9 6

9_5

9_7

Visit 9 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 6's page at Knotilus!

Visit 9 6's page at the original Knot Atlas!

9 6 Quick Notes

9 6 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_3,12,4,13 X_5,14,6,15 X_7,16,8,17 X_9,18,10,1 X_15,6,16,7 X_17,8,18,9 X_13,10,14,11 X_11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 6, -4, 7, -5, 8, -9, 2, -8, 3, -6, 4, -7, 5
Dowker-Thistlethwaite code	4 12 14 16 18 2 10 6 8
Conway Notation	[522]

Minimum Braid Representative:

Length is 10, width is 3.

Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	3
3-genus	3
Bridge index	2
Super bridge index	$\{4,6\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[-16][5]
Hyperbolic Volume	7.2036
A-Polynomial	See Data:9 6/A-polynomial

[edit Notes for 9 6'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 3}
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 3}
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 3}
Rasmussen s-Invariant	-6

[edit Notes for 9 6'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 2 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +2 t^{-3} }
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 2 z^6+8 z^4+7 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	{ 27, -6 }
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^{-3} - q^{-4} +3 q^{-5} -3 q^{-6} +4 q^{-7} -5 q^{-8} +4 q^{-9} -3 q^{-10} +2 q^{-11} - q^{-12} }
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^4 a^{10}-3 z^2 a^{10}-a^{10}+z^6 a^8+4 z^4 a^8+3 z^2 a^8-a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+3 a^6}
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^3 a^{15}-z a^{15}+2 z^4 a^{14}-2 z^2 a^{14}+2 z^5 a^{13}-z^3 a^{13}+2 z^6 a^{12}-2 z^4 a^{12}+z^2 a^{12}+2 z^7 a^{11}-5 z^5 a^{11}+6 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-3 z^2 a^{10}+a^{10}+3 z^7 a^9-10 z^5 a^9+8 z^3 a^9-z a^9+z^8 a^8-3 z^6 a^8+z^4 a^8+z^2 a^8-a^8+z^7 a^7-3 z^5 a^7+2 z a^7+z^6 a^6-5 z^4 a^6+7 z^2 a^6-3 a^6}
The A2 invariant	$-q^{36}-2q^{26}-q^{22}+q^{20}+2q^{18}+q^{16}+2q^{14}+q^{10}$
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^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{186}-2 q^{184}+4 q^{182}-4 q^{180}+4 q^{178}-2 q^{176}-q^{174}+3 q^{172}-4 q^{170}+4 q^{168}-5 q^{166}+3 q^{164}-3 q^{162}-q^{160}+3 q^{158}-4 q^{156}+5 q^{154}-4 q^{152}+2 q^{150}-4 q^{146}+4 q^{144}-2 q^{142}-q^{140}+6 q^{138}-5 q^{136}+2 q^{134}+3 q^{132}-6 q^{130}+9 q^{128}-10 q^{126}+3 q^{124}-5 q^{120}+10 q^{118}-12 q^{116}+6 q^{114}-3 q^{112}-2 q^{110}+3 q^{108}-9 q^{106}+6 q^{104}-4 q^{102}+4 q^{98}-6 q^{96}+4 q^{94}+3 q^{92}-5 q^{90}+7 q^{88}-6 q^{86}+2 q^{84}+5 q^{82}-7 q^{80}+11 q^{78}-7 q^{76}+5 q^{74}+2 q^{72}-4 q^{70}+7 q^{68}-5 q^{66}+5 q^{64}+2 q^{58}-q^{56}+2 q^{54}+q^{50}}

Further Quantum Invariants

Further quantum knot invariants for 9_6.

