9 35

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 7 t-13+7 t^{-1} }[/math]
Conway polynomial [math]\displaystyle{ 7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature { 27, -2 }
Jones polynomial [math]\displaystyle{ q^{-1} -2 q^{-2} +3 q^{-3} -4 q^{-4} +5 q^{-5} -3 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9} - q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^{10}+z^2 a^8-a^8+3 z^2 a^6+3 a^6+2 z^2 a^4+z^2 a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+12 z^3 a^{11}-8 z a^{11}+z^8 a^{10}-4 z^6 a^{10}+3 z^4 a^{10}+z^2 a^{10}+a^{10}+4 z^7 a^9-18 z^5 a^9+23 z^3 a^9-9 z a^9+z^8 a^8+z^6 a^8-15 z^4 a^8+16 z^2 a^8-a^8+3 z^7 a^7-8 z^5 a^7+3 z^3 a^7-z a^7+5 z^6 a^6-15 z^4 a^6+12 z^2 a^6-3 a^6+4 z^5 a^5-6 z^3 a^5+3 z^4 a^4-2 z^2 a^4+2 z^3 a^3+z^2 a^2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{32}-q^{30}-2 q^{26}-q^{24}+q^{22}+q^{20}+3 q^{18}+2 q^{16}+q^{14}-q^{10}+q^8-q^4+q^2 }[/math]
The G2 invariant [math]\displaystyle{ q^{156}+3 q^{152}-3 q^{150}+2 q^{148}-q^{146}-2 q^{144}+7 q^{142}-9 q^{140}+6 q^{138}-2 q^{136}-2 q^{134}+8 q^{132}-12 q^{130}+5 q^{128}-2 q^{126}-3 q^{124}+3 q^{122}-10 q^{120}-2 q^{118}+4 q^{116}-2 q^{114}+q^{112}-8 q^{110}-2 q^{108}+6 q^{106}-6 q^{104}+5 q^{102}-11 q^{100}+6 q^{98}+8 q^{96}-3 q^{94}+8 q^{92}-10 q^{90}+12 q^{88}+4 q^{86}-5 q^{84}+7 q^{82}-5 q^{80}+5 q^{78}+7 q^{76}-3 q^{74}+2 q^{72}+q^{70}-2 q^{68}+4 q^{66}-6 q^{64}+3 q^{62}-2 q^{60}-2 q^{58}+4 q^{56}-4 q^{54}+3 q^{52}-2 q^{50}+q^{48}-q^{46}-q^{44}+2 q^{42}-3 q^{40}+3 q^{38}+q^{36}+q^{34}-q^{30}+2 q^{28}-2 q^{26}+2 q^{24}-q^{22}-q^{16}+q^{14}-q^{12}+q^{10} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_35.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{21}-2 q^{17}+q^{15}+q^{13}+2 q^{11}+q^9-q^7+q^5-q^3+q }[/math]
2 [math]\displaystyle{ q^{60}-q^{56}+2 q^{54}+2 q^{52}-3 q^{50}+q^{46}-4 q^{44}-3 q^{42}+q^{40}-q^{38}-2 q^{36}+3 q^{34}+3 q^{32}+3 q^{26}-q^{24}-2 q^{22}+q^{20}+q^{18}-2 q^{16}+q^{14}+3 q^{12}-q^{10}+q^8-q^4+q^2 }[/math]
3 [math]\displaystyle{ -q^{117}+q^{113}+q^{111}-2 q^{109}-3 q^{107}+5 q^{103}+2 q^{101}-4 q^{99}-4 q^{97}+4 q^{95}+7 q^{93}+3 q^{91}-4 q^{89}-3 q^{87}+4 q^{85}+4 q^{83}-q^{81}-8 q^{79}-3 q^{77}+2 q^{75}+q^{73}-7 q^{71}-4 q^{69}+q^{67}+6 q^{65}-2 q^{63}-4 q^{61}+4 q^{59}+9 q^{57}-q^{55}-8 q^{53}+6 q^{49}+2 q^{47}-6 q^{45}-4 q^{43}-q^{41}+6 q^{39}+4 q^{37}-3 q^{35}-6 q^{33}+4 q^{31}+8 q^{29}-5 q^{25}+3 q^{21}-q^{19}-q^{17}+2 q^{13}+q^{11}-q^5+q^3 }[/math]
