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{{Rolfsen Knot Page| |
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n = 10 | |
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k = 128 | |
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KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,-4,5,10,-2,-3,7,-8,9,-6,4,-5,3,-9,8,-7,6/goTop.html | |
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braid_table = <table cellspacing=0 cellpadding=0 border=0> |
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{{Knot Navigation Links|ext=gif}} |
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{{Rolfsen Knot Page Header|n=10|k=128|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,-4,5,10,-2,-3,7,-8,9,-6,4,-5,3,-9,8,-7,6/goTop.html}} |
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<br style="clear:both" /> |
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{{:{{PAGENAME}} Further Notes and Views}} |
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{{Knot Presentations}} |
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<center><table border=1 cellpadding=10><tr align=center valign=top> |
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<td> |
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[[Braid Representatives|Minimum Braid Representative]]: |
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<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:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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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:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.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:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr> |
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr> |
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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> |
<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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braid_crossings = 11 | |
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braid_width = 4 | |
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[[Invariants from Braid Theory|Length]] is 11, width is 4. |
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braid_index = 4 | |
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same_alexander = | |
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[[Invariants from Braid Theory|Braid index]] is 4. |
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same_jones = | |
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</td> |
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khovanov_table = <table border=1> |
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<td> |
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[[Lightly Documented Features|A Morse Link Presentation]]: |
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[[Image:{{PAGENAME}}_ML.gif]] |
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</tr></table></center> |
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{{3D Invariants}} |
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{{4D Invariants}} |
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{{Polynomial Invariants}} |
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=== "Similar" Knots (within the Atlas) === |
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Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]: |
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{...} |
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Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>): |
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{...} |
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{{Vassiliev Invariants}} |
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{{Khovanov Homology|table=<table border=1> |
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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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</table></td> |
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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>1</td><td>-1</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>1</td><td>-1</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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</table> |
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coloured_jones_2 = <math>q^{29}-q^{28}-q^{27}+2 q^{26}-2 q^{24}+2 q^{23}-2 q^{21}+q^{20}-q^{19}-q^{16}+2 q^{15}-q^{14}-2 q^{13}+4 q^{12}-q^{11}-2 q^{10}+3 q^9-q^7+q^6</math> | |