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^{25}+q^{23}-q^{21}+q^{19}-q^{17}-q^{15}+q^{13}+2 q^9+q^5}
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^{68}-q^{66}-q^{64}+2 q^{62}-q^{60}+3 q^{56}-3 q^{54}-q^{52}+3 q^{50}-2 q^{48}-2 q^{46}+q^{44}+q^{42}-q^{40}-q^{38}+2 q^{36}-q^{34}-3 q^{32}+2 q^{30}-3 q^{26}+2 q^{24}+3 q^{22}-2 q^{20}+q^{18}+3 q^{16}+q^{10}}
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^{129}+q^{127}+q^{125}-2 q^{121}-q^{119}+2 q^{117}+q^{115}-q^{111}-q^{109}-q^{107}+2 q^{105}+3 q^{103}-3 q^{101}-5 q^{99}+2 q^{97}+7 q^{95}+q^{93}-5 q^{91}-2 q^{89}+4 q^{87}+3 q^{85}-4 q^{81}-2 q^{79}+4 q^{77}+q^{75}-4 q^{73}-5 q^{71}+4 q^{69}+3 q^{67}-3 q^{65}-5 q^{63}+4 q^{61}+5 q^{59}-q^{57}-6 q^{55}-2 q^{53}+5 q^{51}+3 q^{49}-5 q^{47}-6 q^{45}+q^{43}+6 q^{41}+2 q^{39}-6 q^{37}-2 q^{35}+4 q^{33}+5 q^{31}-q^{29}-2 q^{27}+q^{25}+3 q^{23}+q^{21}+q^{15}}
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^{208}-q^{206}-q^{204}+4 q^{198}-q^{196}-2 q^{194}-3 q^{192}-3 q^{190}+8 q^{188}+3 q^{186}+q^{184}-6 q^{182}-10 q^{180}+6 q^{178}+8 q^{176}+9 q^{174}-6 q^{172}-20 q^{170}-3 q^{168}+10 q^{166}+22 q^{164}+5 q^{162}-24 q^{160}-17 q^{158}+q^{156}+25 q^{154}+16 q^{152}-13 q^{150}-17 q^{148}-12 q^{146}+11 q^{144}+17 q^{142}+2 q^{140}-5 q^{138}-12 q^{136}-3 q^{134}+5 q^{132}+8 q^{130}+7 q^{128}-5 q^{126}-11 q^{124}-q^{122}+12 q^{120}+10 q^{118}-3 q^{116}-15 q^{114}-4 q^{112}+15 q^{110}+10 q^{108}-6 q^{106}-20 q^{104}-8 q^{102}+16 q^{100}+14 q^{98}-19 q^{94}-14 q^{92}+9 q^{90}+14 q^{88}+10 q^{86}-9 q^{84}-14 q^{82}-3 q^{80}+2 q^{78}+15 q^{76}+6 q^{74}-6 q^{72}-9 q^{70}-13 q^{68}+5 q^{66}+10 q^{64}+7 q^{62}-16 q^{58}-6 q^{56}+3 q^{54}+10 q^{52}+9 q^{50}-6 q^{48}-6 q^{46}-4 q^{44}+3 q^{42}+8 q^{40}+q^{38}-2 q^{34}+3 q^{30}+q^{28}+q^{26}+q^{20}}
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^{305}+q^{303}+q^{301}-2 q^{295}-2 q^{293}+q^{291}+4 q^{289}+2 q^{287}+q^{285}-4 q^{283}-8 q^{281}-3 q^{279}+5 q^{277}+11 q^{275}+7 q^{273}-3 q^{271}-14 q^{269}-14 q^{267}+3 q^{265}+21 q^{263}+19 q^{261}-2 q^{259}-25 q^{257}-31 q^{255}-5 q^{253}+35 q^{251}+44 q^{249}+11 q^{247}-40 q^{245}-59 q^{243}-24 q^{241}+40 q^{239}+78 q^{237}+42 q^{235}-35 q^{233}-86 q^{231}-58 q^{229}+20 q^{227}+82 q^{225}+73 q^{223}-q^{221}-69 q^{219}-75 q^{217}-20 q^{215}+44 q^{213}+63 q^{211}+33 q^{209}-17 q^{207}-47 q^{205}-38 q^{203}-2 q^{201}+25 q^{199}+31 q^{197}+19 q^{195}-7 q^{193}-23 q^{191}-24 q^{189}-6 q^{187}+17 q^{185}+26 q^{183}+13 q^{181}-14 q^{179}-31 q^{177}-15 q^{175}+19 q^{173}+32 q^{171}+15 q^{169}-21 q^{167}-42 q^{165}-16 q^{163}+33 q^{161}+48 q^{159}+22 q^{157}-27 q^{155}-58 q^{153}-30 