9 18

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 4 t^2-10 t+13-10 t^{-1} +4 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 4 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 41, -4 }
Jones polynomial [math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+z^2 a^8+2 z^4 a^6+4 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-5 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-3 z^5 a^{11}+z^8 a^{10}+z^6 a^{10}-2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+4 z^7 a^9-5 z^5 a^9+z^3 a^9+z^8 a^8+2 z^6 a^8-2 z^4 a^8+2 z^7 a^7+z^5 a^7-4 z^3 a^7+2 z a^7+3 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]
The A2 invariant [math]\displaystyle{ -q^{34}-2 q^{28}+q^{26}+q^{20}-q^{18}+2 q^{16}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
The G2 invariant [math]\displaystyle{ q^{176}-q^{174}+3 q^{172}-4 q^{170}+3 q^{168}-2 q^{166}-2 q^{164}+10 q^{162}-15 q^{160}+18 q^{158}-15 q^{156}+5 q^{154}+9 q^{152}-26 q^{150}+36 q^{148}-37 q^{146}+23 q^{144}-2 q^{142}-24 q^{140}+37 q^{138}-38 q^{136}+27 q^{134}-6 q^{132}-16 q^{130}+24 q^{128}-23 q^{126}+5 q^{124}+16 q^{122}-29 q^{120}+31 q^{118}-16 q^{116}-7 q^{114}+34 q^{112}-51 q^{110}+54 q^{108}-40 q^{106}+9 q^{104}+23 q^{102}-48 q^{100}+58 q^{98}-47 q^{96}+23 q^{94}+4 q^{92}-26 q^{90}+32 q^{88}-25 q^{86}+4 q^{84}+16 q^{82}-23 q^{80}+20 q^{78}-2 q^{76}-18 q^{74}+35 q^{72}-36 q^{70}+29 q^{68}-12 q^{66}-11 q^{64}+29 q^{62}-34 q^{60}+33 q^{58}-18 q^{56}+6 q^{54}+6 q^{52}-14 q^{50}+16 q^{48}-13 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_18.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{23}+q^{21}-2 q^{19}+2 q^{17}-q^{15}+q^{11}-q^9+3 q^7-q^5+q^3 }[/math]
2 [math]\displaystyle{ q^{64}-q^{62}-q^{60}+4 q^{58}-2 q^{56}-6 q^{54}+8 q^{52}+q^{50}-10 q^{48}+8 q^{46}+4 q^{44}-11 q^{42}+2 q^{40}+5 q^{38}-4 q^{36}-3 q^{34}+3 q^{32}+5 q^{30}-8 q^{28}-q^{26}+11 q^{24}-8 q^{22}-4 q^{20}+11 q^{18}-3 q^{16}-3 q^{14}+5 q^{12}-q^8+q^6 }[/math]
3 [math]\displaystyle{ -q^{123}+q^{121}+q^{119}-q^{117}-3 q^{115}+2 q^{113}+7 q^{111}-q^{109}-13 q^{107}-3 q^{105}+18 q^{103}+11 q^{101}-22 q^{99}-21 q^{97}+21 q^{95}+31 q^{93}-14 q^{91}-39 q^{89}+9 q^{87}+42 q^{85}+q^{83}-42 q^{81}-10 q^{79}+38 q^{77}+15 q^{75}-30 q^{73}-18 q^{71}+19 q^{69}+21 q^{67}-8 q^{65}-23 q^{63}-7 q^{61}+22 q^{59}+18 q^{57}-20 q^{55}-33 q^{53}+17 q^{51}+41 q^{49}-9 q^{47}-45 q^{45}+2 q^{43}+41 q^{41}+6 q^{39}-35 q^{37}-11 q^{35}+27 q^{33}+10 q^{31}-14 q^{29}-9 q^{27}+10 q^{25}+7 q^{23}-4 q^{21}-3 q^{19}+3 q^{17}+2 q^{15}-q^{11}+q^9 }[/math]