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coloured_jones_3 = <math>-q^{55}+2 q^{53}+2 q^{52}-4 q^{51}-3 q^{50}+3 q^{49}+7 q^{48}-4 q^{47}-9 q^{46}+3 q^{45}+12 q^{44}-3 q^{43}-12 q^{42}+2 q^{41}+14 q^{40}-3 q^{39}-13 q^{38}+3 q^{37}+12 q^{36}-3 q^{35}-12 q^{34}+3 q^{33}+9 q^{32}-q^{31}-9 q^{30}+2 q^{29}+5 q^{28}-6 q^{26}+2 q^{25}+2 q^{24}-q^{23}-3 q^{22}+4 q^{21}+q^{20}-3 q^{19}-2 q^{18}+4 q^{17}+2 q^{16}-2 q^{15}-2 q^{14}+2 q^{13}+q^{12}-q^{10}+q^9</math> | |
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{{Display Coloured Jones|J2=<math>q^{29}-q^{28}-q^{27}+2 q^{26}-2 q^{24}+2 q^{23}-2 q^{21}+q^{20}-q^{19}-q^{16}+2 q^{15}-q^{14}-2 q^{13}+4 q^{12}-q^{11}-2 q^{10}+3 q^9-q^7+q^6</math>|J3=<math>-q^{55}+2 q^{53}+2 q^{52}-4 q^{51}-3 q^{50}+3 q^{49}+7 q^{48}-4 q^{47}-9 q^{46}+3 q^{45}+12 q^{44}-3 q^{43}-12 q^{42}+2 q^{41}+14 q^{40}-3 q^{39}-13 q^{38}+3 q^{37}+12 q^{36}-3 q^{35}-12 q^{34}+3 q^{33}+9 q^{32}-q^{31}-9 q^{30}+2 q^{29}+5 q^{28}-6 q^{26}+2 q^{25}+2 q^{24}-q^{23}-3 q^{22}+4 q^{21}+q^{20}-3 q^{19}-2 q^{18}+4 q^{17}+2 q^{16}-2 q^{15}-2 q^{14}+2 q^{13}+q^{12}-q^{10}+q^9</math>|J4=<math>q^{90}-q^{89}-q^{87}-q^{86}+3 q^{85}-q^{84}+4 q^{83}-2 q^{82}-6 q^{81}+q^{80}-3 q^{79}+12 q^{78}+3 q^{77}-8 q^{76}-6 q^{75}-11 q^{74}+19 q^{73}+10 q^{72}-6 q^{71}-9 q^{70}-19 q^{69}+20 q^{68}+13 q^{67}-3 q^{66}-8 q^{65}-23 q^{64}+20 q^{63}+14 q^{62}-2 q^{61}-7 q^{60}-22 q^{59}+16 q^{58}+15 q^{57}-q^{56}-6 q^{55}-21 q^{54}+9 q^{53}+15 q^{52}+3 q^{51}-3 q^{50}-20 q^{49}+q^{48}+13 q^{47}+8 q^{46}+2 q^{45}-17 q^{44}-7 q^{43}+8 q^{42}+8 q^{41}+7 q^{40}-10 q^{39}-10 q^{38}+4 q^{37}+4 q^{36}+6 q^{35}-4 q^{34}-7 q^{33}+5 q^{32}+2 q^{30}-3 q^{29}-5 q^{28}+7 q^{27}+q^{26}+q^{25}-3 q^{24}-5 q^{23}+5 q^{22}+q^{21}+2 q^{20}-q^{19}-3 q^{18}+2 q^{17}+q^{15}-q^{13}+q^{12}</math>|J5=<math>-q^{132}+q^{130}+2 q^{129}+q^{128}-4 q^{126}-5 q^{125}-q^{124}+4 q^{123}+7 q^{122}+7 q^{121}-10 q^{119}-13 q^{118}-6 q^{117}+8 q^{116}+18 q^{115}+16 q^{114}-2 q^{113}-22 q^{112}-25 q^{111}-7 q^{110}+22 q^{109}+33 q^{108}+15 q^{107}-19 q^{106}-38 q^{105}-22 q^{104}+17 q^{103}+41 q^{102}+24 q^{101}-15 q^{100}-40 q^{99}-27 q^{98}+14 q^{97}+41 q^{96}+25 q^{95}-14 q^{94}-40 q^{93}-25 q^{92}+14 q^{91}+40 q^{90}+24 q^{89}-14 q^{88}-40 q^{87}-22 q^{86}+14 q^{85}+36 q^{84}+23 q^{83}-11 q^{82}-35 q^{81}-19 q^{80}+9 q^{79}+27 q^{78}+21 q^{77}-4 q^{76}-24 q^{75}-17 q^{74}+14 q^{72}+16 q^{71}+5 q^{70}-8 q^{69}-10 q^{68}-8 q^{67}-q^{66}+6 q^{65}+6 q^{64}+8 q^{63}+2 q^{62}-6 q^{61}-10 q^{60}-8 q^{59}-q^{58}+11 q^{57}+13 q^{56}+5 q^{55}-7 q^{54}-14 q^{53}-11 q^{52}+4 q^{51}+12 q^{50}+11 q^{49}-8 q^{47}-11 q^{46}-q^{45}+5 q^{44}+6 q^{43}+q^{42}-3 q^{41}-5 q^{40}+q^{39}+4 q^{38}+3 q^{37}-2 q^{36}-4 q^{35}-4 q^{34}+2 q^{33}+4 q^{32}+4 q^{31}-3 q^{29}-4 q^{28}+q^{27}+q^{26}+2 q^{25}+2 q^{24}-q^{23}-2 q^{22}+q^{21}+q^{18}-q^{16}+q^{15}</math>|J6=<math>q^{183}-q^{182}-q^{179}-q^{178}-q^{177}+3 q^{176}+q^{175}+4 q^{174}+2 q^{173}-q^{172}-7 q^{171}-8 q^{170}-4 q^{169}-3 q^{168}+13 q^{167}+15 q^{166}+15 q^{165}-2 q^{164}-13 q^{163}-26 q^{162}-30 q^{161}+q^{160}+21 q^{159}+46 q^{158}+31 q^{157}+13 q^{156}-38 q^{155}-68 q^{154}-40 q^{153}-5 q^{152}+60 q^{151}+69 q^{150}+64 q^{149}-22 q^{148}-85 q^{147}-79 q^{146}-44 q^{145}+50 q^{144}+84 q^{143}+103 q^{142}-q^{141}-81 q^{140}-94 q^{139}-66 q^{138}+37 q^{137}+83 q^{136}+118 q^{135}+6 q^{134}-75 q^{133}-95 q^{132}-72 q^{131}+33 q^{130}+81 q^{129}+123 q^{128}+4 q^{127}-74 q^{126}-95 q^{125}-74 q^{124}+33 q^{123}+82 q^{122}+122 q^{121}+3 q^{120}-69 q^{119}-94 q^{118}-74 q^{117}+30 q^{116}+78 q^{115}+117 q^{114}+6 q^{113}-58 q^{112}-88 q^{111}-74 q^{110}+20 q^{109}+64 q^{108}+109 q^{107}+16 q^{106}-34 q^{105}-75 q^{104}-76 q^{103}-q^{102}+41 q^{101}+94 q^{100}+28 q^{99}-q^{98}-50 q^{97}-71 q^{96}-25 q^{95}+11 q^{94}+67 q^{93}+29 q^{92}+30 q^{91}-17 q^{90}-50 q^{89}-33 q^{88}-15 q^{87}+30 q^{86}+11 q^{85}+38 q^{84}+8 q^{83}-18 q^{82}-16 q^{81}-16 q^{80}+4 q^{79}-14 q^{78}+20 q^{77}+6 q^{76}-2 q^{75}+7 q^{74}+4 q^{73}+7 q^{72}-19 q^{71}-12 q^{69}-12 q^{68}+11 q^{67}+15 q^{66}+18 q^{65}-3 q^{64}+q^{63}-16 q^{62}-22 q^{61}+2 q^{60}+6 q^{59}+14 q^{58}+5 q^{57}+8 q^{56}-7 q^{55}-16 q^{54}+2 q^{53}-q^{52}+6 q^{51}+q^{50}+5 q^{49}-5 q^{48}-10 q^{47}+6 q^{46}+6 q^{44}+q^{43}+3 q^{42}-6 q^{41}-9 q^{40}+4 q^{39}-q^{38}+4 q^{37}+3 q^{36}+4 q^{35}-3 q^{34}-5 q^{33}+2 q^{32}-2 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>|J7=Not Available}} |