q^{151}+30 q^{149}+62 q^{147}+39 q^{145}-18 q^{143}-62 q^{141}-51 q^{139}+6 q^{137}+55 q^{135}+56 q^{133}+9 q^{131}-45 q^{129}-58 q^{127}-26 q^{125}+25 q^{123}+49 q^{121}+35 q^{119}-7 q^{117}-35 q^{115}-34 q^{113}-12 q^{111}+13 q^{109}+26 q^{107}+23 q^{105}+6 q^{103}-11 q^{101}-18 q^{99}-21 q^{97}-8 q^{95}+10 q^{93}+22 q^{91}+18 q^{89}+5 q^{87}-14 q^{85}-28 q^{83}-17 q^{81}+3 q^{79}+19 q^{77}+22 q^{75}+9 q^{73}-11 q^{71}-20 q^{69}-14 q^{67}+13 q^{63}+14 q^{61}+5 q^{59}-3 q^{57}-9 q^{55}-6 q^{53}+q^{51}+6 q^{49}+4 q^{47}+3 q^{45}-2 q^{41}+2 q^{37}+q^{35}+q^{33}+q^{31}+q^{25}}
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^{420}-q^{418}-q^{416}+2 q^{410}+2 q^{406}-3 q^{404}-3 q^{402}-q^{398}+4 q^{396}+4 q^{394}+8 q^{392}-5 q^{390}-8 q^{388}-7 q^{386}-8 q^{384}+3 q^{382}+11 q^{380}+21 q^{378}-8 q^{374}-15 q^{372}-19 q^{370}-2 q^{368}+18 q^{366}+33 q^{364}+2 q^{362}-13 q^{360}-25 q^{358}-26 q^{356}+5 q^{354}+37 q^{352}+46 q^{350}-5 q^{348}-42 q^{346}-58 q^{344}-38 q^{342}+32 q^{340}+96 q^{338}+96 q^{336}-3 q^{334}-97 q^{332}-141 q^{330}-96 q^{328}+43 q^{326}+172 q^{324}+194 q^{322}+59 q^{320}-113 q^{318}-223 q^{316}-201 q^{314}-26 q^{312}+176 q^{310}+263 q^{308}+163 q^{306}-27 q^{304}-195 q^{302}-248 q^{300}-132 q^{298}+61 q^{296}+204 q^{294}+197 q^{292}+88 q^{290}-59 q^{288}-167 q^{286}-155 q^{284}-58 q^{282}+60 q^{280}+114 q^{278}+108 q^{276}+48 q^{274}-36 q^{272}-85 q^{270}-84 q^{268}-36 q^{266}+14 q^{264}+60 q^{262}+69 q^{260}+36 q^{258}-19 q^{256}-65 q^{254}-56 q^{252}-15 q^{250}+38 q^{248}+65 q^{246}+45 q^{244}-19 q^{242}-77 q^{240}-65 q^{238}+q^{236}+71 q^{234}+92 q^{232}+48 q^{230}-49 q^{228}-124 q^{226}-98 q^{224}+6 q^{222}+109 q^{220}+140 q^{218}+79 q^{216}-55 q^{214}-163 q^{212}-152 q^{210}-27 q^{208}+113 q^{206}+179 q^{204}+137 q^{202}-10 q^{200}-157 q^{198}-196 q^{196}-98 q^{194}+56 q^{192}+175 q^{190}+196 q^{188}+79 q^{186}-87 q^{184}-194 q^{182}-169 q^{180}-52 q^{178}+95 q^{176}+196 q^{174}+158 q^{172}+30 q^{170}-112 q^{168}-168 q^{166}-135 q^{164}-28 q^{162}+103 q^{160}+149 q^{158}+108 q^{156}+6 q^{154}-71 q^{152}-115 q^{150}-92 q^{148}-13 q^{146}+49 q^{144}+79 q^{142}+53 q^{140}+32 q^{138}-15 q^{136}-46 q^{134}-42 q^{132}-33 q^{130}-9 q^{128}+36 q^{124}+40 q^{122}+31 q^{120}+14 q^{118}-15 q^{116}-35 q^{114}-57 q^{112}-24 q^{110}+4 q^{108}+34 q^{106}+48 q^{104}+36 q^{102}+9 q^{100}-37 q^{98}-41 q^{96}-36 q^{94}-11 q^{92}+17 q^{90}+36 q^{88}+34 q^{86}+5 q^{84}-8 q^{82}-24 q^{80}-22 q^{78}-12 q^{76}+7 q^{74}+18 q^{72}+10 q^{70}+9 q^{68}-q^{66}-7 q^{64}-9 q^{62}-2 q^{60}+4 q^{58}+2 q^{56}+5 q^{54}+3 q^{52}+q^{50}-2 q^{48}+2 q^{44}+q^{40}+q^{38}+q^{36}+q^{30}}

