10 6

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -2 z^6-6 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 37, -4 }
Jones polynomial [math]\displaystyle{ 1- q^{-1} +3 q^{-2} -4 q^{-3} +5 q^{-4} -6 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^8+3 z^2 a^8+a^8-z^6 a^6-4 z^4 a^6-4 z^2 a^6-a^6-z^6 a^4-4 z^4 a^4-4 z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+3 a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+2 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+2 z^7 a^9-4 z^5 a^9+4 z^3 a^9+2 z^8 a^8-7 z^6 a^8+12 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+5 z^5 a^7-2 z^3 a^7+3 z^8 a^6-12 z^6 a^6+18 z^4 a^6-10 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+8 z^5 a^5-10 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4-3 z^4 a^4+5 z^2 a^4-2 a^4+z^7 a^3-3 z^5 a^3+2 z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 a^2 }[/math]
The A2 invariant [math]\displaystyle{ q^{30}-q^{22}+q^{20}-q^{18}-q^{14}-2 q^{12}+q^{10}+2 q^6+q^4+q^2+1 }[/math]
The G2 invariant [math]\displaystyle{ q^{162}-q^{160}+2 q^{158}-3 q^{156}+q^{154}-3 q^{150}+5 q^{148}-6 q^{146}+6 q^{144}-4 q^{142}+4 q^{138}-7 q^{136}+9 q^{134}-8 q^{132}+6 q^{130}-4 q^{128}+7 q^{124}-9 q^{122}+13 q^{120}-10 q^{118}+6 q^{116}-7 q^{112}+9 q^{110}-9 q^{108}+5 q^{106}+5 q^{104}-9 q^{102}+7 q^{100}-q^{98}-7 q^{96}+14 q^{94}-17 q^{92}+11 q^{90}-3 q^{88}-7 q^{86}+18 q^{84}-20 q^{82}+18 q^{80}-11 q^{78}-2 q^{76}+9 q^{74}-15 q^{72}+15 q^{70}-14 q^{68}+5 q^{66}+5 q^{64}-11 q^{62}+10 q^{60}-7 q^{58}-4 q^{56}+10 q^{54}-13 q^{52}+5 q^{50}-9 q^{46}+18 q^{44}-17 q^{42}+10 q^{40}-q^{38}-8 q^{36}+14 q^{34}-13 q^{32}+11 q^{30}-4 q^{28}+2 q^{26}+4 q^{24}-5 q^{22}+7 q^{20}-4 q^{18}+4 q^{16}+2 q^{10}-q^8+2 q^6+q^2 }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_6.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{21}-q^{19}+q^{17}-2 q^{15}+q^{13}-q^9+q^7-q^5+2 q^3+ q^{-1} }[/math]
2 [math]\displaystyle{ q^{58}-q^{56}-q^{54}+2 q^{52}-q^{50}-q^{48}+3 q^{46}-3 q^{44}-3 q^{42}+6 q^{40}-2 q^{38}-3 q^{36}+5 q^{34}+q^{32}-2 q^{30}-q^{28}+2 q^{26}-q^{24}-4 q^{22}+3 q^{20}+q^{18}-5 q^{16}+3 q^{14}+4 q^{12}-4 q^{10}+4 q^6-2 q^4-q^2+2+ q^{-6} }[/math]
