9 18
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![]() (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 18's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X3,12,4,13 X5,14,6,15 X9,18,10,1 X17,6,18,7 X7,16,8,17 X15,8,16,9 X13,10,14,11 X11,2,12,3 |
| Gauss code | -1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 2 10 8 6 |
| Conway Notation | [3222] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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![]() [{11, 5}, {1, 9}, {8, 10}, {9, 11}, {10, 6}, {5, 7}, {4, 8}, {6, 3}, {2, 4}, {3, 1}, {7, 2}] |
[edit Notes on presentations of 9 18]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 18"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3,12,4,13 X5,14,6,15 X9,18,10,1 X17,6,18,7 X7,16,8,17 X15,8,16,9 X13,10,14,11 X11,2,12,3 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 14 16 18 2 10 8 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[3222] |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(4,\{-1,-1,-1,-2,1,-2,-2,-2,-3,2,-3\})} |
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[{11, 5}, {1, 9}, {8, 10}, {9, 11}, {10, 6}, {5, 7}, {4, 8}, {6, 3}, {2, 4}, {3, 1}, {7, 2}] |
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
| Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 t^2-10 t+13-10 t^{-1} +4 t^{-2} } |
| Conway polynomial | |
| 2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
| Determinant and Signature | { 41, -4 } |
| Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11} } |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^{10}-a^{10}+z^4 a^8+z^2 a^8+2 z^4 a^6+4 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4} |
| Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-5 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-3 z^5 a^{11}+z^8 a^{10}+z^6 a^{10}-2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+4 z^7 a^9-5 z^5 a^9+z^3 a^9+z^8 a^8+2 z^6 a^8-2 z^4 a^8+2 z^7 a^7+z^5 a^7-4 z^3 a^7+2 z a^7+3 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4} |
| The A2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{34}-2 q^{28}+q^{26}+q^{20}-q^{18}+2 q^{16}+q^{12}+2 q^{10}-q^8+q^6} |