A2 Invariants.

Weight	Invariant
1,0	$-q^{36}-2q^{26}-q^{22}+q^{20}+2q^{18}+q^{16}+2q^{14}+q^{10}$
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^{100}-2 q^{98}+4 q^{96}-6 q^{94}+7 q^{92}-10 q^{90}+12 q^{88}-10 q^{86}+11 q^{84}-10 q^{82}+10 q^{80}-10 q^{78}+7 q^{76}-10 q^{74}+8 q^{72}-4 q^{70}-q^{68}+10 q^{66}-16 q^{64}+26 q^{62}-31 q^{60}+38 q^{58}-40 q^{56}+36 q^{54}-35 q^{52}+24 q^{50}-22 q^{48}+2 q^{46}+2 q^{44}-18 q^{42}+20 q^{40}-24 q^{38}+28 q^{36}-18 q^{34}+22 q^{32}-8 q^{30}+12 q^{28}-2 q^{26}+4 q^{24}+q^{20}}
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^{90}-q^{84}+q^{80}+q^{78}+q^{76}-q^{74}+q^{70}-2 q^{66}+q^{62}-2 q^{60}-2 q^{58}-q^{56}-3 q^{52}-2 q^{50}-q^{48}-3 q^{46}-q^{44}+q^{42}+q^{40}+4 q^{36}+4 q^{34}+2 q^{32}+q^{30}+3 q^{28}+2 q^{26}+q^{20}}

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^{82}-q^{80}+2 q^{76}-2 q^{74}-q^{72}+3 q^{70}-q^{68}-2 q^{66}+3 q^{64}-2 q^{60}+q^{58}-q^{56}-q^{54}-2 q^{52}+q^{50}-4 q^{46}-q^{44}-q^{42}-4 q^{40}-q^{38}+3 q^{36}+4 q^{32}+4 q^{30}+2 q^{28}+3 q^{26}+2 q^{24}+q^{20}}
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^{47}-q^{43}+q^{41}-q^{39}-2 q^{35}-q^{33}-q^{31}+2 q^{27}+q^{25}+3 q^{23}+q^{21}+2 q^{19}+q^{15}}
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^{132}-2 q^{130}+3 q^{128}-q^{126}-3 q^{124}+6 q^{122}-8 q^{120}+4 q^{118}+2 q^{116}-5 q^{114}+9 q^{112}-5 q^{110}+2 q^{108}+2 q^{106}-5 q^{104}-q^{102}+4 q^{100}-11 q^{98}+9 q^{96}-9 q^{92}+17 q^{90}-16 q^{88}+15 q^{86}-6 q^{84}+3 q^{82}+7 q^{80}-6 q^{78}+11 q^{76}-7 q^{74}+9 q^{72}-12 q^{70}+7 q^{68}-17 q^{66}-5 q^{64}-10 q^{62}-18 q^{60}+2 q^{58}-22 q^{56}+11 q^{54}-8 q^{52}+9 q^{50}+12 q^{48}+2 q^{46}+19 q^{44}+3 q^{42}+10 q^{40}+6 q^{38}+2 q^{36}+4 q^{34}+q^{30}}

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^{104}-q^{100}+q^{98}+2 q^{96}-q^{94}-q^{92}+2 q^{90}+q^{88}-q^{86}+2 q^{82}-q^{80}-2 q^{78}+q^{76}-q^{74}-3 q^{72}-5 q^{66}-3 q^{64}-2 q^{62}-4 q^{60}-6 q^{58}-3 q^{56}-q^{54}-q^{52}+q^{50}+4 q^{48}+5 q^{46}+5 q^{44}+7 q^{42}+4 q^{40}+4 q^{38}+3 q^{36}+2 q^{34}+q^{30}}

1,0,0,0

-q^{58}-q^{54}-q^{48}-2q^{44}-q^{42}-2q^{40}+2q^{34}+2q^{32}+2q^{30}+3q^{28}+q^{26}+2q^{24}+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^{82}+q^{80}-2 q^{78}+2 q^{76}-2 q^{74}+3 q^{72}-3 q^{70}+3 q^{68}-2 q^{66}+q^{64}-2 q^{60}+3 q^{58}-5 q^{56}+5 q^{54}-6 q^{52}+5 q^{50}-6 q^{48}+4 q^{46}-3 q^{44}+q^{42}-q^{38}+3 q^{36}-2 q^{34}+4 q^{32}-2 q^{30}+4 q^{28}-q^{26}+2 q^{24}+q^{20}}

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^{132}-q^{128}-q^{126}+q^{124}+2 q^{122}-2 q^{118}-2 q^{116}+q^{114}+3 q^{112}+q^{110}-2 q^{108}-2 q^{106}+q^{104}+3 q^{102}-3 q^{98}-q^{96}+2 q^{94}+q^{92}-3 q^{90}-2 q^{88}+q^{86}+2 q^{84}-q^{82}-q^{80}+q^{76}-q^{74}-3 q^{72}-2 q^{70}+q^{68}+q^{66}-3 q^{64}-4 q^{62}+4 q^{58}+2 q^{56}-q^{54}-2 q^{52}+3 q^{50}+4 q^{48}+2 q^{46}-q^{44}+q^{42}+2 q^{40}+2 q^{38}+q^{30}}

