9 9

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-4 t^2+6 t-7+6 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+8 z^4+8 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 31, -6 }
Jones polynomial	[math]\displaystyle{ q^{-3} - q^{-4} +3 q^{-5} -4 q^{-6} +5 q^{-7} -5 q^{-8} +5 q^{-9} -4 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}-2 a^{10}+z^6 a^8+4 z^4 a^8+4 z^2 a^8+a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+2 a^6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^3 a^{15}-z a^{15}+2 z^4 a^{14}-z^2 a^{14}+3 z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-2 z^5 a^{11}+z^8 a^{10}-z^6 a^{10}+2 z^4 a^{10}-6 z^2 a^{10}+2 a^{10}+3 z^7 a^9-8 z^5 a^9+5 z^3 a^9-2 z a^9+z^8 a^8-3 z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z a^7+z^6 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{36}-q^{32}-q^{30}-q^{26}+2 q^{24}+q^{20}+q^{18}+2 q^{14}+q^{10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{188}-2 q^{184}+5 q^{182}-6 q^{180}+6 q^{178}-6 q^{176}+2 q^{174}+3 q^{172}-7 q^{170}+12 q^{168}-11 q^{166}+9 q^{164}-5 q^{162}-2 q^{160}+6 q^{158}-11 q^{156}+11 q^{154}-7 q^{152}+3 q^{148}-7 q^{146}+5 q^{144}-q^{142}-8 q^{140}+9 q^{138}-12 q^{136}+5 q^{134}+5 q^{132}-15 q^{130}+20 q^{128}-18 q^{126}+10 q^{124}+q^{122}-12 q^{120}+18 q^{118}-18 q^{116}+14 q^{114}-4 q^{112}-4 q^{110}+11 q^{108}-10 q^{106}+7 q^{104}-2 q^{102}-5 q^{100}+8 q^{98}-8 q^{96}+2 q^{94}+6 q^{92}-11 q^{90}+16 q^{88}-11 q^{86}+q^{84}+7 q^{82}-11 q^{80}+16 q^{78}-12 q^{76}+6 q^{74}+2 q^{72}-5 q^{70}+10 q^{68}-7 q^{66}+5 q^{64}+2 q^{58}-2 q^{56}+2 q^{54}+q^{50} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_9.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{25}+q^{23}-2 q^{21}+q^{19}+q^{13}-q^{11}+2 q^9+q^5 }[/math]
2	[math]\displaystyle{ q^{68}-q^{66}+3 q^{62}-3 q^{60}-q^{58}+5 q^{56}-5 q^{54}-2 q^{52}+5 q^{50}-2 q^{48}-2 q^{46}+q^{44}+q^{42}-3 q^{40}-2 q^{38}+4 q^{36}-4 q^{32}+4 q^{30}+2 q^{28}-5 q^{26}+2 q^{24}+3 q^{22}-3 q^{20}+q^{18}+3 q^{16}+q^{10} }[/math]
3	[math]\displaystyle{ -q^{129}+q^{127}-q^{123}-q^{121}+2 q^{119}+2 q^{117}-3 q^{115}-q^{113}+5 q^{111}-7 q^{107}-q^{105}+11 q^{103}+2 q^{101}-13 q^{99}-2 q^{97}+12 q^{95}+6 q^{93}-10 q^{91}-6 q^{89}+5 q^{87}+8 q^{85}-q^{83}-8 q^{81}-5 q^{79}+6 q^{77}+8 q^{75}-6 q^{73}-11 q^{71}+4 q^{69}+10 q^{67}-2 q^{65}-12 q^{63}-q^{61}+11 q^{59}+3 q^{57}-11 q^{55}-6 q^{53}+9 q^{51}+10 q^{49}-6 q^{47}-10 q^{45}+3 q^{43}+10 q^{41}+q^{39}-9 q^{37}-3 q^{35}+5 q^{33}+5 q^{31}-2 q^{29}-2 q^{27}+q^{25}+3 q^{23}+q^{21}+q^{15} }[/math]