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coloured_jones_4 = <math>q^{90}-q^{89}-q^{87}-q^{86}+3 q^{85}-q^{84}+4 q^{83}-2 q^{82}-6 q^{81}+q^{80}-3 q^{79}+12 q^{78}+3 q^{77}-8 q^{76}-6 q^{75}-11 q^{74}+19 q^{73}+10 q^{72}-6 q^{71}-9 q^{70}-19 q^{69}+20 q^{68}+13 q^{67}-3 q^{66}-8 q^{65}-23 q^{64}+20 q^{63}+14 q^{62}-2 q^{61}-7 q^{60}-22 q^{59}+16 q^{58}+15 q^{57}-q^{56}-6 q^{55}-21 q^{54}+9 q^{53}+15 q^{52}+3 q^{51}-3 q^{50}-20 q^{49}+q^{48}+13 q^{47}+8 q^{46}+2 q^{45}-17 q^{44}-7 q^{43}+8 q^{42}+8 q^{41}+7 q^{40}-10 q^{39}-10 q^{38}+4 q^{37}+4 q^{36}+6 q^{35}-4 q^{34}-7 q^{33}+5 q^{32}+2 q^{30}-3 q^{29}-5 q^{28}+7 q^{27}+q^{26}+q^{25}-3 q^{24}-5 q^{23}+5 q^{22}+q^{21}+2 q^{20}-q^{19}-3 q^{18}+2 q^{17}+q^{15}-q^{13}+q^{12}</math> | |
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coloured_jones_5 = <math>-q^{132}+q^{130}+2 q^{129}+q^{128}-4 q^{126}-5 q^{125}-q^{124}+4 q^{123}+7 q^{122}+7 q^{121}-10 q^{119}-13 q^{118}-6 q^{117}+8 q^{116}+18 q^{115}+16 q^{114}-2 q^{113}-22 q^{112}-25 q^{111}-7 q^{110}+22 q^{109}+33 q^{108}+15 q^{107}-19 q^{106}-38 q^{105}-22 q^{104}+17 q^{103}+41 q^{102}+24 q^{101}-15 q^{100}-40 q^{99}-27 q^{98}+14 q^{97}+41 q^{96}+25 q^{95}-14 q^{94}-40 q^{93}-25 q^{92}+14 q^{91}+40 q^{90}+24 q^{89}-14 q^{88}-40 q^{87}-22 q^{86}+14 q^{85}+36 q^{84}+23 q^{83}-11 q^{82}-35 q^{81}-19 q^{80}+9 q^{79}+27 q^{78}+21 q^{77}-4 q^{76}-24 q^{75}-17 q^{74}+14 q^{72}+16 q^{71}+5 q^{70}-8 q^{69}-10 q^{68}-8 q^{67}-q^{66}+6 q^{65}+6 q^{64}+8 q^{63}+2 q^{62}-6 q^{61}-10 q^{60}-8 q^{59}-q^{58}+11 q^{57}+13 q^{56}+5 q^{55}-7 q^{54}-14 q^{53}-11 q^{52}+4 q^{51}+12 q^{50}+11 q^{49}-8 q^{47}-11 q^{46}-q^{45}+5 q^{44}+6 q^{43}+q^{42}-3 q^{41}-5 q^{40}+q^{39}+4 q^{38}+3 q^{37}-2 q^{36}-4 q^{35}-4 q^{34}+2 q^{33}+4 q^{32}+4 q^{31}-3 q^{29}-4 q^{28}+q^{27}+q^{26}+2 q^{25}+2 q^{24}-q^{23}-2 q^{22}+q^{21}+q^{18}-q^{16}+q^{15}</math> | |
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{{Computer Talk Header}} |
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coloured_jones_6 = <math>q^{183}-q^{182}-q^{179}-q^{178}-q^{177}+3 q^{176}+q^{175}+4 q^{174}+2 q^{173}-q^{172}-7 q^{171}-8 q^{170}-4 q^{169}-3 q^{168}+13 q^{167}+15 q^{166}+15 q^{165}-2 q^{164}-13 q^{163}-26 q^{162}-30 q^{161}+q^{160}+21 q^{159}+46 q^{158}+31 q^{157}+13 q^{156}-38 q^{155}-68 q^{154}-40 q^{153}-5 q^{152}+60 q^{151}+69 q^{150}+64 q^{149}-22 q^{148}-85 q^{147}-79 q^{146}-44 q^{145}+50 q^{144}+84 q^{143}+103 q^{142}-q^{141}-81 q^{140}-94 q^{139}-66 q^{138}+37 q^{137}+83 q^{136}+118 q^{135}+6 q^{134}-75 q^{133}-95 q^{132}-72 q^{131}+33 q^{130}+81 q^{129}+123 q^{128}+4 q^{127}-74 q^{126}-95 q^{125}-74 q^{124}+33 q^{123}+82 q^{122}+122 q^{121}+3 q^{120}-69 q^{119}-94 q^{118}-74 q^{117}+30 q^{116}+78 q^{115}+117 q^{114}+6 q^{113}-58 q^{112}-88 q^{111}-74 q^{110}+20 q^{109}+64 q^{108}+109 q^{107}+16 q^{106}-34 q^{105}-75 q^{104}-76 q^{103}-q^{102}+41 q^{101}+94 q^{100}+28 q^{99}-q^{98}-50 q^{97}-71 q^{96}-25 q^{95}+11 q^{94}+67 q^{93}+29 q^{92}+30 q^{91}-17 q^{90}-50 q^{89}-33 q^{88}-15 q^{87}+30 q^{86}+11 q^{85}+38 q^{84}+8 q^{83}-18 q^{82}-16 q^{81}-16 q^{80}+4 q^{79}-14 q^{78}+20 q^{77}+6 q^{76}-2 q^{75}+7 q^{74}+4 q^{73}+7 q^{72}-19 q^{71}-12 q^{69}-12 q^{68}+11 q^{67}+15 q^{66}+18 q^{65}-3 q^{64}+q^{63}-16 q^{62}-22 q^{61}+2 q^{60}+6 q^{59}+14 q^{58}+5 q^{57}+8 q^{56}-7 q^{55}-16 q^{54}+2 q^{53}-q^{52}+6 q^{51}+q^{50}+5 q^{49}-5 q^{48}-10 q^{47}+6 q^{46}+6 q^{44}+q^{43}+3 q^{42}-6 q^{41}-9 q^{40}+4 q^{39}-q^{38}+4 q^{37}+3 q^{36}+4 q^{35}-3 q^{34}-5 q^{33}+2 q^{32}-2 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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coloured_jones_7 = | |
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<table> |
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computer_talk = |
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<tr valign=top> |
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<table> |
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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<tr valign=top> |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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</tr> |
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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 colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr> |
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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>PD[Knot[10, 128]]</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[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[9, 17, 10, 16], X[5, 15, 6, 14], |
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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, 128]]</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[8, 4, 9, 3], X[9, 17, 10, 16], X[5, 15, 6, 14], |