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^{114}-q^{112}+q^{110}-q^{108}+2 q^{106}-2 q^{104}+q^{102}-2 q^{100}+3 q^{98}-2 q^{96}+q^{94}-2 q^{92}+2 q^{90}+q^{84}-2 q^{82}+3 q^{80}-4 q^{78}+3 q^{76}-6 q^{74}+3 q^{72}-5 q^{70}+3 q^{68}-5 q^{66}+2 q^{64}-4 q^{62}-3 q^{58}-2 q^{56}-2 q^{52}+2 q^{50}+6 q^{46}+q^{44}+6 q^{42}+q^{40}+5 q^{38}+q^{36}+2 q^{34}+q^{30}}

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^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{186}-2 q^{184}+4 q^{182}-4 q^{180}+4 q^{178}-2 q^{176}-q^{174}+3 q^{172}-4 q^{170}+4 q^{168}-5 q^{166}+3 q^{164}-3 q^{162}-q^{160}+3 q^{158}-4 q^{156}+5 q^{154}-4 q^{152}+2 q^{150}-4 q^{146}+4 q^{144}-2 q^{142}-q^{140}+6 q^{138}-5 q^{136}+2 q^{134}+3 q^{132}-6 q^{130}+9 q^{128}-10 q^{126}+3 q^{124}-5 q^{120}+10 q^{118}-12 q^{116}+6 q^{114}-3 q^{112}-2 q^{110}+3 q^{108}-9 q^{106}+6 q^{104}-4 q^{102}+4 q^{98}-6 q^{96}+4 q^{94}+3 q^{92}-5 q^{90}+7 q^{88}-6 q^{86}+2 q^{84}+5 q^{82}-7 q^{80}+11 q^{78}-7 q^{76}+5 q^{74}+2 q^{72}-4 q^{70}+7 q^{68}-5 q^{66}+5 q^{64}+2 q^{58}-q^{56}+2 q^{54}+q^{50}}

.

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["9 6"];

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 2 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +2 t^{-3} }

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 2 z^6+8 z^4+7 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]= { 27, -6 }

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^{-3} - q^{-4} +3 q^{-5} -3 q^{-6} +4 q^{-7} -5 q^{-8} +4 q^{-9} -3 q^{-10} +2 q^{-11} - q^{-12} }

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^4 a^{10}-3 z^2 a^{10}-a^{10}+z^6 a^8+4 z^4 a^8+3 z^2 a^8-a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+3 a^6}

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 z^3 a^{15}-z a^{15}+2 z^4 a^{14}-2 z^2 a^{14}+2 z^5 a^{13}-z^3 a^{13}+2 z^6 a^{12}-2 z^4 a^{12}+z^2 a^{12}+2 z^7 a^{11}-5 z^5 a^{11}+6 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-3 z^2 a^{10}+a^{10}+3 z^7 a^9-10 z^5 a^9+8 z^3 a^9-z a^9+z^8 a^8-3 z^6 a^8+z^4 a^8+z^2 a^8-a^8+z^7 a^7-3 z^5 a^7+2 z a^7+z^6 a^6-5 z^4 a^6+7 z^2 a^6-3 a^6}

"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}} ): {...}

Vassiliev invariants

V₂ and V₃:

(7, -18)

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 28}

-144

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 392}

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{2834}{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{382}{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 -4032}

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 -6912}

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 -1184}

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 -784}

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{10976}{3}}

10368

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{79352}{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{10696}{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{1559017}{30}}

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{13882}{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{797474}{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{5687}{18}}

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{64777}{30}}

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=} -6 is the signature of 9 6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

