10 91

10_90

10_92

(KnotPlot image)

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[edit 10 91 Quick Notes]

[edit 10 91 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_20,6,1,5 X_16,9,17,10 X_10,3,11,4 X_2,18,3,17 X_14,7,15,8 X_8,15,9,16 X_12,20,13,19 X_18,12,19,11 X_4,13,5,14
Gauss code	1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5, -9, 8, -2
Dowker-Thistlethwaite code	6 10 20 14 16 18 4 8 2 12
Conway Notation	[.3.2.20]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 10, width is 3,

Braid index is 3

[{3, 6}, {2, 5}, {1, 3}, {7, 14}, {6, 11}, {8, 13}, {9, 7}, {10, 8}, {4, 9}, {5, 10}, {14, 12}, {11, 2}, {13, 4}, {12, 1}]

[edit Notes on presentations of 10 91]

Computer Talk

The above data is available with the Mathematica package 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.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 91"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= X₆₂₇₁ X_20,6,1,5 X_16,9,17,10 X_10,3,11,4 X_2,18,3,17 X_14,7,15,8 X_8,15,9,16 X_12,20,13,19 X_18,12,19,11 X_4,13,5,14

In[5]:= GaussCode[K]

Out[5]= 1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5, -9, 8, -2

In[6]:= DTCode[K]

Out[6]= 6 10 20 14 16 18 4 8 2 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [.3.2.20]

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]= 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}(3,\{-1,-1,-1,2,-1,2,2,-1,2,2\})}

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]= { 3, 10, 3 }

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[{3, 6}, {2, 5}, {1, 3}, {7, 14}, {6, 11}, {8, 13}, {9, 7}, {10, 8}, {4, 9}, {5, 10}, {14, 12}, {11, 2}, {13, 4}, {12, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	13.487
A-Polynomial	See Data:10 91/A-polynomial

[edit Notes for 10 91's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	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}
Topological 4 genus	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}
Concordance genus	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}
Rasmussen s-Invariant	0

[edit Notes for 10 91's four dimensional invariants]

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 t^4-4 t^3+9 t^2-14 t+17-14 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-4} }
Conway 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 z^8+4 z^6+5 z^4+2 z^2+1}
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	{ 73, 0 }
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^5+3 q^4-6 q^3+9 q^2-11 q+13-11 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }
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^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-4 z^4 a^{-2} +13 z^4-5 a^2 z^2-5 z^2 a^{-2} +12 z^2-2 a^2-2 a^{-2} +5}
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 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+5 z^8 a^{-2} +9 z^8+4 a^3 z^7+a z^7+2 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-7 a^2 z^6-13 z^6 a^{-2} +3 z^6 a^{-4} -26 z^6+a^5 z^5-6 a^3 z^5-7 a z^5-13 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+7 a^2 z^4+16 z^4 a^{-2} -6 z^4 a^{-4} +35 z^4-2 a^5 z^3+9 a z^3+18 z^3 a^{-1} +9 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-7 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -19 z^2+a^5 z-4 a z-6 z a^{-1} -3 z a^{-3} +2 a^2+2 a^{-2} +5}
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^{14}+q^{12}-2 q^{10}+q^8-q^4+4 q^2-1+4 q^{-2} - q^{-4} + q^{-8} -2 q^{-10} + q^{-12} - q^{-14} }
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^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-5 q^{70}-3 q^{68}+16 q^{66}-27 q^{64}+36 q^{62}-38 q^{60}+23 q^{58}-q^{56}-34 q^{54}+70 q^{52}-92 q^{50}+95 q^{48}-66 q^{46}+4 q^{44}+68 q^{42}-130 q^{40}+151 q^{38}-122 q^{36}+46 q^{34}+44 q^{32}-116 q^{30}+138 q^{28}-93 q^{26}+9 q^{24}+77 q^{22}-122 q^{20}+97 q^{18}-19 q^{16}-82 q^{14}+161 q^{12}-175 q^{10}+132 q^8-27 q^6-94 q^4+190 q^2-223+188 q^{-2} -92 q^{-4} -27 q^{-6} +132 q^{-8} -176 q^{-10} +165 q^{-12} -89 q^{-14} -10 q^{-16} +93 q^{-18} -126 q^{-20} +87 q^{-22} -3 q^{-24} -84 q^{-26} +137 q^{-28} -124 q^{-30} +54 q^{-32} +40 q^{-34} -124 q^{-36} +160 q^{-38} -142 q^{-40} +74 q^{-42} +7 q^{-44} -77 q^{-46} +108 q^{-48} -102 q^{-50} +72 q^{-52} -28 q^{-54} -10 q^{-56} +32 q^{-58} -42 q^{-60} +36 q^{-62} -23 q^{-64} +12 q^{-66} -6 q^{-70} +7 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }

Further Quantum Invariants

Further quantum knot invariants for 10_91.