4 [math]\displaystyle{ q^{200}-q^{198}-q^{196}+q^{194}+3 q^{190}-4 q^{188}-5 q^{186}+3 q^{184}+4 q^{182}+15 q^{180}-7 q^{178}-23 q^{176}-10 q^{174}+7 q^{172}+50 q^{170}+14 q^{168}-39 q^{166}-55 q^{164}-30 q^{162}+83 q^{160}+77 q^{158}-4 q^{156}-94 q^{154}-117 q^{152}+55 q^{150}+132 q^{148}+88 q^{146}-68 q^{144}-192 q^{142}-29 q^{140}+122 q^{138}+164 q^{136}+6 q^{134}-197 q^{132}-104 q^{130}+64 q^{128}+181 q^{126}+66 q^{124}-146 q^{122}-124 q^{120}+9 q^{118}+142 q^{116}+87 q^{114}-72 q^{112}-111 q^{110}-38 q^{108}+83 q^{106}+93 q^{104}+10 q^{102}-86 q^{100}-88 q^{98}+4 q^{96}+96 q^{94}+110 q^{92}-43 q^{90}-137 q^{88}-92 q^{86}+73 q^{84}+195 q^{82}+30 q^{80}-137 q^{78}-171 q^{76}+6 q^{74}+209 q^{72}+94 q^{70}-69 q^{68}-172 q^{66}-64 q^{64}+138 q^{62}+99 q^{60}+6 q^{58}-106 q^{56}-76 q^{54}+56 q^{52}+52 q^{50}+30 q^{48}-37 q^{46}-43 q^{44}+15 q^{42}+14 q^{40}+19 q^{38}-9 q^{36}-15 q^{34}+7 q^{32}+2 q^{30}+7 q^{28}-2 q^{26}-4 q^{24}+3 q^{22}+2 q^{18}-q^{14}+q^{12} }[/math]
5 [math]\displaystyle{ -q^{295}+q^{293}+q^{291}-q^{289}-q^{283}+2 q^{281}+4 q^{279}-3 q^{277}-7 q^{275}-4 q^{273}-q^{271}+11 q^{269}+20 q^{267}+8 q^{265}-20 q^{263}-39 q^{261}-29 q^{259}+14 q^{257}+67 q^{255}+74 q^{253}+13 q^{251}-86 q^{249}-137 q^{247}-76 q^{245}+73 q^{243}+199 q^{241}+182 q^{239}-5 q^{237}-239 q^{235}-308 q^{233}-121 q^{231}+209 q^{229}+422 q^{227}+305 q^{225}-100 q^{223}-487 q^{221}-502 q^{219}-79 q^{217}+462 q^{215}+665 q^{213}+314 q^{211}-353 q^{209}-771 q^{207}-532 q^{205}+180 q^{203}+772 q^{201}+718 q^{199}+25 q^{197}-709 q^{195}-827 q^{193}-204 q^{191}+583 q^{189}+846 q^{187}+349 q^{185}-443 q^{183}-803 q^{181}-430 q^{179}+309 q^{177}+714 q^{175}+452 q^{173}-186 q^{171}-605 q^{169}-448 q^{167}+91 q^{165}+500 q^{163}+427 q^{161}-7 q^{159}-389 q^{157}-415 q^{155}-89 q^{153}+291 q^{151}+419 q^{149}+199 q^{147}-177 q^{145}-426 q^{143}-345 q^{141}+44 q^{139}+437 q^{137}+497 q^{135}+130 q^{133}-414 q^{131}-656 q^{129}-324 q^{127}+342 q^{125}+771 q^{123}+538 q^{121}-213 q^{119}-828 q^{117}-719 q^{115}+35 q^{113}+775 q^{111}+850 q^{109}+160 q^{107}-656 q^{105}-871 q^{103}-332 q^{101}+455 q^{99}+800 q^{97}+446 q^{95}-249 q^{93}-656 q^{91}-466 q^{89}+70 q^{87}+461 q^{85}+418 q^{83}+59 q^{81}-290 q^{79}-326 q^{77}-96 q^{75}+139 q^{73}+214 q^{71}+107 q^{69}-55 q^{67}-129 q^{65}-76 q^{63}+13 q^{61}+62 q^{59}+49 q^{57}-27 q^{53}-23 q^{51}-3 q^{49}+15 q^{47}+11 q^{45}-q^{43}-6 q^{41}-q^{37}+4 q^{35}+5 q^{33}-2 q^{31}-2 q^{29}+2 q^{27}+2 q^{21}-q^{17}+q^{15} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{34}-2 q^{28}+q^{26}+q^{20}-q^{18}+2 q^{16}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