1

0

-9

2

-11

1

0

-13

3

2

1

-15

2

1

-1

-17

2

3

-1

-19

1

2

1

-21

1

2

-1

-23

1

-25

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=-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 i=-5}
$r=-9$	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 {\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=-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 {\mathbb Z}\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=-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 {\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=-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}^{2}\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}^{3}\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=-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^{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}^{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 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}	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=-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}\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=-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}_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=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}}	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^{-6} - q^{-7} +4 q^{-9} -3 q^{-10} -3 q^{-11} +9 q^{-12} -4 q^{-13} -8 q^{-14} +12 q^{-15} -2 q^{-16} -13 q^{-17} +14 q^{-18} + q^{-19} -16 q^{-20} +14 q^{-21} +3 q^{-22} -16 q^{-23} +11 q^{-24} +3 q^{-25} -11 q^{-26} +7 q^{-27} + q^{-28} -5 q^{-29} +4 q^{-30} -2 q^{-32} + q^{-33} }
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^{-9} - q^{-10} + q^{-12} +3 q^{-13} -3 q^{-14} -3 q^{-15} +2 q^{-16} +9 q^{-17} -4 q^{-18} -9 q^{-19} -2 q^{-20} +17 q^{-21} -14 q^{-23} -9 q^{-24} +18 q^{-25} +8 q^{-26} -12 q^{-27} -16 q^{-28} +14 q^{-29} +13 q^{-30} -6 q^{-31} -17 q^{-32} +5 q^{-33} +15 q^{-34} -16 q^{-36} -4 q^{-37} +16 q^{-38} +5 q^{-39} -13 q^{-40} -10 q^{-41} +14 q^{-42} +9 q^{-43} -10 q^{-44} -9 q^{-45} +8 q^{-46} +6 q^{-47} -4 q^{-48} -3 q^{-49} +3 q^{-50} - q^{-51} -2 q^{-52} +3 q^{-53} +2 q^{-54} -4 q^{-55} -2 q^{-56} +3 q^{-57} +3 q^{-58} -3 q^{-59} - q^{-60} +2 q^{-62} - 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^{-12} - q^{-13} + q^{-15} +3 q^{-17} -4 q^{-18} -2 q^{-19} +3 q^{-20} + q^{-21} +10 q^{-22} -9 q^{-23} -9 q^{-24} + q^{-25} + q^{-26} +25 q^{-27} -8 q^{-28} -16 q^{-29} -8 q^{-30} -9 q^{-31} +41 q^{-32} - q^{-33} -13 q^{-34} -13 q^{-35} -27 q^{-36} +45 q^{-37} +2 q^{-38} - q^{-39} -4 q^{-40} -40 q^{-41} +40 q^{-42} -9 q^{-43} +4 q^{-44} +15 q^{-45} -36 q^{-46} +35 q^{-47} -32 q^{-48} - q^{-49} +34 q^{-50} -22 q^{-51} +37 q^{-52} -56 q^{-53} -13 q^{-54} +48 q^{-55} -6 q^{-56} +42 q^{-57} -75 q^{-58} -24 q^{-59} +60 q^{-60} +7 q^{-61} +44 q^{-62} -88 q^{-63} -34 q^{-64} +66 q^{-65} +19 q^{-66} +45 q^{-67} -91 q^{-68} -42 q^{-69} +57 q^{-70} +26 q^{-71} +52 q^{-72} -76 q^{-73} -48 q^{-74} +34 q^{-75} +21 q^{-76} +56 q^{-77} -47 q^{-78} -39 q^{-79} +10 q^{-80} +3 q^{-81} +49 q^{-82} -18 q^{-83} -22 q^{-84} -2 q^{-85} -10 q^{-86} +32 q^{-87} -4 q^{-88} -7 q^{-89} -3 q^{-90} -12 q^{-91} +16 q^{-92} - q^{-95} -7 q^{-96} +5 q^{-97} + q^{-99} -2 q^{-101} + q^{-102} }