| The G2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{176}-q^{174}+3 q^{172}-4 q^{170}+3 q^{168}-2 q^{166}-2 q^{164}+10 q^{162}-15 q^{160}+18 q^{158}-15 q^{156}+5 q^{154}+9 q^{152}-26 q^{150}+36 q^{148}-37 q^{146}+23 q^{144}-2 q^{142}-24 q^{140}+37 q^{138}-38 q^{136}+27 q^{134}-6 q^{132}-16 q^{130}+24 q^{128}-23 q^{126}+5 q^{124}+16 q^{122}-29 q^{120}+31 q^{118}-16 q^{116}-7 q^{114}+34 q^{112}-51 q^{110}+54 q^{108}-40 q^{106}+9 q^{104}+23 q^{102}-48 q^{100}+58 q^{98}-47 q^{96}+23 q^{94}+4 q^{92}-26 q^{90}+32 q^{88}-25 q^{86}+4 q^{84}+16 q^{82}-23 q^{80}+20 q^{78}-2 q^{76}-18 q^{74}+35 q^{72}-36 q^{70}+29 q^{68}-12 q^{66}-11 q^{64}+29 q^{62}-34 q^{60}+33 q^{58}-18 q^{56}+6 q^{54}+6 q^{52}-14 q^{50}+16 q^{48}-13 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30}} |
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{23}+q^{21}-2 q^{19}+2 q^{17}-q^{15}+q^{11}-q^9+3 q^7-q^5+q^3} |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{64}-q^{62}-q^{60}+4 q^{58}-2 q^{56}-6 q^{54}+8 q^{52}+q^{50}-10 q^{48}+8 q^{46}+4 q^{44}-11 q^{42}+2 q^{40}+5 q^{38}-4 q^{36}-3 q^{34}+3 q^{32}+5 q^{30}-8 q^{28}-q^{26}+11 q^{24}-8 q^{22}-4 q^{20}+11 q^{18}-3 q^{16}-3 q^{14}+5 q^{12}-q^8+q^6} |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{123}+q^{121}+q^{119}-q^{117}-3 q^{115}+2 q^{113}+7 q^{111}-q^{109}-13 q^{107}-3 q^{105}+18 q^{103}+11 q^{101}-22 q^{99}-21 q^{97}+21 q^{95}+31 q^{93}-14 q^{91}-39 q^{89}+9 q^{87}+42 q^{85}+q^{83}-42 q^{81}-10 q^{79}+38 q^{77}+15 q^{75}-30 q^{73}-18 q^{71}+19 q^{69}+21 q^{67}-8 q^{65}-23 q^{63}-7 q^{61}+22 q^{59}+18 q^{57}-20 q^{55}-33 q^{53}+17 q^{51}+41 q^{49}-9 q^{47}-45 q^{45}+2 q^{43}+41 q^{41}+6 q^{39}-35 q^{37}-11 q^{35}+27 q^{33}+10 q^{31}-14 q^{29}-9 q^{27}+10 q^{25}+7 q^{23}-4 q^{21}-3 q^{19}+3 q^{17}+2 q^{15}-q^{11}+q^9} |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{200}-q^{198}-q^{196}+q^{194}+3 q^{190}-4 q^{188}-5 q^{186}+3 q^{184}+4 q^{182}+15 q^{180}-7 q^{178}-23 q^{176}-10 q^{174}+7 q^{172}+50 q^{170}+14 q^{168}-39 q^{166}-55 q^{164}-30 q^{162}+83 q^{160}+77 q^{158}-4 q^{156}-94 q^{154}-117 q^{152}+55 q^{150}+132 q^{148}+88 q^{146}-68 q^{144}-192 q^{142}-29 q^{140}+122 q^{138}+164 q^{136}+6 q^{134}-197 q^{132}-104 q^{130}+64 q^{128}+181 q^{126}+66 q^{124}-146 q^{122}-124 q^{120}+9 q^{118}+142 q^{116}+87 q^{114}-72 q^{112}-111 q^{110}-38 q^{108}+83 q^{106}+93 q^{104}+10 q^{102}-86 q^{100}-88 q^{98}+4 q^{96}+96 q^{94}+110 q^{92}-43 q^{90}-137 q^{88}-92 q^{86}+73 q^{84}+195 q^{82}+30 q^{80}-137 q^{78}-171 q^{76}+6 q^{74}+209 q^{72}+94 q^{70}-69 q^{68}-172 q^{66}-64 q^{64}+138 q^{62}+99 q^{60}+6 q^{58}-106 q^{56}-76 q^{54}+56 q^{52}+52 q^{50}+30 q^{48}-37 q^{46}-43 q^{44}+15 q^{42}+14 q^{40}+19 q^{38}-9 q^{36}-15 q^{34}+7 q^{32}+2 q^{30}+7 q^{28}-2 q^{26}-4 q^{24}+3 q^{22}+2 q^{18}-q^{14}+q^{12}} |