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^{11}+2 q^9-3 q^7+3 q^5-2 q^3+2 q+2 q^{-1} -2 q^{-3} +3 q^{-5} -3 q^{-7} +2 q^{-9} - q^{-11} }
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^{32}-2 q^{30}-q^{28}+6 q^{26}-6 q^{24}-5 q^{22}+17 q^{20}-8 q^{18}-17 q^{16}+24 q^{14}-25 q^{10}+17 q^8+10 q^6-18 q^4+q^2+14+ q^{-2} -17 q^{-4} +10 q^{-6} +17 q^{-8} -25 q^{-10} + q^{-12} +24 q^{-14} -18 q^{-16} -8 q^{-18} +18 q^{-20} -5 q^{-22} -8 q^{-24} +6 q^{-26} -2 q^{-30} + q^{-32} }
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^{63}+2 q^{61}+q^{59}-2 q^{57}-3 q^{55}+3 q^{53}+5 q^{51}-9 q^{49}-7 q^{47}+18 q^{45}+18 q^{43}-28 q^{41}-43 q^{39}+32 q^{37}+77 q^{35}-19 q^{33}-111 q^{31}-16 q^{29}+135 q^{27}+61 q^{25}-134 q^{23}-107 q^{21}+113 q^{19}+136 q^{17}-74 q^{15}-147 q^{13}+33 q^{11}+134 q^9+11 q^7-114 q^5-43 q^3+82 q+78 q^{-1} -47 q^{-3} -108 q^{-5} +18 q^{-7} +130 q^{-9} +22 q^{-11} -147 q^{-13} -60 q^{-15} +144 q^{-17} +100 q^{-19} -123 q^{-21} -127 q^{-23} +81 q^{-25} +138 q^{-27} -33 q^{-29} -122 q^{-31} -12 q^{-33} +92 q^{-35} +34 q^{-37} -55 q^{-39} -38 q^{-41} +23 q^{-43} +30 q^{-45} -7 q^{-47} -17 q^{-49} +2 q^{-51} +7 q^{-53} -3 q^{-57} +2 q^{-61} - q^{-63} }
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^{104}-2 q^{102}-q^{100}+2 q^{98}-q^{96}+6 q^{94}-3 q^{92}-q^{90}+3 q^{88}-15 q^{86}+4 q^{84}-9 q^{82}+19 q^{80}+48 q^{78}-16 q^{76}-39 q^{74}-104 q^{72}-6 q^{70}+168 q^{68}+139 q^{66}+21 q^{64}-307 q^{62}-297 q^{60}+114 q^{58}+432 q^{56}+462 q^{54}-242 q^{52}-737 q^{50}-435 q^{48}+355 q^{46}+1046 q^{44}+398 q^{42}-712 q^{40}-1097 q^{38}-314 q^{36}+1080 q^{34}+1080 q^{32}-83 q^{30}-1165 q^{28}-957 q^{26}+511 q^{24}+1151 q^{22}+518 q^{20}-669 q^{18}-1032 q^{16}-79 q^{14}+741 q^{12}+699 q^{10}-144 q^8-745 q^6-410 q^4+316 q^2+684+232 q^{-2} -476 q^{-4} -699 q^{-6} -32 q^{-8} +739 q^{-10} +644 q^{-12} -221 q^{-14} -1039 q^{-16} -505 q^{-18} +672 q^{-20} +1106 q^{-22} +288 q^{-24} -1105 q^{-26} -1051 q^{-28} +173 q^{-30} +1194 q^{-32} +915 q^{-34} -570 q^{-36} -1158 q^{-38} -528 q^{-40} +619 q^{-42} +1068 q^{-44} +183 q^{-46} -600 q^{-48} -745 q^{-50} -112 q^{-52} +579 q^{-54} +439 q^{-56} +35 q^{-58} -384 q^{-60} -324 q^{-62} +69 q^{-64} +201 q^{-66} +194 q^{-68} -39 q^{-70} -150 q^{-72} -56 q^{-74} +6 q^{-76} +81 q^{-78} +24 q^{-80} -26 q^{-82} -13 q^{-84} -15 q^{-86} +15 q^{-88} +6 q^{-90} -5 q^{-92} + q^{-94} -3 q^{-96} +3 q^{-98} -2 q^{-102} + q^{-104} }
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^{155}+2 q^{153}+q^{151}-2 q^{149}+q^{147}-2 q^{145}-6 q^{143}-q^{141}+7 q^{139}+7 q^{137}+14 q^{135}+9 q^{133}-21 q^{131}-42 q^{129}-34 q^{127}+6 q^{125}+67 q^{123}+114 q^{121}+68 q^{119}-86 q^{117}-230 q^{115}-229 q^{113}-10 q^{111}+337 q^{109}+541 q^{107}+312 q^{105}-327 q^{103}-914 q^{101}-900 q^{99}-60 q^{97}+1158 q^{95}+1774 q^{93}+986 q^{91}-942 q^{89}-2613 q^{87}-2479 q^{85}-132 q^{83}+2956 q^{81}+4253 q^{79}+2154 q^{77}-2281 q^{75}-5656 q^{73}-4835 q^{71}+270 q^{69}+6021 q^{67}+7522 q^{65}+2799 q^{63}-4910 q^{61}-9312 q^{59}-6264 q^{57}+2355 q^{55}+9646 q^{53}+9218 q^{51}+970 q^{49}-8359 q^{47}-10865 q^{45}-4266 q^{43}+5890 q^{41}+10936 q^{39}+6697 q^{37}-2954 q^{35}-9629 q^{33}-7863 q^{31}+307 q^{29}+7487 q^{27}+7789 q^{25}+1633 q^{23}-5202 q^{21}-6905 q^{19}-2664 q^{17}+3225 q^{15}+5673 q^{13}+3104 q^{11}-1790 q^9-4664 q^7-3236 q^5+875 q^3+4027 q+3531 q^{-1} -167 q^{-3} -3868 q^{-5} -4244 q^{-7} -626 q^{-9} +3982 q^{-11} +5379 q^{-13} +1868 q^{-15} -3940 q^{-17} -6868 q^{-19} -3681 q^{-21} +3417 q^{-23} +8205 q^{-25} +5965 q^{-27} -1999 q^{-29} -8937 q^{-31} -8380 q^{-33} -285 q^{-35} +8536 q^{-37} +10289 q^{-39} +3193 q^{-41} -6785 q^{-43} -11078 q^{-45} -6071 q^{-47} +3866 q^{-49} +10342 q^{-51} +8163 q^{-53} -435 q^{-55} -8081 q^{-57} -8823 q^{-59} -2710 q^{-61} +4870 q^{-63} +7921 q^{-65} +4688 q^{-67} -1549 q^{-69} -5752 q^{-71} -5199 q^{-73} -1054 q^{-75} +3147 q^{-77} +4391 q^{-79} +2384 q^{-81} -845 q^{-83} -2851 q^{-85} -2536 q^{-87} -611 q^{-89} +1307 q^{-91} +1917 q^{-93} +1124 q^{-95} -203 q^{-97} -1060 q^{-99} -1016 q^{-101} -324 q^{-103} +405 q^{-105} +645 q^{-107} +391 q^{-109} -35 q^{-111} -300 q^{-113} -280 q^{-115} -85 q^{-117} +108 q^{-119} +145 q^{-121} +67 q^{-123} -18 q^{-125} -54 q^{-127} -42 q^{-129} -4 q^{-131} +25 q^{-133} +14 q^{-135} -2 q^{-137} -3 q^{-139} -3 q^{-141} -3 q^{-143} +2 q^{-145} +3 q^{-147} -3 q^{-149} +2 q^{-153} - q^{-155} }
6	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^{216}-2 q^{214}-q^{212}+2 q^{210}-q^{208}+2 q^{206}+2 q^{204}+10 q^{202}-5 q^{200}-17 q^{198}-6 q^{196}-15 q^{194}+22 q^{190}+66 q^{188}+35 q^{186}-26 q^{184}-53 q^{182}-116 q^{180}-105 q^{178}-10 q^{176}+211 q^{174}+268 q^{172}+173 q^{170}+q^{168}-370 q^{166}-611 q^{164}-514 q^{162}+164 q^{160}+824 q^{158}+1177 q^{156}+1011 q^{154}-149 q^{152}-1661 q^{150}-2616 q^{148}-1812 q^{146}+264 q^{144}+2928 q^{142}+4822 q^{140}+3798 q^{138}-158 q^{136}-5401 q^{134}-8227 q^{132}-6897 q^{130}-564 q^{128}+8415 q^{126}+14040 q^{124}+12017 q^{122}+1420 q^{120}-12120 q^{118}-21811 q^{116}-19675 q^{114}-3864 q^{112}+17532 q^{110}+32243 q^{108}+29010 q^{106}+7550 q^{104}-23624 q^{102}-45160 q^{100}-41132 q^{98}-10684 q^{96}+31190 q^{94}+58901 q^{92}+54223 q^{90}+13506 q^{88}-40305 q^{86}-74157 q^{84}-64849 q^{82}-13776 q^{80}+49834 q^{78}+88138 q^{76}+72211 q^{74}+10264 q^{72}-61061 q^{70}-97033 q^{68}-73330 q^{66}-3373 q^{64}+71176 q^{62}+100362 q^{60}+67117 q^{58}-7919 q^{56}-76638 q^{54}-95921 q^{52}-54917 q^{50}+19905 q^{48}+77417 q^{46}+83879 q^{44}+37456 q^{42}-28664 q^{40}-71967 q^{38}-66850 q^{36}-19669 q^{34}+33992 q^{32}+61163 q^{30}+46783 q^{28}+5305 q^{26}-34721 q^{24}-47784 q^{22}-28341 q^{20}+5852 q^{18}+32118 q^{16}+33749 q^{14}+13526 q^{12}-13663 q^{10}-28520 q^8-22157 q^6-1010 q^4+20049 q^2+25575+12761 q^{-2} -10407 q^{-4} -26842 q^{-6} -24716 q^{-8} -3554 q^{-10} +23004 q^{-12} +35485 q^{-14} +23930 q^{-16} -7415 q^{-18} -37620 q^{-20} -45703 q^{-22} -21294 q^{-24} +21842 q^{-26} +54957 q^{-28} +54079 q^{-30} +14814 q^{-32} -38933 q^{-34} -72716 q^{-36} -58950 q^{-38} -3884 q^{-40} +58180 q^{-42} +86227 q^{-44} +58431 q^{-46} -9267 q^{-48} -75600 q^{-50} -94134 q^{-52} -52644 q^{-54} +23575 q^{-56} +86248 q^{-58} +94811 q^{-60} +44229 q^{-62} -34894 q^{-64} -89869 q^{-66} -88817 q^{-68} -33787 q^{-70} +39509 q^{-72} +85832 q^{-74} +78917 q^{-76} +24663 q^{-78} -38975 q^{-80} -75574 q^{-82} -65579 q^{-84} -19254 q^{-86} +33738 q^{-88} +62390 q^{-90} +52059 q^{-92} +15336 q^{-94} -25610 q^{-96} -47167 q^{-98} -40662 q^{-100} -12962 q^{-102} +17501 q^{-104} +33234 q^{-106} +29980 q^{-108} +11451 q^{-110} -9670 q^{-112} -22494 q^{-114} -21164 q^{-116} -9504 q^{-118} +4152 q^{-120} +13708 q^{-122} +14400 q^{-124} +7972 q^{-126} -1388 q^{-128} -7710 q^{-130} -9017 q^{-132} -6002 q^{-134} -348 q^{-136} +4107 q^{-138} +5613 q^{-140} +3770 q^{-142} +821 q^{-144} -1881 q^{-146} -3184 q^{-148} -2387 q^{-150} -647 q^{-152} +988 q^{-154} +1497 q^{-156} +1301 q^{-158} +468 q^{-160} -464 q^{-162} -779 q^{-164} -598 q^{-166} -136 q^{-168} +137 q^{-170} +336 q^{-172} +284 q^{-174} +40 q^{-176} -102 q^{-178} -129 q^{-180} -57 q^{-182} -31 q^{-184} +38 q^{-186} +60 q^{-188} +14 q^{-190} -11 q^{-192} -17 q^{-194} +2 q^{-196} -10 q^{-198} +11 q^{-202} -2 q^{-206} -3 q^{-208} +3 q^{-210} -2 q^{-214} + q^{-216} }