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^{-15} - q^{-16} + q^{-18} +2 q^{-21} -3 q^{-22} -2 q^{-23} +3 q^{-24} +3 q^{-25} + q^{-26} +4 q^{-27} -8 q^{-28} -9 q^{-29} +9 q^{-31} +9 q^{-32} +13 q^{-33} -9 q^{-34} -22 q^{-35} -14 q^{-36} +3 q^{-37} +18 q^{-38} +33 q^{-39} +4 q^{-40} -25 q^{-41} -30 q^{-42} -17 q^{-43} +7 q^{-44} +47 q^{-45} +23 q^{-46} -12 q^{-47} -26 q^{-48} -29 q^{-49} -11 q^{-50} +34 q^{-51} +26 q^{-52} -5 q^{-53} -9 q^{-54} -12 q^{-55} -8 q^{-56} +21 q^{-57} + q^{-58} -27 q^{-59} -10 q^{-60} +16 q^{-61} +34 q^{-62} +35 q^{-63} -23 q^{-64} -78 q^{-65} -42 q^{-66} +29 q^{-67} +88 q^{-68} +82 q^{-69} -24 q^{-70} -127 q^{-71} -99 q^{-72} +18 q^{-73} +132 q^{-74} +139 q^{-75} - q^{-76} -159 q^{-77} -159 q^{-78} -10 q^{-79} +163 q^{-80} +188 q^{-81} +25 q^{-82} -174 q^{-83} -208 q^{-84} -36 q^{-85} +184 q^{-86} +224 q^{-87} +42 q^{-88} -187 q^{-89} -242 q^{-90} -52 q^{-91} +201 q^{-92} +251 q^{-93} +55 q^{-94} -196 q^{-95} -265 q^{-96} -70 q^{-97} +202 q^{-98} +267 q^{-99} +81 q^{-100} -184 q^{-101} -271 q^{-102} -97 q^{-103} +166 q^{-104} +258 q^{-105} +111 q^{-106} -134 q^{-107} -241 q^{-108} -116 q^{-109} +102 q^{-110} +203 q^{-111} +117 q^{-112} -66 q^{-113} -167 q^{-114} -107 q^{-115} +40 q^{-116} +125 q^{-117} +89 q^{-118} -15 q^{-119} -90 q^{-120} -71 q^{-121} +2 q^{-122} +61 q^{-123} +54 q^{-124} +4 q^{-125} -39 q^{-126} -38 q^{-127} -7 q^{-128} +21 q^{-129} +28 q^{-130} +10 q^{-131} -16 q^{-132} -17 q^{-133} -5 q^{-134} +3 q^{-135} +11 q^{-136} +10 q^{-137} -5 q^{-138} -7 q^{-139} - q^{-140} -3 q^{-141} +3 q^{-142} +5 q^{-143} - q^{-144} -2 q^{-145} - q^{-147} +2 q^{-149} - q^{-150} }
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^{-18} - q^{-19} + q^{-21} - q^{-24} +3 q^{-25} -3 q^{-26} -2 q^{-27} +4 q^{-28} +2 q^{-29} +2 q^{-30} -4 q^{-31} +5 q^{-32} -9 q^{-33} -9 q^{-34} +6 q^{-35} +8 q^{-36} +12 q^{-37} -3 q^{-38} +13 q^{-39} -20 q^{-40} -28 q^{-41} -4 q^{-42} +6 q^{-43} +28 q^{-44} +10 q^{-45} +42 q^{-46} -18 q^{-47} -47 q^{-48} -32 q^{-49} -19 q^{-50} +23 q^{-51} +14 q^{-52} +88 q^{-53} +9 q^{-54} -35 q^{-55} -46 q^{-56} -49 q^{-57} -5 q^{-58} -19 q^{-59} +110 q^{-60} +29 q^{-61} -6 q^{-62} -29 q^{-63} -40 q^{-64} -9 q^{-65} -55 q^{-66} +101 q^{-67} +5 q^{-68} -15 q^{-69} -33 q^{-70} -9 q^{-71} +38 q^{-72} -34 q^{-73} +127 q^{-74} -25 q^{-75} -77 q^{-76} -112 q^{-77} -32 q^{-78} +82 q^{-79} +43 q^{-80} +229 q^{-81} +16 q^{-82} -123 q^{-83} -243 q^{-84} -139 q^{-85} +49 q^{-86} +99 q^{-87} +371 q^{-88} +144 q^{-89} -85 q^{-90} -344 q^{-91} -286 q^{-92} -68 q^{-93} +74 q^{-94} +478 q^{-95} +310 q^{-96} +32 q^{-97} -365 q^{-98} -405 q^{-99} -222 q^{-100} -24 q^{-101} +517 q^{-102} +457 q^{-103} +179 q^{-104} -323 q^{-105} -471 q^{-106} -362 q^{-107} -149 q^{-108} +506 q^{-109} +565 q^{-110} +313 q^{-111} -262 q^{-112} -502 q^{-113} -465 q^{-114} -253 q^{-115} +480 q^{-116} +640 q^{-117} +410 q^{-118} -218 q^{-119} -523 q^{-120} -535 q^{-121} -319 q^{-122} +468 q^{-123} +698 q^{-124} +474 q^{-125} -198 q^{-126} -550 q^{-127} -588 q^{-128} -360 q^{-129} +459 q^{-130} +744 q^{-131} +529 q^{-132} -165 q^{-133} -559 q^{-134} -634 q^{-135} -410 q^{-136} +411 q^{-137} +743 q^{-138} +578 q^{-139} -81 q^{-140} -499 q^{-141} -628 q^{-142} -464 q^{-143} +293 q^{-144} +646 q^{-145} +566 q^{-146} +27 q^{-147} -352 q^{-148} -519 q^{-149} -457 q^{-150} +150 q^{-151} +453 q^{-152} +450 q^{-153} +80 q^{-154} -184 q^{-155} -329 q^{-156} -357 q^{-157} +63 q^{-158} +251 q^{-159} +275 q^{-160} +58 q^{-161} -74 q^{-162} -157 q^{-163} -222 q^{-164} +41 q^{-165} +122 q^{-166} +136 q^{-167} +13 q^{-168} -30 q^{-169} -63 q^{-170} -123 q^{-171} +41 q^{-172} +58 q^{-173} +66 q^{-174} -7 q^{-175} -14 q^{-176} -26 q^{-177} -72 q^{-178} +32 q^{-179} +26 q^{-180} +35 q^{-181} -6 q^{-182} -2 q^{-183} -11 q^{-184} -41 q^{-185} +17 q^{-186} +6 q^{-187} +18 q^{-188} -2 q^{-189} +5 q^{-190} -3 q^{-191} -20 q^{-192} +7 q^{-193} -2 q^{-194} +7 q^{-195} - q^{-196} +4 q^{-197} -7 q^{-199} +3 q^{-200} -2 q^{-201} +2 q^{-202} + q^{-204} -2 q^{-206} + q^{-207} }
