {2, 11}, {12, 3}, {8, 2}, {7, 1}] |
In[14]:=
|
Draw[ap]
|
Out[14]=
|
-Graphics- |
Three dimensional invariants
|
Four dimensional invariants
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Polynomial invariants
A1 Invariants.
Weight | Invariant |
---|---|
1 | |
2 | |
3 |
A2 Invariants.
Weight | Invariant |
---|---|
1,0 | |
1,1 | |
2,0 |
A3 Invariants.
Weight | Invariant |
---|---|
0,1,0 | |
1,0,0 |
A4 Invariants.
Weight | Invariant |
---|---|
0,1,0,0 | |
1,0,0,0 |
B2 Invariants.
Weight | Invariant |
---|---|
0,1 | |
1,0 |
D4 Invariants.
Weight | Invariant |
---|---|
1,0,0,0 |
G2 Invariants.
Weight | Invariant |
---|---|
1,0 |
.
KnotTheory`
, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
|
K = Knot["10 142"];
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In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
In[6]:=
|
Alexander[K, 2][t]
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 15, 6 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, ): {}
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["10 142"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
|
{ , } |
In[5]:=
|
DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
|
{} |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{} |
Vassiliev invariants
V2 and V3: | (8, 21) |
V2,1 through V6,9: |
|
V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 6 is the signature of 10 142. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
2 | |
3 | |
4 | |
5 | |
6 |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
|