4	[math]\displaystyle{ q^{208}-q^{206}+q^{202}-q^{200}+2 q^{198}-3 q^{196}-q^{194}+2 q^{192}-2 q^{190}+5 q^{188}-4 q^{186}+q^{184}+4 q^{182}-9 q^{180}+q^{178}-3 q^{176}+13 q^{174}+12 q^{172}-19 q^{170}-13 q^{168}-10 q^{166}+26 q^{164}+32 q^{162}-18 q^{160}-31 q^{158}-28 q^{156}+25 q^{154}+46 q^{152}+2 q^{150}-24 q^{148}-37 q^{146}+2 q^{144}+34 q^{142}+21 q^{140}+q^{138}-26 q^{136}-20 q^{134}+2 q^{132}+21 q^{130}+23 q^{128}-5 q^{126}-26 q^{124}-19 q^{122}+15 q^{120}+29 q^{118}+8 q^{116}-26 q^{114}-29 q^{112}+14 q^{110}+29 q^{108}+13 q^{106}-27 q^{104}-33 q^{102}+11 q^{100}+27 q^{98}+25 q^{96}-19 q^{94}-39 q^{92}-5 q^{90}+18 q^{88}+37 q^{86}+q^{84}-33 q^{82}-22 q^{80}-6 q^{78}+35 q^{76}+24 q^{74}-10 q^{72}-23 q^{70}-26 q^{68}+14 q^{66}+26 q^{64}+13 q^{62}-5 q^{60}-26 q^{58}-7 q^{56}+8 q^{54}+14 q^{52}+9 q^{50}-10 q^{48}-8 q^{46}-4 q^{44}+4 q^{42}+8 q^{40}-2 q^{34}+3 q^{30}+q^{28}+q^{26}+q^{20} }[/math]
5	[math]\displaystyle{ -q^{305}+q^{303}-q^{299}+q^{297}-q^{293}+2 q^{291}+2 q^{289}-2 q^{287}-q^{285}-q^{283}-3 q^{281}+2 q^{279}+4 q^{277}+4 q^{275}+q^{273}-3 q^{271}-8 q^{269}-12 q^{267}+16 q^{263}+24 q^{261}+9 q^{259}-22 q^{257}-43 q^{255}-27 q^{253}+27 q^{251}+72 q^{249}+52 q^{247}-26 q^{245}-94 q^{243}-86 q^{241}+6 q^{239}+113 q^{237}+124 q^{235}+21 q^{233}-114 q^{231}-154 q^{229}-57 q^{227}+88 q^{225}+165 q^{223}+98 q^{221}-56 q^{219}-151 q^{217}-117 q^{215}+2 q^{213}+112 q^{211}+125 q^{209}+39 q^{207}-63 q^{205}-102 q^{203}-68 q^{201}+7 q^{199}+75 q^{197}+80 q^{195}+32 q^{193}-36 q^{191}-78 q^{189}-60 q^{187}+8 q^{185}+73 q^{183}+75 q^{181}+9 q^{179}-67 q^{177}-78 q^{175}-13 q^{173}+67 q^{171}+82 q^{169}+14 q^{167}-73 q^{165}-88 q^{163}-11 q^{161}+82 q^{159}+95 q^{157}+19 q^{155}-88 q^{153}-112 q^{151}-28 q^{149}+84 q^{147}+124 q^{145}+49 q^{143}-71 q^{141}-129 q^{139}-77 q^{137}+41 q^{135}+124 q^{133}+100 q^{131}-6 q^{129}-102 q^{127}-115 q^{125}-38 q^{123}+67 q^{121}+115 q^{119}+70 q^{117}-22 q^{115}-92 q^{113}-94 q^{111}-27 q^{109}+57 q^{107}+93 q^{105}+59 q^{103}-14 q^{101}-71 q^{99}-78 q^{97}-27 q^{95}+40 q^{93}+70 q^{91}+47 q^{89}-q^{87}-46 q^{85}-55 q^{83}-21 q^{81}+21 q^{79}+41 