3 [math]\displaystyle{ q^{111}-q^{109}-q^{107}+2 q^{103}+q^{101}-2 q^{99}-2 q^{97}+2 q^{93}+q^{91}-2 q^{87}-2 q^{85}+q^{83}+7 q^{81}+2 q^{79}-8 q^{77}-8 q^{75}+9 q^{73}+10 q^{71}-7 q^{69}-11 q^{67}+2 q^{65}+11 q^{63}+q^{61}-7 q^{59}-4 q^{57}+3 q^{55}+6 q^{53}-q^{51}-8 q^{49}+9 q^{45}+2 q^{43}-11 q^{41}-2 q^{39}+11 q^{37}+7 q^{35}-11 q^{33}-9 q^{31}+6 q^{29}+11 q^{27}-2 q^{25}-12 q^{23}-4 q^{21}+10 q^{19}+7 q^{17}-5 q^{15}-8 q^{13}+2 q^{11}+9 q^9+q^7-4 q^5-2 q^3+3 q+ q^{-1} - q^{-3} - q^{-5} + q^{-7} + q^{-15} }[/math]
4 [math]\displaystyle{ q^{180}-q^{178}-q^{176}+4 q^{170}-q^{168}-2 q^{166}-3 q^{164}-4 q^{162}+8 q^{160}+4 q^{158}+2 q^{156}-5 q^{154}-13 q^{152}+4 q^{150}+7 q^{148}+13 q^{146}+q^{144}-21 q^{142}-10 q^{140}-q^{138}+26 q^{136}+24 q^{134}-14 q^{132}-27 q^{130}-30 q^{128}+18 q^{126}+46 q^{124}+15 q^{122}-19 q^{120}-57 q^{118}-13 q^{116}+43 q^{114}+39 q^{112}+8 q^{110}-47 q^{108}-31 q^{106}+13 q^{104}+29 q^{102}+26 q^{100}-15 q^{98}-24 q^{96}-9 q^{94}+7 q^{92}+19 q^{90}+7 q^{88}-9 q^{86}-18 q^{84}-5 q^{82}+15 q^{80}+21 q^{78}-6 q^{76}-28 q^{74}-10 q^{72}+20 q^{70}+36 q^{68}-4 q^{66}-42 q^{64}-22 q^{62}+17 q^{60}+48 q^{58}+11 q^{56}-37 q^{54}-35 q^{52}-7 q^{50}+42 q^{48}+30 q^{46}-9 q^{44}-27 q^{42}-31 q^{40}+11 q^{38}+25 q^{36}+20 q^{34}+5 q^{32}-31 q^{30}-17 q^{28}-q^{26}+19 q^{24}+27 q^{22}-6 q^{20}-15 q^{18}-19 q^{16}-q^{14}+21 q^{12}+8 q^{10}+q^8-12 q^6-8 q^4+6 q^2+4+6 q^{-2} -2 q^{-4} -4 q^{-6} + q^{-8} - q^{-10} +2 q^{-12} - q^{-16} + q^{-18} - q^{-20} + q^{-28} }[/math]
5 [math]\displaystyle{ q^{265}-q^{263}-q^{261}+2 q^{255}+2 q^{253}-q^{251}-4 q^{249}-2 q^{247}-q^{245}+3 q^{243}+8 q^{241}+4 q^{239}-4 q^{237}-9 q^{235}-8 q^{233}-2 q^{231}+10 q^{229}+15 q^{227}+5 q^{225}-9 q^{223}-18 q^{221}-12 q^{219}+4 q^{217}+25 q^{215}+27 q^{213}-q^{211}-31 q^{209}-41 q^{207}-16 q^{205}+34 q^{203}+66 q^{201}+40 q^{199}-33 q^{197}-91 q^{195}-80 q^{193}+14 q^{191}+112 q^{189}+127 q^{187}+29 q^{185}-116 q^{183}-176 q^{181}-82 q^{179}+98 q^{177}+200 q^{175}+144 q^{173}-48 q^{171}-209 q^{169}-186 q^{167}-4 q^{165}+176 q^{163}+198 q^{161}+56 q^{159}-119 q^{157}-185 q^{155}-88 q^{153}+65 q^{151}+140 q^{149}+95 q^{147}-10 q^{145}-90 q^{143}-88 q^{141}-19 q^{139}+45 q^{137}+65 q^{135}+33 q^{133}-13 q^{131}-49 q^{129}-43 q^{127}-4 q^{125}+38 q^{123}+49 q^{121}+15 q^{119}-42 q^{117}-61 q^{115}-19 q^{113}+55 q^{111}+86 