| 5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{295}+q^{293}+q^{291}-q^{289}-q^{283}+2 q^{281}+4 q^{279}-3 q^{277}-7 q^{275}-4 q^{273}-q^{271}+11 q^{269}+20 q^{267}+8 q^{265}-20 q^{263}-39 q^{261}-29 q^{259}+14 q^{257}+67 q^{255}+74 q^{253}+13 q^{251}-86 q^{249}-137 q^{247}-76 q^{245}+73 q^{243}+199 q^{241}+182 q^{239}-5 q^{237}-239 q^{235}-308 q^{233}-121 q^{231}+209 q^{229}+422 q^{227}+305 q^{225}-100 q^{223}-487 q^{221}-502 q^{219}-79 q^{217}+462 q^{215}+665 q^{213}+314 q^{211}-353 q^{209}-771 q^{207}-532 q^{205}+180 q^{203}+772 q^{201}+718 q^{199}+25 q^{197}-709 q^{195}-827 q^{193}-204 q^{191}+583 q^{189}+846 q^{187}+349 q^{185}-443 q^{183}-803 q^{181}-430 q^{179}+309 q^{177}+714 q^{175}+452 q^{173}-186 q^{171}-605 q^{169}-448 q^{167}+91 q^{165}+500 q^{163}+427 q^{161}-7 q^{159}-389 q^{157}-415 q^{155}-89 q^{153}+291 q^{151}+419 q^{149}+199 q^{147}-177 q^{145}-426 q^{143}-345 q^{141}+44 q^{139}+437 q^{137}+497 q^{135}+130 q^{133}-414 q^{131}-656 q^{129}-324 q^{127}+342 q^{125}+771 q^{123}+538 q^{121}-213 q^{119}-828 q^{117}-719 q^{115}+35 q^{113}+775 q^{111}+850 q^{109}+160 q^{107}-656 q^{105}-871 q^{103}-332 q^{101}+455 q^{99}+800 q^{97}+446 q^{95}-249 q^{93}-656 q^{91}-466 q^{89}+70 q^{87}+461 q^{85}+418 q^{83}+59 q^{81}-290 q^{79}-326 q^{77}-96 q^{75}+139 q^{73}+214 q^{71}+107 q^{69}-55 q^{67}-129 q^{65}-76 q^{63}+13 q^{61}+62 q^{59}+49 q^{57}-27 q^{53}-23 q^{51}-3 q^{49}+15 q^{47}+11 q^{45}-q^{43}-6 q^{41}-q^{37}+4 q^{35}+5 q^{33}-2 q^{31}-2 q^{29}+2 q^{27}+2 q^{21}-q^{17}+q^{15}} |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{34}-2 q^{28}+q^{26}+q^{20}-q^{18}+2 q^{16}+q^{12}+2 q^{10}-q^8+q^6} |
| 1,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{92}-2 q^{90}+6 q^{88}-12 q^{86}+23 q^{84}-40 q^{82}+60 q^{80}-84 q^{78}+110 q^{76}-132 q^{74}+142 q^{72}-136 q^{70}+114 q^{68}-76 q^{66}+18 q^{64}+52 q^{62}-119 q^{60}+184 q^{58}-236 q^{56}+268 q^{54}-280 q^{52}+260 q^{50}-224 q^{48}+162 q^{46}-97 q^{44}+24 q^{42}+38 q^{40}-92 q^{38}+123 q^{36}-134 q^{34}+136 q^{32}-120 q^{30}+103 q^{28}-72 q^{26}+56 q^{24}-34 q^{22}+23 q^{20}-10 q^{18}+6 q^{16}-2 q^{14}+q^{12}} |
| 2,0 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{74}-q^{72}+q^{70}+2 q^{68}-4 q^{66}+2 q^{64}+4 q^{62}-8 q^{60}+3 q^{58}+5 q^{56}-10 q^{54}+q^{52}+5 q^{50}-6 q^{48}-2 q^{46}+2 q^{44}-3 q^{40}-2 q^{38}+7 q^{36}-2 q^{34}-5 q^{32}+10 q^{30}-6 q^{26}+9 q^{24}+q^{22}-3 q^{20}+4 q^{18}+q^{16}-q^{14}+q^{12}} |