1,1 [math]\displaystyle{ q^{92}-2 q^{90}+6 q^{88}-12 q^{86}+23 q^{84}-40 q^{82}+60 q^{80}-84 q^{78}+110 q^{76}-132 q^{74}+142 q^{72}-136 q^{70}+114 q^{68}-76 q^{66}+18 q^{64}+52 q^{62}-119 q^{60}+184 q^{58}-236 q^{56}+268 q^{54}-280 q^{52}+260 q^{50}-224 q^{48}+162 q^{46}-97 q^{44}+24 q^{42}+38 q^{40}-92 q^{38}+123 q^{36}-134 q^{34}+136 q^{32}-120 q^{30}+103 q^{28}-72 q^{26}+56 q^{24}-34 q^{22}+23 q^{20}-10 q^{18}+6 q^{16}-2 q^{14}+q^{12} }[/math]
2,0 [math]\displaystyle{ q^{86}+2 q^{78}-4 q^{74}-q^{72}+4 q^{70}+2 q^{68}-4 q^{66}-q^{64}+5 q^{62}+q^{60}-7 q^{58}-2 q^{56}+2 q^{54}-3 q^{52}-3 q^{50}+q^{48}+q^{46}-3 q^{44}+q^{42}+3 q^{40}-4 q^{38}-q^{36}+7 q^{34}+2 q^{32}-5 q^{30}+3 q^{28}+7 q^{26}+q^{24}-4 q^{22}+2 q^{20}+4 q^{18}-q^{16}-q^{14}+q^{12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{74}-q^{72}+q^{70}+2 q^{68}-4 q^{66}+2 q^{64}+4 q^{62}-8 q^{60}+3 q^{58}+5 q^{56}-10 q^{54}+q^{52}+5 q^{50}-6 q^{48}-2 q^{46}+2 q^{44}-3 q^{40}-2 q^{38}+7 q^{36}-2 q^{34}-5 q^{32}+10 q^{30}-6 q^{26}+9 q^{24}+q^{22}-3 q^{20}+4 q^{18}+q^{16}-q^{14}+q^{12} }[/math]
1,0,0 [math]\displaystyle{ -q^{45}-q^{41}-2 q^{37}+q^{35}-q^{33}+q^{31}+q^{27}+2 q^{21}+2 q^{17}+2 q^{13}-q^{11}+q^9 }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{96}-q^{92}+2 q^{90}+3 q^{88}-2 q^{86}-q^{84}+5 q^{82}+3 q^{80}-6 q^{78}-4 q^{76}+4 q^{74}-3 q^{72}-10 q^{70}+4 q^{66}-5 q^{64}-3 q^{62}+4 q^{60}-q^{58}-6 q^{56}+q^{54}+4 q^{52}-5 q^{50}-2 q^{48}+9 q^{46}+2 q^{44}-5 q^{42}+4 q^{40}+8 q^{38}-2 q^{34}+4 q^{32}+5 q^{30}-q^{28}+3 q^{24}+q^{22}-q^{20}+q^{18} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{56}-q^{52}-q^{50}-2 q^{46}+q^{44}-q^{42}+q^{38}+q^{34}+q^{30}+2 q^{26}+2 q^{22}+q^{20}+2 q^{16}-q^{14}+q^{12} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{74}+q^{72}-3 q^{70}+4 q^{68}-6 q^{66}+8 q^{64}-8 q^{62}+8 q^{60}-7 q^{58}+5 q^{56}-2 q^{54}-3 q^{52}+7 q^{50}-12 q^{48}+14 q^{46}-16 q^{44}+16 q^{42}-15 q^{40}+12 q^{38}-7 q^{36}+4 q^{34}+q^{32}-4 q^{30}+8 q^{28}-8 q^{26}+9 q^{24}-7 q^{22}+7 q^{20}-4 q^{18}+3 q^{16}-q^{14}+q^{12} }[/math]