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^{14}+q^{12}-2 q^{10}+q^8-q^4+4 q^2-1+4 q^{-2} - q^{-4} + q^{-8} -2 q^{-10} + q^{-12} - q^{-14} }
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^{44}-4 q^{42}+12 q^{40}-28 q^{38}+52 q^{36}-86 q^{34}+134 q^{32}-200 q^{30}+273 q^{28}-356 q^{26}+454 q^{24}-534 q^{22}+577 q^{20}-576 q^{18}+508 q^{16}-358 q^{14}+124 q^{12}+156 q^{10}-458 q^8+750 q^6-996 q^4+1162 q^2-1226+1198 q^{-2} -1050 q^{-4} +834 q^{-6} -548 q^{-8} +246 q^{-10} +58 q^{-12} -320 q^{-14} +494 q^{-16} -598 q^{-18} +623 q^{-20} -584 q^{-22} +494 q^{-24} -390 q^{-26} +293 q^{-28} -202 q^{-30} +128 q^{-32} -80 q^{-34} +46 q^{-36} -22 q^{-38} +10 q^{-40} -4 q^{-42} + q^{-44} }
2,0	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^{38}-q^{36}-q^{34}+2 q^{32}-2 q^{30}-3 q^{28}+4 q^{26}+3 q^{24}-4 q^{22}-3 q^{20}+7 q^{18}+3 q^{16}-12 q^{14}+q^{12}+8 q^{10}-7 q^8-6 q^6+8 q^4+4 q^2-4+5 q^{-2} +9 q^{-4} -3 q^{-6} -5 q^{-8} +10 q^{-10} + q^{-12} -11 q^{-14} +4 q^{-16} +6 q^{-18} -5 q^{-20} -6 q^{-22} +3 q^{-24} +3 q^{-26} -4 q^{-28} -2 q^{-30} +3 q^{-32} - q^{-36} + q^{-38} }