q^{77}+33 q^{75}+4 q^{73}-24 q^{71}-28 q^{69}-14 q^{67}+5 q^{65}+18 q^{63}+15 q^{61}+2 q^{59}-6 q^{57}-10 q^{55}-6 q^{53}+2 q^{51}+6 q^{49}+3 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^{414}-q^{412}-q^{408}+2 q^{406}-3 q^{404}-2 q^{402}+5 q^{400}+q^{396}+3 q^{392}-8 q^{390}-8 q^{388}+6 q^{386}+q^{384}+6 q^{382}+9 q^{380}+14 q^{378}-14 q^{376}-24 q^{374}-8 q^{372}-13 q^{370}+13 q^{368}+37 q^{366}+50 q^{364}-6 q^{362}-56 q^{360}-58 q^{358}-59 q^{356}+18 q^{354}+106 q^{352}+143 q^{350}+38 q^{348}-103 q^{346}-176 q^{344}-183 q^{342}-13 q^{340}+207 q^{338}+326 q^{336}+176 q^{334}-108 q^{332}-327 q^{330}-405 q^{328}-157 q^{326}+232 q^{324}+516 q^{322}+422 q^{320}+45 q^{318}-352 q^{316}-602 q^{314}-417 q^{312}+50 q^{310}+508 q^{308}+601 q^{306}+322 q^{304}-124 q^{302}-538 q^{300}-570 q^{298}-253 q^{296}+219 q^{294}+491 q^{292}+462 q^{290}+190 q^{288}-207 q^{286}-421 q^{284}-384 q^{282}-117 q^{280}+152 q^{278}+318 q^{276}+312 q^{274}+117 q^{272}-106 q^{270}-264 q^{268}-250 q^{266}-127 q^{264}+75 q^{262}+229 q^{260}+240 q^{258}+105 q^{256}-102 q^{254}-222 q^{252}-213 q^{250}-36 q^{248}+147 q^{246}+239 q^{244}+141 q^{242}-73 q^{240}-216 q^{238}-204 q^{236}-6 q^{234}+181 q^{232}+262 q^{230}+118 q^{228}-139 q^{226}-297 q^{224}-236 q^{222}+27 q^{220}+262 q^{218}+360 q^{216}+168 q^{214}-165 q^{212}-396 q^{210}-344 q^{208}-43 q^{206}+271 q^{204}+463 q^{202}+310 q^{200}-59 q^{198}-400 q^{196}-459 q^{194}-225 q^{192}+129 q^{190}+456 q^{188}+453 q^{186}+166 q^{184}-233 q^{182}-447 q^{180}-400 q^{178}-137 q^{176}+257 q^{174}+447 q^{172}+372 q^{170}+62 q^{168}-229 q^{166}-396 q^{164}-354 q^{162}-64 q^{160}+214 q^{158}+365 q^{156}+278 q^{154}+94 q^{152}-154 q^{150}-324 q^{148}-268 q^{146}-99 q^{144}+121 q^{142}+229 q^{140}+261 q^{138}+127 q^{136}-76 q^{134}-194 q^{132}-221 q^{130}-122 q^{128}+156 q^{124}+192 q^{122}+121 q^{120}+14 q^{118}-99 q^{116}-142 q^{114}-135 q^{112}-24 q^{110}+62 q^{108}+106 q^{106}+101 q^{104}+43 q^{102}-23 q^{100}-86 q^{98}-69 q^{96}-38 q^{94}+7 q^{92}+44 q^{90}+55 q^{88}+36 q^{86}-7 q^{84}-19 q^{82}-31 q^{80}-24 q^{78}-8 q^{76}+13 q^{74}+20 q^{72}+8 q^{70}+7 q^{68}-3 q^{66}-8 q^{64}-9 q^{62}-q^{60}+4 q^{58}+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}-q^{32}-q^{30}-q^{26}+2 q^{24}+q^{20}+q^{18}+2 q^{14}+q^{10} }[/math]