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X[15, 7, 16, 6], X[13, 1, 14, 20], X[19, 11, 20, 10], |
X[15, 7, 16, 6], X[13, 1, 14, 20], X[19, 11, 20, 10], |
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X[11, 19, 12, 18], X[17, 13, 18, 12], X[2, 8, 3, 7]]</nowiki></ |
X[11, 19, 12, 18], X[17, 13, 18, 12], X[2, 8, 3, 7]]</nowiki></code></td></tr> |
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</table> |
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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[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 128]]</nowiki></pre></td></tr> |
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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, 128]]</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, 10, -2, -3, 7, -8, 9, -6, 4, -5, 3, -9, |
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8, -7, 6]</nowiki></ |
8, -7, 6]</nowiki></code></td></tr> |
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</table> |
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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[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>DTCode[Knot[10, 128]]</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, 128]]</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>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>br = BR[Knot[10, 128]]</nowiki></pre></td></tr> |
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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, 8, -14, 2, -16, -18, -20, -6, -12, -10]</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[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{First[br], Crossings[br]}</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[6]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{4, 11}</nowiki></pre></td></tr> |
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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, 128]]</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[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>4</nowiki></pre></td></tr> |
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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, 2, 2, 3, -2, 3}]</nowiki></code></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[10, 128]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_128_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr> |
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</table> |
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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[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[10, 128]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr> |
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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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<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>alex = Alexander[Knot[10, 128]][t]</nowiki></pre></td></tr> |
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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, 128]]</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, 128]]]</nowiki></code></td></tr> |
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<tr align=left><td></td><td>[[Image:10_128_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, 128]]&) /@ { |
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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, 1}</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, 128]][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 1 2 3 |
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1 + -- - -- + - + t - 3 t + 2 t |
1 + -- - -- + - + 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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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[10, 128]][z]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</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>Conway[Knot[10, 128]][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[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 + 7 z + 9 z + 2 z</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[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[Knot[10, 128]], KnotSignature[Knot[10, 128]]}</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[13]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{11, 6}</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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<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> 3 4 5 6 7 8 9 10 |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td> |