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^{34}-2 q^{32}+q^{30}+4 q^{28}-7 q^{26}+2 q^{24}+9 q^{22}-15 q^{20}+2 q^{18}+15 q^{16}-21 q^{14}-q^{12}+16 q^{10}-14 q^8-2 q^6+13 q^4+3 q^2+2 q^{-2} +13 q^{-4} -3 q^{-6} -14 q^{-8} +15 q^{-10} - q^{-12} -21 q^{-14} +15 q^{-16} +2 q^{-18} -15 q^{-20} +10 q^{-22} +2 q^{-24} -6 q^{-26} +4 q^{-28} -2 q^{-32} + q^{-34} }
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^{17}+q^{15}-3 q^{13}+2 q^{11}-3 q^9+2 q^7-q^5+3 q^3+2 q+2 q^{-1} +3 q^{-3} - q^{-5} +2 q^{-7} -3 q^{-9} +2 q^{-11} -3 q^{-13} + q^{-15} - q^{-17} }
1,0,1	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^{56}-4 q^{54}+10 q^{52}-14 q^{50}+7 q^{48}+16 q^{46}-49 q^{44}+71 q^{42}-56 q^{40}-13 q^{38}+107 q^{36}-182 q^{34}+175 q^{32}-43 q^{30}-161 q^{28}+356 q^{26}-413 q^{24}+264 q^{22}+51 q^{20}-409 q^{18}+607 q^{16}-567 q^{14}+273 q^{12}+91 q^{10}-359 q^8+410 q^6-263 q^4+82 q^2+28+54 q^{-2} -201 q^{-4} +340 q^{-6} -272 q^{-8} +25 q^{-10} +331 q^{-12} -591 q^{-14} +606 q^{-16} -387 q^{-18} +6 q^{-20} +308 q^{-22} -460 q^{-24} +390 q^{-26} -188 q^{-28} -33 q^{-30} +176 q^{-32} -192 q^{-34} +126 q^{-36} -29 q^{-38} -42 q^{-40} +61 q^{-42} -46 q^{-44} +18 q^{-46} +4 q^{-48} -10 q^{-50} +8 q^{-52} -4 q^{-54} + q^{-56} }

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^{40}-q^{38}+4 q^{34}-2 q^{32}-3 q^{30}+8 q^{28}-2 q^{26}-11 q^{24}+5 q^{22}+6 q^{20}-16 q^{18}-9 q^{16}+9 q^{14}-2 q^{12}-17 q^{10}+7 q^8+21 q^6-8 q^4+5 q^2+29+4 q^{-2} -11 q^{-4} +18 q^{-6} +4 q^{-8} -19 q^{-10} -5 q^{-12} +8 q^{-14} -8 q^{-16} -15 q^{-18} +7 q^{-20} +7 q^{-22} -8 q^{-24} - q^{-26} +8 q^{-28} -3 q^{-30} -2 q^{-32} +3 q^{-34} - q^{-36} - q^{-38} + q^{-40} }

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^{20}+q^{18}-3 q^{16}+q^{14}-2 q^{12}-q^{10}+q^8-q^6+4 q^4+q^2+5+ q^{-2} +4 q^{-4} - q^{-6} + q^{-8} - q^{-10} -2 q^{-12} + q^{-14} -3 q^{-16} + q^{-18} - q^{-20} }

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^{34}+2 q^{32}-5 q^{30}+8 q^{28}-11 q^{26}+16 q^{24}-21 q^{22}+23 q^{20}-24 q^{18}+21 q^{16}-15 q^{14}+7 q^{12}+4 q^{10}-16 q^8+28 q^6-37 q^4+45 q^2-46+46 q^{-2} -37 q^{-4} +29 q^{-6} -16 q^{-8} +5 q^{-10} +7 q^{-12} -15 q^{-14} +21 q^{-16} -24 q^{-18} +23 q^{-20} -22 q^{-22} +16 q^{-24} -12 q^{-26} +8 q^{-28} -4 q^{-30} +2 q^{-32} - q^{-34} }

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^{56}-2 q^{52}-2 q^{50}+3 q^{48}+6 q^{46}-q^{44}-9 q^{42}-6 q^{40}+9 q^{38}+15 q^{36}-4 q^{34}-21 q^{32}-9 q^{30}+19 q^{28}+20 q^{26}-11 q^{24}-26 q^{22}-3 q^{20}+23 q^{18}+11 q^{16}-17 q^{14}-15 q^{12}+11 q^{10}+17 q^8-4 q^6-15 q^4+4 q^2+18+4 q^{-2} -15 q^{-4} -4 q^{-6} +16 q^{-8} +11 q^{-10} -15 q^{-12} -17 q^{-14} +10 q^{-16} +23 q^{-18} -2 q^{-20} -26 q^{-22} -12 q^{-24} +20 q^{-26} +20 q^{-28} -9 q^{-30} -22 q^{-32} -4 q^{-34} +16 q^{-36} +9 q^{-38} -6 q^{-40} -9 q^{-42} +6 q^{-46} +2 q^{-48} -2 q^{-50} -2 q^{-52} + q^{-56} }