4 [math]\displaystyle{ q^{192}-q^{188}-q^{186}-q^{184}+3 q^{182}+3 q^{180}+q^{178}-2 q^{176}-8 q^{174}-2 q^{172}+4 q^{170}+9 q^{168}+6 q^{166}-8 q^{164}-11 q^{162}-10 q^{160}+3 q^{158}+15 q^{156}+8 q^{154}-q^{152}-17 q^{150}-15 q^{148}+4 q^{146}+15 q^{144}+22 q^{142}+2 q^{140}-17 q^{138}-15 q^{136}-2 q^{134}+23 q^{132}+23 q^{130}-q^{128}-17 q^{126}-21 q^{124}+q^{122}+19 q^{120}+14 q^{118}-7 q^{116}-25 q^{114}-15 q^{112}+9 q^{110}+17 q^{108}+3 q^{106}-15 q^{104}-12 q^{102}+8 q^{100}+17 q^{98}+6 q^{96}-14 q^{94}-12 q^{92}+10 q^{90}+18 q^{88}+2 q^{86}-18 q^{84}-21 q^{82}+6 q^{80}+20 q^{78}+13 q^{76}-6 q^{74}-25 q^{72}-15 q^{70}+6 q^{68}+21 q^{66}+23 q^{64}-4 q^{62}-27 q^{60}-20 q^{58}+6 q^{56}+35 q^{54}+21 q^{52}-14 q^{50}-31 q^{48}-15 q^{46}+20 q^{44}+22 q^{42}-15 q^{38}-11 q^{36}+8 q^{34}+10 q^{32}+2 q^{30}-3 q^{28}-5 q^{26}+5 q^{24}+q^{22}-2 q^{20}-q^{18}-q^{16}+4 q^{14}-q^6+q^4 }[/math]
5 [math]\displaystyle{ -q^{285}+q^{281}+q^{279}+q^{277}-3 q^{273}-4 q^{271}-q^{269}+2 q^{267}+5 q^{265}+8 q^{263}+3 q^{261}-6 q^{259}-12 q^{257}-11 q^{255}-2 q^{253}+12 q^{251}+20 q^{249}+16 q^{247}-18 q^{243}-27 q^{241}-18 q^{239}+4 q^{237}+27 q^{235}+37 q^{233}+20 q^{231}-11 q^{229}-38 q^{227}-44 q^{225}-23 q^{223}+20 q^{221}+50 q^{219}+46 q^{217}+10 q^{215}-36 q^{213}-66 q^{211}-51 q^{209}+3 q^{207}+57 q^{205}+69 q^{203}+39 q^{201}-26 q^{199}-74 q^{197}-67 q^{195}-10 q^{193}+55 q^{191}+87 q^{189}+58 q^{187}-16 q^{185}-76 q^{183}-76 q^{181}-19 q^{179}+57 q^{177}+92 q^{175}+49 q^{173}-28 q^{171}-83 q^{169}-76 q^{167}-9 q^{165}+65 q^{163}+76 q^{161}+27 q^{159}-46 q^{157}-76 q^{155}-44 q^{153}+29 q^{151}+66 q^{149}+43 q^{147}-11 q^{145}-49 q^{143}-37 q^{141}+15 q^{139}+42 q^{137}+23 q^{135}-19 q^{133}-40 q^{131}-15 q^{129}+30 q^{127}+48 q^{125}+13 q^{123}-44 q^{121}-64 q^{119}-24 q^{117}+36 q^{115}+73 q^{113}+47 q^{111}-23 q^{109}-76 q^{107}-69 q^{105}-16 q^{103}+53 q^{101}+85 q^{99}+61 q^{97}-7 q^{95}-75 q^{93}-95 q^{91}-44 q^{89}+46 q^{87}+106 q^{85}+93 q^{83}-3 q^{81}-100 q^{79}-114 q^{77}-40 q^{75}+66 q^{73}+115 q^{71}+63 q^{69}-34 q^{67}-94 q^{65}-66 q^{63}+11 q^{61}+68 q^{59}+60 q^{57}+4 q^{55}-44 q^{53}-42 q^{51}-3 q^{49}+24 q^{47}+25 q^{45}+3 q^{43}-15 q^{41}-14 q^{39}+2 q^{37}+9 q^{35}+5 q^{33}-q^{31}-q^{29}+q^{25}+q^{23}-2 q^{21}-2 q^{19}+q^{17}+3 q^{15}-q^7+q^5 }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{32}-q^{30}-2 q^{26}-q^{24}+q^{22}+q^{20}+3 q^{18}+2 q^{16}+q^{14}-q^{10}+q^8-q^4+q^2 }[/math]
1,1 [math]\displaystyle{ q^{84}+6 q^{80}-6 q^{78}+12 q^{76}-18 q^{74}+18 q^{72}-22 q^{70}+14 q^{68}-16 q^{66}+4 q^{64}+8 q^{62}-7 q^{60}+20 q^{58}-24 q^{56}+30 q^{54}-35 q^{52}+22 q^{50}-32 q^{48}+16 q^{46}-14 q^{44}+2 q^{42}+8 q^{40}-2 q^{38}+15 q^{36}-4 q^{34}+12 q^{32}-10 q^{30}+7 q^{28}-4 q^{26}+4 q^{24}-6 q^{22}+5 q^{20}+2 q^{18}+4 q^{16}-4 q^{14}+3 q^{12}-2 q^{10}+2 q^8-2 q^6+q^4 }[/math]
2,0 [math]\displaystyle{ q^{82}+q^{80}+q^{78}-q^{76}+q^{74}+3 q^{72}+3 q^{70}-q^{68}-3 q^{66}-q^{64}-q^{62}-5 q^{60}-7 q^{58}-4 q^{56}-2 q^{54}-2 q^{52}-3 q^{50}+2 q^{48}+5 q^{46}+6 q^{44}+4 q^{42}+3 q^{40}+2 q^{38}+q^{36}-q^{34}-q^{32}-q^{30}+q^{28}+3 q^{26}-q^{24}-3 q^{22}+2 q^{20}+4 q^{18}+q^{16}-2 q^{14}+2 q^{10}-q^8-q^6+q^4 }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{66}+3 q^{62}+2 q^{60}+q^{58}+q^{56}-3 q^{54}-7 q^{52}-6 q^{50}-7 q^{48}-5 q^{46}+2 q^{44}+2 q^{42}+5 q^{40}+6 q^{38}+4 q^{36}+q^{28}+3 q^{24}+3 q^{22}-q^{20}+q^{18}+q^{16}-2 q^{14}+2 q^{10}-q^8-q^6+q^4 }[/math]