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q - q + 2 q - q + 2 q - 2 q + q - q</nowiki></pre></td></tr> |
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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, 128]}</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[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][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>Out[15]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 128]}</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, 128]], KnotSignature[Knot[10, 128]]}</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[16]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 10 14 16 18 20 22 24 26 28 30 32 |
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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>{11, 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, 128]][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 - q + 2 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[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, 128]}</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, 128]][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 16 18 20 22 24 26 28 30 32 |
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q + q + q + 2 q + q + q + q - q - q - 2 q - q - |
q + q + q + 2 q + q + q + q - q - q - 2 q - q - |
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34 38 |
34 38 |
||
q + q</nowiki></ |
q + q</nowiki></code></td></tr> |
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</table> |
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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[17]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Knot[10, 128]][a, z]</nowiki></pre></td></tr> |
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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, 128]][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 4 2 2 5 z 6 z 6 z z 5 z 5 z z z |
-12 4 2 2 5 z 6 z 6 z z 5 z 5 z z z |
||
a - --- + -- + -- - ---- + ---- + ---- - --- + ---- + ---- + -- + -- |
a - --- + -- + -- - ---- + ---- + ---- - --- + ---- + ---- + -- + -- |
||
10 8 6 10 8 6 10 8 6 8 6 |
10 8 6 10 8 6 10 8 6 8 6 |
||
a a a a a a a a a a a</nowiki></ |
a 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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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[18]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 128]][a, z]</nowiki></pre></td></tr> |
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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, 128]][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 3 |
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-12 4 2 2 6 z 5 z z 11 z 5 z 6 z 11 z |
-12 4 2 2 6 z 5 z z 11 z 5 z 6 z 11 z |
||
a + --- + -- - -- - --- - --- + -- - ----- - ---- + ---- + ----- + |
a + --- + -- - -- - --- - --- + -- - ----- - ---- + ---- + ----- + |
||
Line 166: | Line 207: | ||
---- + -- + --- + ---- + -- + --- + -- |
---- + -- + --- + ---- + -- + --- + -- |
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8 6 11 9 7 10 8 |
8 6 11 9 7 10 8 |
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a a a a a a a</nowiki></ |
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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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[19]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[10, 128]], Vassiliev[3][Knot[10, 128]]}</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</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>{Vassiliev[2][Knot[10, 128]], Vassiliev[3][Knot[10, 128]]}</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>In[20]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[10, 128]][q, t]</nowiki></pre></td></tr> |
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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>{7, 17}</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, 128]][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 11 4 13 4 |
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q + q + q t + q t + q t + q t + q t + q t + 2 q t + |
q + q + q t + q t + q t + q t + q t + q t + 2 q t + |
||
15 4 15 5 17 5 17 6 21 7 |
15 4 15 5 17 5 17 6 21 7 |
||
q t + q t + 2 q t + q t + q t</nowiki></ |
q t + q t + 2 q t + q t + q t</nowiki></code></td></tr> |
||
</table> |