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^{46}-2 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-9 q^{36}+10 q^{34}-13 q^{32}+17 q^{30}-19 q^{28}+17 q^{26}-18 q^{24}+17 q^{22}-16 q^{20}+5 q^{18}-6 q^{16}-2 q^{14}+8 q^{12}-19 q^{10}+22 q^8-23 q^6+39 q^4-30 q^2+40-30 q^{-2} +39 q^{-4} -24 q^{-6} +22 q^{-8} -20 q^{-10} +8 q^{-12} -3 q^{-14} -6 q^{-16} +4 q^{-18} -16 q^{-20} +17 q^{-22} -18 q^{-24} +18 q^{-26} -19 q^{-28} +18 q^{-30} -13 q^{-32} +11 q^{-34} -9 q^{-36} +7 q^{-38} -4 q^{-40} +2 q^{-42} -2 q^{-44} + q^{-46} }

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^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-5 q^{70}-3 q^{68}+16 q^{66}-27 q^{64}+36 q^{62}-38 q^{60}+23 q^{58}-q^{56}-34 q^{54}+70 q^{52}-92 q^{50}+95 q^{48}-66 q^{46}+4 q^{44}+68 q^{42}-130 q^{40}+151 q^{38}-122 q^{36}+46 q^{34}+44 q^{32}-116 q^{30}+138 q^{28}-93 q^{26}+9 q^{24}+77 q^{22}-122 q^{20}+97 q^{18}-19 q^{16}-82 q^{14}+161 q^{12}-175 q^{10}+132 q^8-27 q^6-94 q^4+190 q^2-223+188 q^{-2} -92 q^{-4} -27 q^{-6} +132 q^{-8} -176 q^{-10} +165 q^{-12} -89 q^{-14} -10 q^{-16} +93 q^{-18} -126 q^{-20} +87 q^{-22} -3 q^{-24} -84 q^{-26} +137 q^{-28} -124 q^{-30} +54 q^{-32} +40 q^{-34} -124 q^{-36} +160 q^{-38} -142 q^{-40} +74 q^{-42} +7 q^{-44} -77 q^{-46} +108 q^{-48} -102 q^{-50} +72 q^{-52} -28 q^{-54} -10 q^{-56} +32 q^{-58} -42 q^{-60} +36 q^{-62} -23 q^{-64} +12 q^{-66} -6 q^{-70} +7 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }

.

Computer Talk

The above data is available with the Mathematica package 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 91"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[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 t^4-4 t^3+9 t^2-14 t+17-14 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-4} }

In[5]:= Conway[K][z]

Out[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 z^8+4 z^6+5 z^4+2 z^2+1}

In[6]:= Alexander[K, 2][t]

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]= 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]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 73, 0 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= 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^5+3 q^4-6 q^3+9 q^2-11 q+13-11 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= 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^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-4 z^4 a^{-2} +13 z^4-5 a^2 z^2-5 z^2 a^{-2} +12 z^2-2 a^2-2 a^{-2} +5}

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= 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 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+5 z^8 a^{-2} +9 z^8+4 a^3 z^7+a z^7+2 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-7 a^2 z^6-13 z^6 a^{-2} +3 z^6 a^{-4} -26 z^6+a^5 z^5-6 a^3 z^5-7 a z^5-13 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+7 a^2 z^4+16 z^4 a^{-2} -6 z^4 a^{-4} +35 z^4-2 a^5 z^3+9 a z^3+18 z^3 a^{-1} +9 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-7 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -19 z^2+a^5 z-4 a z-6 z a^{-1} -3 z a^{-3} +2 a^2+2 a^{-2} +5}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, 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\leftrightarrow q^{-1}} ): {10_43,}

Computer Talk

The above data is available with the Mathematica package 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 91"];

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]= { 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^4-4 t^3+9 t^2-14 t+17-14 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-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^5+3 q^4-6 q^3+9 q^2-11 q+13-11 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} } }

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]= {10_43,}

Vassiliev invariants

V₂ and V₃:

(2, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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 8}

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 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 32}

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 \frac{28}{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 -\frac{76}{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 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 32}

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 -32}

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 64}

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 \frac{256}{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 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 \frac{224}{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 -\frac{608}{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 \frac{511}{15}}

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 \frac{812}{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 -\frac{13556}{45}}

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 -\frac{175}{9}}

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 -\frac{1649}{15}}

V_2,1 through V_6,9 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=} 0 is the signature of 10 91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