q^{109}+28 q^{107}-68 q^{105}-110 q^{103}-47 q^{101}+77 q^{99}+142 q^{97}+70 q^{95}-77 q^{93}-158 q^{91}-103 q^{89}+51 q^{87}+172 q^{85}+133 q^{83}-20 q^{81}-155 q^{79}-157 q^{77}-29 q^{75}+121 q^{73}+163 q^{71}+73 q^{69}-68 q^{67}-146 q^{65}-101 q^{63}+11 q^{61}+98 q^{59}+108 q^{57}+39 q^{55}-44 q^{53}-85 q^{51}-63 q^{49}-13 q^{47}+37 q^{45}+65 q^{43}+49 q^{41}+7 q^{39}-38 q^{37}-56 q^{35}-43 q^{33}+47 q^{29}+56 q^{27}+28 q^{25}-16 q^{23}-46 q^{21}-44 q^{19}-7 q^{17}+31 q^{15}+38 q^{13}+22 q^{11}-8 q^9-28 q^7-23 q^5-4 q^3+13 q+18 q^{-1} +9 q^{-3} -5 q^{-5} -9 q^{-7} -7 q^{-9} -2 q^{-11} +6 q^{-13} +6 q^{-15} + q^{-17} - q^{-19} - q^{-21} -3 q^{-23} +2 q^{-27} + q^{-33} - q^{-35} - q^{-37} + q^{-45} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{30}-q^{22}+q^{20}-q^{18}-q^{14}-2 q^{12}+q^{10}+2 q^6+q^4+q^2+1 }[/math]
1,1 [math]\displaystyle{ q^{84}-2 q^{82}+4 q^{80}-8 q^{78}+11 q^{76}-14 q^{74}+18 q^{72}-20 q^{70}+23 q^{68}-22 q^{66}+22 q^{64}-28 q^{62}+26 q^{60}-26 q^{58}+24 q^{56}-20 q^{54}+11 q^{52}+6 q^{50}-20 q^{48}+38 q^{46}-54 q^{44}+68 q^{42}-76 q^{40}+82 q^{38}-79 q^{36}+70 q^{34}-58 q^{32}+40 q^{30}-19 q^{28}-4 q^{26}+24 q^{24}-36 q^{22}+43 q^{20}-50 q^{18}+42 q^{16}-38 q^{14}+28 q^{12}-22 q^{10}+18 q^8-8 q^6+10 q^4-2 q^2+4+ q^{-4} }[/math]
2,0 [math]\displaystyle{ q^{76}-q^{70}+q^{66}-2 q^{60}-q^{58}-3 q^{52}+4 q^{48}+2 q^{46}+q^{42}+4 q^{40}-q^{38}-3 q^{36}+q^{34}-q^{30}-3 q^{24}-q^{22}+2 q^{20}-q^{18}-3 q^{16}+3 q^{12}-q^{10}-q^8+2 q^6+3 q^4+q^2+1+ q^{-2} + q^{-4} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{68}-q^{66}+q^{62}-3 q^{60}+3 q^{56}-3 q^{54}-q^{52}+6 q^{50}-2 q^{48}-2 q^{46}+4 q^{44}-q^{42}-q^{40}+q^{38}+2 q^{36}-2 q^{32}+3 q^{30}-5 q^{26}-7 q^{20}-q^{18}+q^{16}-2 q^{14}+4 q^{12}+3 q^{10}+2 q^8+3 q^6+2 q^4+1 }[/math]
1,0,0 [math]\displaystyle{ q^{39}+q^{35}-q^{33}+q^{31}-q^{29}+q^{27}-q^{25}-q^{21}-2 q^{19}-q^{17}-2 q^{15}+q^{13}+3 q^9+q^7+2 q^5+q^3+q }[/math]

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{68}-q^{66}+2 q^{64}-3 q^{62}+3 q^{60}-4 q^{58}+5 q^{56}-5 q^{54}+5 q^{52}-4 q^{50}+2 q^{48}-2 q^{44}+5 q^{42}-7 q^{40}+9 q^{38}-10 q^{36}+10 q^{34}-10 q^{32}+7 q^{30}-6 q^{28}+3 q^{26}-2 q^{24}-2 q^{22}+3 q^{20}-5 q^{18}+5 q^{16}-4 q^{14}+6 q^{12}-3 q^{10}+4 q^8-q^6+2 q^4+1 }[/math]