1,0,0 [math]\displaystyle{ -q^{43}-q^{41}-q^{39}-2 q^{35}-q^{33}-q^{31}+q^{29}+q^{27}+3 q^{25}+3 q^{23}+2 q^{21}+q^{19}-q^{13}+q^{11}-q^5+q^3 }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{66}-3 q^{62}+2 q^{60}-3 q^{58}+3 q^{56}-3 q^{54}+3 q^{52}-2 q^{50}+q^{48}+q^{46}-2 q^{44}+4 q^{42}-5 q^{40}+6 q^{38}-6 q^{36}+6 q^{34}-4 q^{32}+4 q^{30}-q^{28}+2 q^{26}+q^{24}-q^{22}+3 q^{20}-3 q^{18}+3 q^{16}-2 q^{14}+2 q^{12}-2 q^{10}+q^8-q^6+q^4 }[/math]
1,0 [math]\displaystyle{ q^{108}+3 q^{100}+2 q^{98}-q^{96}-q^{94}+2 q^{92}+2 q^{90}-q^{88}-5 q^{86}-4 q^{84}-q^{82}-q^{80}-5 q^{78}-7 q^{76}-2 q^{74}+2 q^{72}+2 q^{70}-q^{68}+q^{66}+4 q^{64}+5 q^{62}+2 q^{60}+q^{58}+2 q^{56}+3 q^{54}-q^{52}-3 q^{50}-q^{48}+2 q^{46}+q^{44}-q^{42}-2 q^{40}+2 q^{38}+4 q^{36}+q^{34}-2 q^{32}+2 q^{28}+2 q^{26}-q^{24}-2 q^{22}-q^{20}+q^{18}+2 q^{16}-q^{12}-q^{10}+q^6 }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{156}+3 q^{152}-3 q^{150}+2 q^{148}-q^{146}-2 q^{144}+7 q^{142}-9 q^{140}+6 q^{138}-2 q^{136}-2 q^{134}+8 q^{132}-12 q^{130}+5 q^{128}-2 q^{126}-3 q^{124}+3 q^{122}-10 q^{120}-2 q^{118}+4 q^{116}-2 q^{114}+q^{112}-8 q^{110}-2 q^{108}+6 q^{106}-6 q^{104}+5 q^{102}-11 q^{100}+6 q^{98}+8 q^{96}-3 q^{94}+8 q^{92}-10 q^{90}+12 q^{88}+4 q^{86}-5 q^{84}+7 q^{82}-5 q^{80}+5 q^{78}+7 q^{76}-3 q^{74}+2 q^{72}+q^{70}-2 q^{68}+4 q^{66}-6 q^{64}+3 q^{62}-2 q^{60}-2 q^{58}+4 q^{56}-4 q^{54}+3 q^{52}-2 q^{50}+q^{48}-q^{46}-q^{44}+2 q^{42}-3 q^{40}+3 q^{38}+q^{36}+q^{34}-q^{30}+2 q^{28}-2 q^{26}+2 q^{24}-q^{22}-q^{16}+q^{14}-q^{12}+q^{10} }[/math]

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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 35"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 7 t-13+7 t^{-1} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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{ \{3,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 27, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-1} -2 q^{-2} +3 q^{-3} -4 q^{-4} +5 q^{-5} -3 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9} - q^{-10} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^{10}+z^2 a^8-a^8+3 z^2 a^6+3 a^6+2 z^2 a^4+z^2 a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+12 z^3 a^{11}-8 z a^{11}+z^8 a^{10}-4 z^6 a^{10}+3 z^4 a^{10}+z^2 a^{10}+a^{10}+4 z^7 a^9-18 z^5 a^9+23 z^3 a^9-9 z a^9+z^8 a^8+z^6 a^8-15 z^4 a^8+16 z^2 a^8-a^8+3 z^7 a^7-8 z^5 a^7+3 z^3 a^7-z a^7+5 z^6 a^6-15 z^4 a^6+12 z^2 a^6-3 a^6+4 z^5 a^5-6 z^3 a^5+3 z^4 a^4-2 z^2 a^4+2 z^3 a^3+z^2 a^2 }[/math]
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