4

1

-3

5

2

3

6

4

-2

1

7

5

2

-1

5

7

2

-3

4

6

-2

-5

2

5

3

-7

1

4

-3

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	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 i=-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 i=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 r=-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 {\mathbb Z}}
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=-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 {\mathbb Z}^{2}\oplus{\mathbb Z}_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 {\mathbb Z}}
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=-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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}}	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 {\mathbb Z}^{7}}
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=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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}}	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 {\mathbb Z}^{6}}
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=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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}_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 {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the 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+1} -dimensional representation of 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 sl(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 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^{15}-3 q^{14}+2 q^{13}+7 q^{12}-17 q^{11}+5 q^{10}+30 q^9-43 q^8-5 q^7+72 q^6-66 q^5-31 q^4+114 q^3-73 q^2-58 q+132-60 q^{-1} -71 q^{-2} +113 q^{-3} -32 q^{-4} -64 q^{-5} +71 q^{-6} -7 q^{-7} -40 q^{-8} +30 q^{-9} +2 q^{-10} -15 q^{-11} +8 q^{-12} + q^{-13} -3 q^{-14} + q^{-15} }
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^{30}+3 q^{29}-2 q^{28}-3 q^{27}+2 q^{26}+10 q^{25}-7 q^{24}-22 q^{23}+12 q^{22}+47 q^{21}-14 q^{20}-83 q^{19}-5 q^{18}+136 q^{17}+44 q^{16}-187 q^{15}-115 q^{14}+225 q^{13}+215 q^{12}-244 q^{11}-323 q^{10}+229 q^9+438 q^8-200 q^7-527 q^6+142 q^5+607 q^4-92 q^3-639 q^2+16 q+668+33 q^{-1} -635 q^{-2} -109 q^{-3} +597 q^{-4} +158 q^{-5} -512 q^{-6} -210 q^{-7} +417 q^{-8} +231 q^{-9} -302 q^{-10} -233 q^{-11} +197 q^{-12} +204 q^{-13} -107 q^{-14} -159 q^{-15} +46 q^{-16} +109 q^{-17} -15 q^{-18} -63 q^{-19} + q^{-20} +34 q^{-21} -17 q^{-23} + q^{-24} +9 q^{-25} -2 q^{-26} -3 q^{-27} - q^{-28} +3 q^{-29} - q^{-30} }
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^{50}-3 q^{49}+2 q^{48}+3 q^{47}-6 q^{46}+5 q^{45}-9 q^{44}+13 q^{43}+12 q^{42}-36 q^{41}+7 q^{40}-22 q^{39}+63 q^{38}+69 q^{37}-111 q^{36}-55 q^{35}-116 q^{34}+174 q^{33}+302 q^{32}-104 q^{31}-187 q^{30}-509 q^{29}+114 q^{28}+721 q^{27}+300 q^{26}-47 q^{25}-1200 q^{24}-519 q^{23}+866 q^{22}+1083 q^{21}+838 q^{20}-1649 q^{19}-1666 q^{18}+236 q^{17}+1671 q^{16}+2323 q^{15}-1370 q^{14}-2687 q^{13}-988 q^{12}+1617 q^{11}+3716 q^{10}-552 q^9-3121 q^8-2165 q^7+1083 q^6+4534 q^5+313 q^4-3026 q^3-2936 q^2+416 q+4757+1021 q^{-1} -2574 q^{-2} -3304 q^{-3} -310 q^{-4} +4422 q^{-5} +1622 q^{-6} -1731 q^{-7} -3262 q^{-8} -1130 q^{-9} +3469 q^{-10} +1985 q^{-11} -544 q^{-12} -2629 q^{-13} -1770 q^{-14} +2001 q^{-15} +1777 q^{-16} +538 q^{-17} -1466 q^{-18} -1770 q^{-19} +607 q^{-20} +994 q^{-21} +923 q^{-22} -356 q^{-23} -1122 q^{-24} -84 q^{-25} +204 q^{-26} +621 q^{-27} +139 q^{-28} -418 q^{-29} -114 q^{-30} -114 q^{-31} +210 q^{-32} +129 q^{-33} -90 q^{-34} +4 q^{-35} -85 q^{-36} +36 q^{-37} +31 q^{-38} -25 q^{-39} +27 q^{-40} -21 q^{-41} +7 q^{-42} +3 q^{-43} -12 q^{-44} +8 q^{-45} -3 q^{-46} +3 q^{-47} + q^{-48} -3 q^{-49} + q^{-50} }