1,1	[math]\displaystyle{ q^{100}-2 q^{98}+4 q^{96}-6 q^{94}+11 q^{92}-14 q^{90}+18 q^{88}-24 q^{86}+28 q^{84}-30 q^{82}+28 q^{80}-30 q^{78}+27 q^{76}-16 q^{74}+8 q^{72}+4 q^{70}-20 q^{68}+36 q^{66}-52 q^{64}+64 q^{62}-70 q^{60}+72 q^{58}-68 q^{56}+54 q^{54}-47 q^{52}+22 q^{50}-10 q^{48}-10 q^{46}+21 q^{44}-34 q^{42}+42 q^{40}-38 q^{38}+38 q^{36}-28 q^{34}+28 q^{32}-12 q^{30}+12 q^{28}-4 q^{26}+4 q^{24}+q^{20} }[/math]
2,0	[math]\displaystyle{ q^{90}+q^{86}+q^{84}+q^{82}-2 q^{74}-2 q^{72}+q^{70}-3 q^{66}+q^{62}-2 q^{60}-4 q^{58}-q^{56}-q^{54}-3 q^{52}+3 q^{48}+3 q^{42}+q^{40}-q^{38}+2 q^{36}+3 q^{34}+q^{32}+3 q^{28}+2 q^{26}+q^{20} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{82}-q^{80}+3 q^{76}-2 q^{74}-2 q^{72}+5 q^{70}-2 q^{68}-3 q^{66}+4 q^{64}-q^{62}-3 q^{60}+q^{58}-q^{56}-3 q^{54}-3 q^{52}-5 q^{46}+q^{44}+3 q^{42}-3 q^{40}+2 q^{38}+5 q^{36}-q^{34}+3 q^{32}+3 q^{30}+2 q^{26}+2 q^{24}+q^{20} }[/math]
1,0,0	[math]\displaystyle{ -q^{47}-2 q^{43}-2 q^{39}-q^{35}+q^{33}+q^{31}+q^{29}+2 q^{27}+2 q^{23}+2 q^{19}+q^{15} }[/math]
1,0,1	[math]\displaystyle{ q^{132}-2 q^{130}+3 q^{128}-q^{126}-3 q^{124}+9 q^{122}-9 q^{120}+5 q^{118}+5 q^{116}-17 q^{114}+19 q^{112}-11 q^{110}-6 q^{108}+19 q^{106}-26 q^{104}+16 q^{102}+5 q^{100}-18 q^{98}+31 q^{96}-23 q^{94}+3 q^{92}+14 q^{90}-31 q^{88}+27 q^{86}-8 q^{84}+2 q^{82}+17 q^{80}-2 q^{78}-4 q^{76}+3 q^{74}-14 q^{72}-19 q^{70}+10 q^{68}-39 q^{66}+21 q^{64}-20 q^{62}-8 q^{60}+22 q^{58}-30 q^{56}+32 q^{54}-11 q^{52}+9 q^{50}+16 q^{48}-5 q^{46}+20 q^{44}-2 q^{42}+6 q^{40}+4 q^{38}+4 q^{34}+q^{30} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{104}-q^{100}+2 q^{98}+3 q^{96}-q^{94}+3 q^{90}+q^{88}-2 q^{86}+q^{84}+2 q^{82}-3 q^{80}-4 q^{78}-q^{76}-5 q^{74}-9 q^{72}-2 q^{70}-2 q^{68}-6 q^{66}-q^{64}+3 q^{62}-2 q^{58}+3 q^{56}+3 q^{54}+q^{52}+2 q^{50}+5 q^{48}+3 q^{46}+2 q^{44}+5 q^{42}+2 q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+q^{30} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{58}-2 q^{54}-q^{52}-q^{50}-2 q^{48}-q^{44}+q^{42}+2 q^{38}+q^{36}+2 q^{34}+q^{32}+q^{30}+2 q^{28}+2 q^{24}+q^{20} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{82}+q^{80}-2 q^{78}+3 q^{76}-4 q^{74}+4 q^{72}-5 q^{70}+4 q^{68}-3 q^{66}+2 q^{64}+q^{62}-3 q^{60}+5 q^{58}-7 q^{56}+7 q^{54}-9 q^{52}+8 q^{50}-8 q^{48}+5 q^{46}-3 q^{44}+q^{42}+q^{40}-2 q^{38}+5 q^{36}-3 q^{34}+5 q^{32}-3 q^{30}+4 q^{28}-2 q^{26}+2 q^{24}+q^{20} }[/math]