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 6]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[7, 16, 8, 17], X[9, 18, 10, 1], X[15, 6, 16, 7], X[17, 8, 18, 9], X[13, 10, 14, 11], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[9, 6]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 6, -4, 7, -5, 8, -9, 2, -8, 3, -6, 4, -7, 5]
In[4]:=	DTCode[Knot[9, 6]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 2, 10, 6, 8]
In[5]:=	br = BR[Knot[9, 6]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -1, -2, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[9, 6]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[9, 6]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[9, 6]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 6]][t]
Out[10]=	2 4 5 2 3 -5 + -- - -- + - + 5 t - 4 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[9, 6]][z]
Out[11]=	2 4 6 1 + 7 z + 8 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 6]}
In[13]:=	{KnotDet[Knot[9, 6]], KnotSignature[Knot[9, 6]]}
Out[13]=	{27, -6}
In[14]:=	Jones[Knot[9, 6]][q]
Out[14]=	-12 2 3 4 5 4 3 3 -4 -3 -q + --- - --- + -- - -- + -- - -- + -- - q + q 11 10 9 8 7 6 5 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 6]}
In[16]:=	A2Invariant[Knot[9, 6]][q]
Out[16]=	-36 2 -22 -20 2 -16 2 -10 -q - --- - q + q + --- + q + --- + q 26 18 14 q q q
In[17]:=	HOMFLYPT[Knot[9, 6]][a, z]
Out[17]=	6 8 10 6 2 8 2 10 2 6 4 8 4 3 a - a - a + 7 a z + 3 a z - 3 a z + 5 a z + 4 a z - 10 4 6 6 8 6 a z + a z + a z
In[18]:=	Kauffman[Knot[9, 6]][a, z]
Out[18]=	6 8 10 7 9 11 15 6 2 8 2 -3 a - a + a + 2 a z - a z - 2 a z - a z + 7 a z + a z - 10 2 12 2 14 2 9 3 11 3 13 3 15 3 3 a z + a z - 2 a z + 8 a z + 6 a z - a z + a z - 6 4 8 4 10 4 12 4 14 4 7 5 5 a z + a z + 2 a z - 2 a z + 2 a z - 3 a z - 9 5 11 5 13 5 6 6 8 6 10 6 10 a z - 5 a z + 2 a z + a z - 3 a z - 2 a z + 12 6 7 7 9 7 11 7 8 8 10 8 2 a z + a z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 6]], Vassiliev[3][Knot[9, 6]]}
Out[19]=	{7, -18}
In[20]:=	Kh[Knot[9, 6]][q, t]
Out[20]=	-7 -5 1 1 1 2 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 25 9 23 8 21 8 21 7 19 7 19 6 q t q t q t q t q t q t 2 3 2 1 3 2 1 ------ + ------ + ------ + ------ + ------ + ------ + ------ + 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t 1 2 1 ------ + ----- + ---- 11 2 9 2 7 q t q t q t
In[21]:=	ColouredJones[Knot[9, 6], 2][q]
Out[21]=	-33 2 4 5 -28 7 11 3 11 16 3 q - --- + --- - --- + q + --- - --- + --- + --- - --- + --- + 32 30 29 27 26 25 24 23 22 q q q q q q q q q 14 16 -19 14 13 2 12 8 4 9 3 --- - --- + q + --- - --- - --- + --- - --- - --- + --- - --- - 21 20 18 17 16 15 14 13 12 11 q q q q q q q q q q 3 4 -7 -6 --- + -- - q + q 10 9 q q

See/edit the Rolfsen_Splice_Template.

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Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

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