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^{75}+3 q^{74}-2 q^{73}-3 q^{72}+6 q^{71}-q^{70}-6 q^{69}+3 q^{68}-2 q^{67}-2 q^{66}+22 q^{65}+10 q^{64}-35 q^{63}-35 q^{62}-14 q^{61}+34 q^{60}+107 q^{59}+88 q^{58}-72 q^{57}-228 q^{56}-209 q^{55}+14 q^{54}+372 q^{53}+514 q^{52}+182 q^{51}-468 q^{50}-938 q^{49}-678 q^{48}+328 q^{47}+1371 q^{46}+1509 q^{45}+325 q^{44}-1548 q^{43}-2596 q^{42}-1597 q^{41}+1056 q^{40}+3515 q^{39}+3554 q^{38}+459 q^{37}-3840 q^{36}-5798 q^{35}-3089 q^{34}+2962 q^{33}+7757 q^{32}+6696 q^{31}-607 q^{30}-8849 q^{29}-10669 q^{28}-3151 q^{27}+8499 q^{26}+14342 q^{25}+7991 q^{24}-6670 q^{23}-17145 q^{22}-13088 q^{21}+3492 q^{20}+18635 q^{19}+17969 q^{18}+426 q^{17}-18898 q^{16}-21909 q^{15}-4603 q^{14}+18078 q^{13}+24907 q^{12}+8390 q^{11}-16658 q^{10}-26697 q^9-11701 q^8+14891 q^7+27835 q^6+14198 q^5-13147 q^4-28094 q^3-16309 q^2+11273 q+28211+17899 q^{-1} -9449 q^{-2} -27598 q^{-3} -19461 q^{-4} +7162 q^{-5} +26783 q^{-6} +20773 q^{-7} -4555 q^{-8} -25029 q^{-9} -21909 q^{-10} +1273 q^{-11} +22542 q^{-12} +22476 q^{-13} +2280 q^{-14} -18873 q^{-15} -22211 q^{-16} -5907 q^{-17} +14372 q^{-18} +20710 q^{-19} +8955 q^{-20} -9222 q^{-21} -17972 q^{-22} -10953 q^{-23} +4216 q^{-24} +14111 q^{-25} +11461 q^{-26} +107 q^{-27} -9724 q^{-28} -10525 q^{-29} -3075 q^{-30} +5492 q^{-31} +8413 q^{-32} +4509 q^{-33} -2015 q^{-34} -5802 q^{-35} -4576 q^{-36} -259 q^{-37} +3308 q^{-38} +3688 q^{-39} +1360 q^{-40} -1367 q^{-41} -2477 q^{-42} -1556 q^{-43} +220 q^{-44} +1341 q^{-45} +1226 q^{-46} +304 q^{-47} -549 q^{-48} -768 q^{-49} -396 q^{-50} +123 q^{-51} +386 q^{-52} +290 q^{-53} +38 q^{-54} -129 q^{-55} -167 q^{-56} -81 q^{-57} +39 q^{-58} +71 q^{-59} +37 q^{-60} +15 q^{-61} -13 q^{-62} -35 q^{-63} -8 q^{-64} +10 q^{-65} -3 q^{-66} +7 q^{-67} +8 q^{-68} -5 q^{-69} -3 q^{-70} +3 q^{-71} -3 q^{-72} - q^{-73} +3 q^{-74} - q^{-75} }
6	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^{105}-3 q^{104}+2 q^{103}+3 q^{102}-6 q^{101}+q^{100}+2 q^{99}+12 q^{98}-14 q^{97}-8 q^{96}+15 q^{95}-25 q^{94}+7 q^{93}+27 q^{92}+58 q^{91}-36 q^{90}-77 q^{89}-11 q^{88}-97 q^{87}+34 q^{86}+169 q^{85}+302 q^{84}+16 q^{83}-276 q^{82}-284 q^{81}-559 q^{80}-147 q^{79}+484 q^{78}+1234 q^{77}+849 q^{76}-80 q^{75}-793 q^{74}-2194 q^{73}-1887 q^{72}-313 q^{71}+2537 q^{70}+3551 q^{69}+2869 q^{68}+1050 q^{67}-3700 q^{66}-6342 q^{65}-5967 q^{64}-478 q^{63}+4858 q^{62}+9191 q^{61}+10410 q^{60}+2728 q^{59}-7034 q^{58}-15523 q^{57}-14134 q^{56}-6802 q^{55}+7861 q^{54}+23234 q^{53}+23849 q^{52}+11495 q^{51}-12269 q^{50}-29867 q^{49}-37265 q^{48}-19839 q^{47}+16729 q^{46}+45406 q^{45}+52441 q^{44}+24454 q^{43}-19536 q^{42}-65917 q^{41}-72831 q^{40}-29596 q^{39}+35411 q^{38}+89040 q^{37}+88092 q^{36}+34718 q^{35}-59002 q^{34}-119154 q^{33}-102892 q^{32}-19619 q^{31}+87988 q^{30}+143067 q^{29}+113841 q^{28}-8420 q^{27}-127717 q^{26}-165565 q^{25}-95838 q^{24}+46498 q^{23}+161601 q^{22}+180174 q^{21}+59278 q^{20}-99921 q^{19}-193612 q^{18}-157902 q^{17}-8568 q^{16}+147835 q^{15}+213957 q^{14}+113025 q^{13}-60656 q^{12}-192734 q^{11}-191017 q^{10}-51704 q^9+123426 q^8+222040 q^7+143230 q^6-29311 q^5-181179 q^4-203498 q^3-78262 q^2+102264 q+219914+159665 q^{-1} -6143 q^{-2} -168365 q^{-3} -209024 q^{-4} -99321 q^{-5} +81117 q^{-6} +213551 q^{-7} +174522 q^{-8} +21046 q^{-9} -148142 q^{-10} -210655 q^{-11} -125587 q^{-12} +46924 q^{-13} +194108 q^{-14} +187585 q^{-15} +61072 q^{-16} -106664 q^{-17} -196275 q^{-18} -152758 q^{-19} -6737 q^{-20} +146927 q^{-21} +182468 q^{-22} +104375 q^{-23} -40544 q^{-24} -149852 q^{-25} -159220 q^{-26} -64249 q^{-27} +72105 q^{-28} +141464 q^{-29} +123658 q^{-30} +28175 q^{-31} -74816 q^{-32} -125975 q^{-33} -93435 q^{-34} -2444 q^{-35} +71507 q^{-36} +99955 q^{-37} +64147 q^{-38} -3491 q^{-39} -64028 q^{-40} -77508 q^{-41} -40748 q^{-42} +7897 q^{-43} +48882 q^{-44} +54839 q^{-45} +30361 q^{-46} -10217 q^{-47} -36791 q^{-48} -36070 q^{-49} -19814 q^{-50} +7008 q^{-51} +24391 q^{-52} +26333 q^{-53} +11319 q^{-54} -5617 q^{-55} -14610 q^{-56} -16581 q^{-57} -7703 q^{-58} +2995 q^{-59} +10522 q^{-60} +9183 q^{-61} +4074 q^{-62} -1070 q^{-63} -5984 q^{-64} -5680 q^{-65} -2630 q^{-66} +1543 q^{-67} +2850 q^{-68} +2744 q^{-69} +1756 q^{-70} -741 q^{-71} -1724 q^{-72} -1606 q^{-73} -351 q^{-74} +186 q^{-75} +668 q^{-76} +952 q^{-77} +214 q^{-78} -212 q^{-79} -446 q^{-80} -185 q^{-81} -167 q^{-82} +8 q^{-83} +274 q^{-84} +117 q^{-85} +29 q^{-86} -75 q^{-87} -13 q^{-88} -72 q^{-89} -49 q^{-90} +53 q^{-91} +22 q^{-92} +18 q^{-93} -12 q^{-94} +14 q^{-95} -11 q^{-96} -18 q^{-97} +9 q^{-98} +3 q^{-100} -3 q^{-101} +3 q^{-102} + q^{-103} -3 q^{-104} + q^{-105} }