1,0	[math]\displaystyle{ q^{132}-q^{128}-q^{126}+q^{124}+3 q^{122}+q^{120}-3 q^{118}-3 q^{116}+q^{114}+5 q^{112}+q^{110}-4 q^{108}-3 q^{106}+2 q^{104}+4 q^{102}-q^{100}-4 q^{98}-q^{96}+3 q^{94}+q^{92}-4 q^{90}-3 q^{88}+q^{86}+2 q^{84}-2 q^{82}-3 q^{80}+2 q^{76}-q^{74}-4 q^{72}-q^{70}+4 q^{68}+3 q^{66}-3 q^{64}-4 q^{62}+2 q^{60}+6 q^{58}+2 q^{56}-2 q^{54}-3 q^{52}+3 q^{50}+4 q^{48}+q^{46}-2 q^{44}+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}+3 q^{106}-3 q^{104}+2 q^{102}-3 q^{100}+5 q^{98}-3 q^{96}+2 q^{94}-3 q^{92}+2 q^{90}-q^{86}+q^{84}-4 q^{82}+4 q^{80}-6 q^{78}+4 q^{76}-9 q^{74}+4 q^{72}-8 q^{70}+4 q^{68}-7 q^{66}+3 q^{64}-3 q^{62}+2 q^{60}+4 q^{54}-q^{52}+4 q^{50}-q^{48}+6 q^{46}-q^{44}+5 q^{42}-q^{40}+4 q^{38}+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^{188}-2 q^{184}+5 q^{182}-6 q^{180}+6 q^{178}-6 q^{176}+2 q^{174}+3 q^{172}-7 q^{170}+12 q^{168}-11 q^{166}+9 q^{164}-5 q^{162}-2 q^{160}+6 q^{158}-11 q^{156}+11 q^{154}-7 q^{152}+3 q^{148}-7 q^{146}+5 q^{144}-q^{142}-8 q^{140}+9 q^{138}-12 q^{136}+5 q^{134}+5 q^{132}-15 q^{130}+20 q^{128}-18 q^{126}+10 q^{124}+q^{122}-12 q^{120}+18 q^{118}-18 q^{116}+14 q^{114}-4 q^{112}-4 q^{110}+11 q^{108}-10 q^{106}+7 q^{104}-2 q^{102}-5 q^{100}+8 q^{98}-8 q^{96}+2 q^{94}+6 q^{92}-11 q^{90}+16 q^{88}-11 q^{86}+q^{84}+7 q^{82}-11 q^{80}+16 q^{78}-12 q^{76}+6 q^{74}+2 q^{72}-5 q^{70}+10 q^{68}-7 q^{66}+5 q^{64}+2 q^{58}-2 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 9"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-4 t^2+6 t-7+6 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+8 z^4+8 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]=

{ 31, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-3} - q^{-4} +3 q^{-5} -4 q^{-6} +5 q^{-7} -5 q^{-8} +5 q^{-9} -4 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}-2 a^{10}+z^6 a^8+4 z^4 a^8+4 z^2 a^8+a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+2 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}-z^2 a^{14}+3 z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-2 z^5 a^{11}+z^8 a^{10}-z^6 a^{10}+2 z^4 a^{10}-6 z^2 a^{10}+2 a^{10}+3 z^7 a^9-8 z^5 a^9+5 z^3 a^9-2 z a^9+z^8 a^8-3 z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z a^7+z^6 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6 }[/math]