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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[21]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>ColouredJones[Knot[10, 128], 2][q]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</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>ColouredJones[Knot[10, 128], 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 16 |
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q - q + 3 q - 2 q - q + 4 q - 2 q - q + 2 q - q - |
q - q + 3 q - 2 q - q + 4 q - 2 q - q + 2 q - q - |
||
19 20 21 23 24 26 27 28 29 |
19 20 21 23 24 26 27 28 29 |
||
q + q - 2 q + 2 q - 2 q + 2 q - q - q + q</nowiki></ |
q + q - 2 q + 2 q - 2 q + 2 q - q - q + q</nowiki></code></td></tr> |
||
</table> }} |
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</table> |
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{| width=100% |
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|align=left|See/edit the [[Rolfsen_Splice_Template]]. |
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Back to the [[#top|top]]. |
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|align=right|{{Knot Navigation Links|ext=gif}} |
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|} |
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[[Category:Knot Page]] |
Latest revision as of 16:58, 1 September 2005
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(KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 128's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X2837 |
Gauss code | 1, -10, 2, -1, -4, 5, 10, -2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6 |
Dowker-Thistlethwaite code | 4 8 -14 2 -16 -18 -20 -6 -12 -10 |
Conway Notation | [32,3,2-] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
[{5, 8}, {4, 6}, {3, 7}, {1, 5}, {9, 4}, {8, 10}, {2, 9}, {10, 3}, {7, 2}, {6, 1}] |
[edit Notes on presentations of 10 128]
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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 128"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X2837 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -10, 2, -1, -4, 5, 10, -2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 8 -14 2 -16 -18 -20 -6 -12 -10 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[32,3,2-] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 4, 11, 4 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{5, 8}, {4, 6}, {3, 7}, {1, 5}, {9, 4}, {8, 10}, {2, 9}, {10, 3}, {7, 2}, {6, 1}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
A1 Invariants.
Weight | Invariant |
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1 | |
2 | |
3 |
A2 Invariants.
Weight | Invariant |
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1,0 | |
1,1 | |
2,0 |
A3 Invariants.
Weight | Invariant |
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0,1,0 | |
1,0,0 |
A4 Invariants.
Weight | Invariant |
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0,1,0,0 | |
1,0,0,0 |
B2 Invariants.
Weight | Invariant |
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0,1 | |
1,0 |
D4 Invariants.
Weight | Invariant |
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1,0,0,0 |
G2 Invariants.
Weight | Invariant |
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1,0 |
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 128"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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In[5]:=
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Conway[K][z]
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Out[5]=
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In[6]:=
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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.
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Out[6]=
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 11, 6 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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"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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 128"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ , } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{} |
Vassiliev invariants
V2 and V3: | (7, 17) |
V2,1 through V6,9: |
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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 10 128. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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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. |
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