9 6

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+8 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 27, -6 }
Jones polynomial	[math]\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} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\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 }[/math]
Kauffman polynomial (db, data sources)	[math]\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 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{36}-2 q^{26}-q^{22}+q^{20}+2 q^{18}+q^{16}+2 q^{14}+q^{10} }[/math]
The G2 invariant	[math]\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} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_6.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{25}+q^{23}-q^{21}+q^{19}-q^{17}-q^{15}+q^{13}+2 q^9+q^5 }[/math]
2	[math]\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} }[/math]
3	[math]\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} }[/math]
4	[math]\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} }[/math]
5	[math]\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} }[/math]
6	[math]\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} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{36}-2 q^{26}-q^{22}+q^{20}+2 q^{18}+q^{16}+2 q^{14}+q^{10} }[/math]
1,1	[math]\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} }[/math]
2,0	[math]\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} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\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} }[/math]
1,0,0	[math]\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} }[/math]
1,0,1	[math]\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} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\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} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{58}-q^{54}-q^{48}-2 q^{44}-q^{42}-2 q^{40}+2 q^{34}+2 q^{32}+2 q^{30}+3 q^{28}+q^{26}+2 q^{24}+q^{20} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\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} }[/math]
1,0	[math]\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} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\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} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\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} }[/math]

.

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+8 z^4+7 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\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} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\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 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\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 }[/math]