7	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^{140}+3 q^{139}-2 q^{138}-3 q^{137}+6 q^{136}-q^{135}-2 q^{134}-8 q^{133}-q^{132}+24 q^{131}-5 q^{130}-12 q^{129}+9 q^{128}-13 q^{127}-10 q^{126}-28 q^{125}+107 q^{123}+46 q^{122}-22 q^{121}-43 q^{120}-146 q^{119}-98 q^{118}-102 q^{117}+34 q^{116}+439 q^{115}+423 q^{114}+214 q^{113}-160 q^{112}-761 q^{111}-853 q^{110}-796 q^{109}-210 q^{108}+1276 q^{107}+2062 q^{106}+2114 q^{105}+966 q^{104}-1551 q^{103}-3331 q^{102}-4503 q^{101}-3647 q^{100}+359 q^{99}+4648 q^{98}+8226 q^{97}+8480 q^{96}+3476 q^{95}-3482 q^{94}-11393 q^{93}-15909 q^{92}-12223 q^{91}-2920 q^{90}+11191 q^{89}+23390 q^{88}+25430 q^{87}+17552 q^{86}-1768 q^{85}-24994 q^{84}-39645 q^{83}-41137 q^{82}-22049 q^{81}+12339 q^{80}+45867 q^{79}+67945 q^{78}+61275 q^{77}+23695 q^{76}-30157 q^{75}-85168 q^{74}-109777 q^{73}-86340 q^{72}-20184 q^{71}+73631 q^{70}+149589 q^{69}+167802 q^{68}+112693 q^{67}-13360 q^{66}-156006 q^{65}-248112 q^{64}-240273 q^{63}-106294 q^{62}+102642 q^{61}+295626 q^{60}+380223 q^{59}+282191 q^{58}+28014 q^{57}-278825 q^{56}-498532 q^{55}-491176 q^{54}-234937 q^{53}+174113 q^{52}+557006 q^{51}+697131 q^{50}+498890 q^{49}+23069 q^{48}-528379 q^{47}-859848 q^{46}-783000 q^{45}-295636 q^{44}+401905 q^{43}+946025 q^{42}+1045961 q^{41}+611059 q^{40}-189095 q^{39}-940980 q^{38}-1251817 q^{37}-927406 q^{36}-81393 q^{35}+847907 q^{34}+1379085 q^{33}+1208434 q^{32}+372926 q^{31}-688799 q^{30}-1424863 q^{29}-1428063 q^{28}-649734 q^{27}+493157 q^{26}+1401264 q^{25}+1577280 q^{24}+885628 q^{23}-292144 q^{22}-1329981 q^{21}-1660056 q^{20}-1067682 q^{19}+109185 q^{18}+1235684 q^{17}+1691414 q^{16}+1194554 q^{15}+40634 q^{14}-1138216 q^{13}-1688281 q^{12}-1275815 q^{11}-155218 q^{10}+1052033 q^9+1670034 q^8+1324510 q^7+237205 q^6-982038 q^5-1647416 q^4-1356453 q^3-300176 q^2+926937 q+1631196+1384848 q^{-1} +354411 q^{-2} -878768 q^{-3} -1619062 q^{-4} -1419481 q^{-5} -417387 q^{-6} +825039 q^{-7} +1609342 q^{-8} +1464942 q^{-9} +497790 q^{-10} -751578 q^{-11} -1587939 q^{-12} -1517934 q^{-13} -605726 q^{-14} +643232 q^{-15} +1541628 q^{-16} +1569085 q^{-17} +738971 q^{-18} -490362 q^{-19} -1451106 q^{-20} -1600154 q^{-21} -889095 q^{-22} +289369 q^{-23} +1302550 q^{-24} +1589981 q^{-25} +1035402 q^{-26} -49707 q^{-27} -1088279 q^{-28} -1516805 q^{-29} -1151186 q^{-30} -207003 q^{-31} +814744 q^{-32} +1366862 q^{-33} +1206888 q^{-34} +448642 q^{-35} -502674 q^{-36} -1140697 q^{-37} -1180870 q^{-38} -638782 q^{-39} +187587 q^{-40} +855626 q^{-41} +1064750 q^{-42} +747466 q^{-43} +90940 q^{-44} -545478 q^{-45} -871645 q^{-46} -759540 q^{-47} -295978 q^{-48} +252223 q^{-49} +630712 q^{-50} +680531 q^{-51} +407382 q^{-52} -14127 q^{-53} -382931 q^{-54} -536608 q^{-55} -424989 q^{-56} -143221 q^{-57} +167515 q^{-58} +364088 q^{-59} +368655 q^{-60} +215994 q^{-61} -10817 q^{-62} -201155 q^{-63} -271514 q^{-64} -217968 q^{-65} -77732 q^{-66} +74444 q^{-67} +166205 q^{-68} +175493 q^{-69} +107793 q^{-70} +5076 q^{-71} -78580 q^{-72} -116915 q^{-73} -98125 q^{-74} -40710 q^{-75} +19953 q^{-76} +62796 q^{-77} +70393 q^{-78} +46335 q^{-79} +10071 q^{-80} -24744 q^{-81} -41254 q^{-82} -36242 q^{-83} -19084 q^{-84} +3547 q^{-85} +19079 q^{-86} +22466 q^{-87} +17185 q^{-88} +4852 q^{-89} -6219 q^{-90} -11397 q^{-91} -11531 q^{-92} -5868 q^{-93} +431 q^{-94} +4384 q^{-95} +6340 q^{-96} +4485 q^{-97} +1356 q^{-98} -1230 q^{-99} -3135 q^{-100} -2534 q^{-101} -1169 q^{-102} -75 q^{-103} +1210 q^{-104} +1319 q^{-105} +912 q^{-106} +321 q^{-107} -593 q^{-108} -618 q^{-109} -388 q^{-110} -264 q^{-111} +134 q^{-112} +228 q^{-113} +277 q^{-114} +241 q^{-115} -99 q^{-116} -138 q^{-117} -84 q^{-118} -96 q^{-119} +5 q^{-121} +52 q^{-122} +97 q^{-123} -7 q^{-124} -24 q^{-125} -12 q^{-126} -22 q^{-127} +2 q^{-128} -12 q^{-129} +22 q^{-131} + q^{-132} -4 q^{-133} -3 q^{-135} +3 q^{-136} -3 q^{-137} - q^{-138} +3 q^{-139} - q^{-140} }

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).

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Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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