1,0 [math]\displaystyle{ q^{120}-q^{116}-q^{114}+2 q^{112}+3 q^{110}-q^{108}-5 q^{106}-2 q^{104}+6 q^{102}+6 q^{100}-4 q^{98}-9 q^{96}-q^{94}+9 q^{92}+5 q^{90}-7 q^{88}-8 q^{86}+2 q^{84}+7 q^{82}-7 q^{78}-2 q^{76}+5 q^{74}+2 q^{72}-6 q^{70}-4 q^{68}+4 q^{66}+5 q^{64}-3 q^{62}-6 q^{60}+2 q^{58}+7 q^{56}+q^{54}-8 q^{52}-3 q^{50}+8 q^{48}+8 q^{46}-4 q^{44}-8 q^{42}+9 q^{38}+5 q^{36}-3 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+q^{18} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{102}-q^{100}+2 q^{98}-2 q^{96}+4 q^{94}-5 q^{92}+5 q^{90}-6 q^{88}+7 q^{86}-7 q^{84}+5 q^{82}-5 q^{80}+4 q^{78}-3 q^{76}-3 q^{74}+2 q^{72}-5 q^{70}+7 q^{68}-12 q^{66}+10 q^{64}-12 q^{62}+13 q^{60}-13 q^{58}+9 q^{56}-10 q^{54}+9 q^{52}-4 q^{50}+3 q^{48}-q^{46}+7 q^{42}-4 q^{40}+6 q^{38}-6 q^{36}+9 q^{34}-4 q^{32}+6 q^{30}-4 q^{28}+5 q^{26}-q^{24}+2 q^{22}-q^{20}+q^{18} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{176}-q^{174}+3 q^{172}-4 q^{170}+3 q^{168}-2 q^{166}-2 q^{164}+10 q^{162}-15 q^{160}+18 q^{158}-15 q^{156}+5 q^{154}+9 q^{152}-26 q^{150}+36 q^{148}-37 q^{146}+23 q^{144}-2 q^{142}-24 q^{140}+37 q^{138}-38 q^{136}+27 q^{134}-6 q^{132}-16 q^{130}+24 q^{128}-23 q^{126}+5 q^{124}+16 q^{122}-29 q^{120}+31 q^{118}-16 q^{116}-7 q^{114}+34 q^{112}-51 q^{110}+54 q^{108}-40 q^{106}+9 q^{104}+23 q^{102}-48 q^{100}+58 q^{98}-47 q^{96}+23 q^{94}+4 q^{92}-26 q^{90}+32 q^{88}-25 q^{86}+4 q^{84}+16 q^{82}-23 q^{80}+20 q^{78}-2 q^{76}-18 q^{74}+35 q^{72}-36 q^{70}+29 q^{68}-12 q^{66}-11 q^{64}+29 q^{62}-34 q^{60}+33 q^{58}-18 q^{56}+6 q^{54}+6 q^{52}-14 q^{50}+16 q^{48}-13 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/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 18"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 4 t^2-10 t+13-10 t^{-1} +4 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 4 z^4+6 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]= { 41, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+z^2 a^8+2 z^4 a^6+4 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-5 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-3 z^5 a^{11}+z^8 a^{10}+z^6 a^{10}-2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+4 z^7 a^9-5 z^5 a^9+z^3 a^9+z^8 a^8+2 z^6 a^8-2 z^4 a^8+2 z^7 a^7+z^5 a^7-4 z^3 a^7+2 z a^7+3 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]
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