| 1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{45}-q^{41}-2 q^{37}+q^{35}-q^{33}+q^{31}+q^{27}+2 q^{21}+2 q^{17}+2 q^{13}-q^{11}+q^9} |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{96}-q^{92}+2 q^{90}+3 q^{88}-2 q^{86}-q^{84}+5 q^{82}+3 q^{80}-6 q^{78}-4 q^{76}+4 q^{74}-3 q^{72}-10 q^{70}+4 q^{66}-5 q^{64}-3 q^{62}+4 q^{60}-q^{58}-6 q^{56}+q^{54}+4 q^{52}-5 q^{50}-2 q^{48}+9 q^{46}+2 q^{44}-5 q^{42}+4 q^{40}+8 q^{38}-2 q^{34}+4 q^{32}+5 q^{30}-q^{28}+3 q^{24}+q^{22}-q^{20}+q^{18}} |
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{56}-q^{52}-q^{50}-2 q^{46}+q^{44}-q^{42}+q^{38}+q^{34}+q^{30}+2 q^{26}+2 q^{22}+q^{20}+2 q^{16}-q^{14}+q^{12}} |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{74}+q^{72}-3 q^{70}+4 q^{68}-6 q^{66}+8 q^{64}-8 q^{62}+8 q^{60}-7 q^{58}+5 q^{56}-2 q^{54}-3 q^{52}+7 q^{50}-12 q^{48}+14 q^{46}-16 q^{44}+16 q^{42}-15 q^{40}+12 q^{38}-7 q^{36}+4 q^{34}+q^{32}-4 q^{30}+8 q^{28}-8 q^{26}+9 q^{24}-7 q^{22}+7 q^{20}-4 q^{18}+3 q^{16}-q^{14}+q^{12}} |
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{120}-q^{116}-q^{114}+2 q^{112}+3 q^{110}-q^{108}-5 q^{106}-2 q^{104}+6 q^{102}+6 q^{100}-4 q^{98}-9 q^{96}-q^{94}+9 q^{92}+5 q^{90}-7 q^{88}-8 q^{86}+2 q^{84}+7 q^{82}-7 q^{78}-2 q^{76}+5 q^{74}+2 q^{72}-6 q^{70}-4 q^{68}+4 q^{66}+5 q^{64}-3 q^{62}-6 q^{60}+2 q^{58}+7 q^{56}+q^{54}-8 q^{52}-3 q^{50}+8 q^{48}+8 q^{46}-4 q^{44}-8 q^{42}+9 q^{38}+5 q^{36}-3 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+q^{18}} |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{102}-q^{100}+2 q^{98}-2 q^{96}+4 q^{94}-5 q^{92}+5 q^{90}-6 q^{88}+7 q^{86}-7 q^{84}+5 q^{82}-5 q^{80}+4 q^{78}-3 q^{76}-3 q^{74}+2 q^{72}-5 q^{70}+7 q^{68}-12 q^{66}+10 q^{64}-12 q^{62}+13 q^{60}-13 q^{58}+9 q^{56}-10 q^{54}+9 q^{52}-4 q^{50}+3 q^{48}-q^{46}+7 q^{42}-4 q^{40}+6 q^{38}-6 q^{36}+9 q^{34}-4 q^{32}+6 q^{30}-4 q^{28}+5 q^{26}-q^{24}+2 q^{22}-q^{20}+q^{18}} |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{176}-q^{174}+3 q^{172}-4 q^{170}+3 q^{168}-2 q^{166}-2 q^{164}+10 q^{162}-15 q^{160}+18 q^{158}-15 q^{156}+5 q^{154}+9 q^{152}-26 q^{150}+36 q^{148}-37 q^{146}+23 q^{144}-2 q^{142}-24 q^{140}+37 q^{138}-38 q^{136}+27 q^{134}-6 q^{132}-16 q^{130}+24 q^{128}-23 q^{126}+5 q^{124}+16 q^{122}-29 q^{120}+31 q^{118}-16 q^{116}-7 q^{114}+34 q^{112}-51 q^{110}+54 q^{108}-40 q^{106}+9 q^{104}+23 q^{102}-48 q^{100}+58 q^{98}-47 q^{96}+23 q^{94}+4 q^{92}-26 q^{90}+32 q^{88}-25 q^{86}+4 q^{84}+16 q^{82}-23 q^{80}+20 q^{78}-2 q^{76}-18 q^{74}+35 q^{72}-36 q^{70}+29 q^{68}-12 q^{66}-11 q^{64}+29 q^{62}-34 q^{60}+33 q^{58}-18 q^{56}+6 q^{54}+6 q^{52}-14 q^{50}+16 q^{48}-13 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30}} |
.