1,0 [math]\displaystyle{ q^{110}-q^{106}-q^{104}+q^{102}+2 q^{100}-q^{98}-3 q^{96}-2 q^{94}+2 q^{92}+4 q^{90}-4 q^{86}-3 q^{84}+3 q^{82}+6 q^{80}-5 q^{76}-2 q^{74}+4 q^{72}+3 q^{70}-3 q^{68}-3 q^{66}+2 q^{64}+4 q^{62}-3 q^{58}+3 q^{54}+q^{52}-3 q^{50}-q^{48}+2 q^{46}+2 q^{44}-3 q^{42}-5 q^{40}+5 q^{36}+q^{34}-6 q^{32}-6 q^{30}+2 q^{28}+5 q^{26}-4 q^{22}-2 q^{20}+4 q^{18}+3 q^{16}+q^{14}-q^{12}+q^{10}+2 q^8+2 q^6+ q^{-2} }[/math]

D4 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{162}-q^{160}+2 q^{158}-3 q^{156}+q^{154}-3 q^{150}+5 q^{148}-6 q^{146}+6 q^{144}-4 q^{142}+4 q^{138}-7 q^{136}+9 q^{134}-8 q^{132}+6 q^{130}-4 q^{128}+7 q^{124}-9 q^{122}+13 q^{120}-10 q^{118}+6 q^{116}-7 q^{112}+9 q^{110}-9 q^{108}+5 q^{106}+5 q^{104}-9 q^{102}+7 q^{100}-q^{98}-7 q^{96}+14 q^{94}-17 q^{92}+11 q^{90}-3 q^{88}-7 q^{86}+18 q^{84}-20 q^{82}+18 q^{80}-11 q^{78}-2 q^{76}+9 q^{74}-15 q^{72}+15 q^{70}-14 q^{68}+5 q^{66}+5 q^{64}-11 q^{62}+10 q^{60}-7 q^{58}-4 q^{56}+10 q^{54}-13 q^{52}+5 q^{50}-9 q^{46}+18 q^{44}-17 q^{42}+10 q^{40}-q^{38}-8 q^{36}+14 q^{34}-13 q^{32}+11 q^{30}-4 q^{28}+2 q^{26}+4 q^{24}-5 q^{22}+7 q^{20}-4 q^{18}+4 q^{16}+2 q^{10}-q^8+2 q^6+q^2 }[/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["10 6"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -2 z^6-6 z^4-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]= { 37, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 1- q^{-1} +3 q^{-2} -4 q^{-3} +5 q^{-4} -6 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} -2 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{ z^4 a^8+3 z^2 a^8+a^8-z^6 a^6-4 z^4 a^6-4 z^2 a^6-a^6-z^6 a^4-4 z^4 a^4-4 z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+3 a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+2 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+2 z^7 a^9-4 z^5 a^9+4 z^3 a^9+2 z^8 a^8-7 z^6 a^8+12 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+5 z^5 a^7-2 z^3 a^7+3 z^8 a^6-12 z^6 a^6+18 z^4 a^6-10 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+8 z^5 a^5-10 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4-3 z^4 a^4+5 z^2 a^4-2 a^4+z^7 a^3-3 z^5 a^3+2 z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 a^2 }[/math]
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