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["9 18"];
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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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 t^2-10 t+13-10 t^{-1} +4 t^{-2} } |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 41, -4 } |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11} } |
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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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^{10}-a^{10}+z^4 a^8+z^2 a^8+2 z^4 a^6+4 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4} |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-5 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-3 z^5 a^{11}+z^8 a^{10}+z^6 a^{10}-2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+4 z^7 a^9-5 z^5 a^9+z^3 a^9+z^8 a^8+2 z^6 a^8-2 z^4 a^8+2 z^7 a^7+z^5 a^7-4 z^3 a^7+2 z a^7+3 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a246,}
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["9 18"];
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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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{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 t^2-10 t+13-10 t^{-1} +4 t^{-2} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11} } } |
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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{K11a246,} |
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: | (6, -15) |
| 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -4 is the signature of 9 18. 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
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n} |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-4} -2 q^{-5} + q^{-6} +6 q^{-7} -10 q^{-8} + q^{-9} +20 q^{-10} -25 q^{-11} -3 q^{-12} +39 q^{-13} -37 q^{-14} -10 q^{-15} +52 q^{-16} -39 q^{-17} -16 q^{-18} +51 q^{-19} -30 q^{-20} -19 q^{-21} +38 q^{-22} -15 q^{-23} -15 q^{-24} +20 q^{-25} -4 q^{-26} -8 q^{-27} +6 q^{-28} -2 q^{-30} + q^{-31} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} -2 q^{-7} + q^{-8} +2 q^{-9} +2 q^{-10} -8 q^{-11} +13 q^{-13} +5 q^{-14} -27 q^{-15} -5 q^{-16} +37 q^{-17} +22 q^{-18} -65 q^{-19} -29 q^{-20} +78 q^{-21} +57 q^{-22} -104 q^{-23} -76 q^{-24} +114 q^{-25} +107 q^{-26} -128 q^{-27} -126 q^{-28} +127 q^{-29} +145 q^{-30} -124 q^{-31} -155 q^{-32} +111 q^{-33} +160 q^{-34} -95 q^{-35} -157 q^{-36} +74 q^{-37} +148 q^{-38} -50 q^{-39} -134 q^{-40} +26 q^{-41} +116 q^{-42} -7 q^{-43} -93 q^{-44} -7 q^{-45} +68 q^{-46} +18 q^{-47} -48 q^{-48} -17 q^{-49} +26 q^{-50} +17 q^{-51} -15 q^{-52} -10 q^{-53} +5 q^{-54} +7 q^{-55} -3 q^{-56} -2 q^{-57} +2 q^{-59} - q^{-60} } |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-8} -2 q^{-9} + q^{-10} +2 q^{-11} -2 q^{-12} +4 q^{-13} -9 q^{-14} +3 q^{-15} +11 q^{-16} -7 q^{-17} +9 q^{-18} -31 q^{-19} +9 q^{-20} +39 q^{-21} -12 q^{-22} +10 q^{-23} -89 q^{-24} +15 q^{-25} +106 q^{-26} +10 q^{-27} +14 q^{-28} -221 q^{-29} -15 q^{-30} +218 q^{-31} +103 q^{-32} +53 q^{-33} -423 q^{-34} -123 q^{-35} +321 q^{-36} +266 q^{-37} +168 q^{-38} -626 q^{-39} -300 q^{-40} +355 q^{-41} +433 q^{-42} +333 q^{-43} -748 q^{-44} -465 q^{-45} +310 q^{-46} +527 q^{-47} +486 q^{-48} -762 q^{-49} -557 q^{-50} +218 q^{-51} +529 q^{-52} +582 q^{-53} -679 q^{-54} -567 q^{-55} +97 q^{-56} +456 q^{-57} +621 q^{-58} -520 q^{-59} -512 q^{-60} -36 q^{-61} +323 q^{-62} +599 q^{-63} -308 q^{-64} -397 q^{-65} -153 q^{-66} +155 q^{-67} +506 q^{-68} -105 q^{-69} -239 q^{-70} -195 q^{-71} +4 q^{-72} +343 q^{-73} +19 q^{-74} -83 q^{-75} -151 q^{-76} -73 q^{-77} +171 q^{-78} +42 q^{-79} +7 q^{-80} -70 q^{-81} -67 q^{-82} +58 q^{-83} +17 q^{-84} +23 q^{-85} -17 q^{-86} -31 q^{-87} +15 q^{-88} +10 q^{-90} - q^{-91} -9 q^{-92} +4 q^{-93} - q^{-94} +2 q^{-95} -2 q^{-97} + q^{-98} } |
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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