10 7

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -3 t^2+11 t-15+11 t^{-1} -3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ -3 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 43, -2 }
Jones polynomial [math]\displaystyle{ q-2+4 q^{-1} -5 q^{-2} +7 q^{-3} -7 q^{-4} +6 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4+a^4-z^4 a^2-z^2 a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+8 z^3 a^9-3 z a^9+2 z^8 a^8-7 z^6 a^8+6 z^4 a^8-3 z^2 a^8+a^8+z^9 a^7-z^7 a^7-6 z^5 a^7+10 z^3 a^7-5 z a^7+4 z^8 a^6-15 z^6 a^6+20 z^4 a^6-10 z^2 a^6+2 a^6+z^9 a^5-z^7 a^5-2 z^5 a^5+6 z^3 a^5-2 z a^5+2 z^8 a^4-5 z^6 a^4+8 z^4 a^4-4 z^2 a^4+a^4+2 z^7 a^3-2 z^5 a^3+z^3 a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1 }[/math]
The A2 invariant [math]\displaystyle{ q^{28}+q^{22}-2 q^{20}-q^{18}-q^{14}+q^{12}+q^8+q^6-q^4+2 q^2+ q^{-4} }[/math]
The G2 invariant [math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-3 q^{132}-q^{130}+6 q^{128}-10 q^{126}+12 q^{124}-12 q^{122}+8 q^{120}+2 q^{118}-12 q^{116}+22 q^{114}-24 q^{112}+21 q^{110}-8 q^{108}-7 q^{106}+20 q^{104}-23 q^{102}+22 q^{100}-9 q^{98}-4 q^{96}+14 q^{94}-16 q^{92}+8 q^{90}+2 q^{88}-14 q^{86}+19 q^{84}-16 q^{82}+q^{80}+10 q^{78}-23 q^{76}+27 q^{74}-25 q^{72}+11 q^{70}+2 q^{68}-19 q^{66}+29 q^{64}-30 q^{62}+21 q^{60}-6 q^{58}-8 q^{56}+18 q^{54}-20 q^{52}+15 q^{50}-3 q^{48}-6 q^{46}+11 q^{44}-8 q^{42}+q^{40}+10 q^{38}-14 q^{36}+14 q^{34}-7 q^{32}-2 q^{30}+9 q^{28}-14 q^{26}+16 q^{24}-13 q^{22}+9 q^{20}-2 q^{18}-5 q^{16}+10 q^{14}-12 q^{12}+14 q^{10}-10 q^8+6 q^6-6 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_7.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{19}-q^{17}+q^{15}-2 q^{13}+q^{11}-q^9+2 q^5-q^3+2 q- q^{-1} + q^{-3} }[/math]
2 [math]\displaystyle{ q^{54}-q^{52}-q^{50}+3 q^{48}-2 q^{46}-4 q^{44}+5 q^{42}+q^{40}-6 q^{38}+4 q^{36}+4 q^{34}-6 q^{32}+q^{30}+5 q^{28}-4 q^{26}-q^{24}+3 q^{22}-4 q^{18}-q^{16}+6 q^{14}-4 q^{12}-3 q^{10}+7 q^8-2 q^6-3 q^4+5 q^2-1- q^{-2} +3 q^{-4} - q^{-6} - q^{-8} + q^{-10} }[/math]
3 [math]\displaystyle{ q^{105}-q^{103}-q^{101}+q^{99}+2 q^{97}-2 q^{95}-5 q^{93}+2 q^{91}+8 q^{89}-10 q^{85}-3 q^{83}+12 q^{81}+9 q^{79}-11 q^{77}-13 q^{75}+4 q^{73}+16 q^{71}+q^{69}-15 q^{67}-8 q^{65}+12 q^{63}+14 q^{61}-7 q^{59}-17 q^{57}+3 q^{55}+17 q^{53}-16 q^{49}-3 q^{47}+13 q^{45}+4 q^{43}-9 q^{41}-7 q^{39}+9 q^{37}+8 q^{35}-13 q^{31}-4 q^{29}+13 q^{27}+9 q^{25}-12 q^{23}-13 q^{21}+7 q^{19}+13 q^{17}-2 q^{15}-11 q^{13}-q^{11}+6 q^9+4 q^7-q^5-q^3-2 q+ q^{-1} +3 q^{-3} +2 q^{-5} -2 q^{-7} -3 q^{-9} + q^{-11} +3 q^{-13} - q^{-17} - q^{-19} + q^{-21} }[/math]
4 [math]\displaystyle{ q^{172}-q^{170}-q^{168}+q^{166}+2 q^{162}-4 q^{160}-3 q^{158}+4 q^{156}+3 q^{154}+8 q^{152}-8 q^{150}-13 q^{148}+2 q^{146}+8 q^{144}+23 q^{142}-4 q^{140}-26 q^{138}-16 q^{136}-2 q^{134}+39 q^{132}+20 q^{130}-15 q^{128}-29 q^{126}-32 q^{124}+24 q^{122}+34 q^{120}+21 q^{118}-5 q^{116}-47 q^{114}-18 q^{112}+3 q^{110}+36 q^{108}+44 q^{106}-16 q^{104}-42 q^{102}-48 q^{100}+13 q^{98}+72 q^{96}+31 q^{94}-28 q^{92}-73 q^{90}-22 q^{88}+62 q^{86}+55 q^{84}-4 q^{82}-66 q^{80}-36 q^{78}+38 q^{76}+47 q^{74}+7 q^{72}-43 q^{70}-32 q^{68}+17 q^{66}+36 q^{64}+12 q^{62}-22 q^{60}-30 q^{58}-10 q^{56}+29 q^{54}+30 q^{52}+12 q^{50}-34 q^{48}-53 q^{46}+8 q^{44}+46 q^{42}+58 q^{40}-11 q^{38}-82 q^{36}-38 q^{34}+27 q^{32}+85 q^{30}+34 q^{28}-63 q^{26}-62 q^{24}-20 q^{22}+62 q^{20}+60 q^{18}-14 q^{16}-42 q^{14}-46 q^{12}+18 q^{10}+45 q^8+15 q^6-6 q^4-38 q^2-8+18 q^{-2} +15 q^{-4} +11 q^{-6} -18 q^{-8} -10 q^{-10} + q^{-12} +5 q^{-14} +12 q^{-16} -4 q^{-18} -4 q^{-20} -3 q^{-22} - q^{-24} +5 q^{-26} - q^{-32} - q^{-34} + q^{-36} }[/math]
5 [math]\displaystyle{ q^{255}-q^{253}-q^{251}+q^{249}-2 q^{241}-2 q^{239}+4 q^{237}+6 q^{235}+q^{233}-4 q^{231}-9 q^{229}-8 q^{227}+4 q^{225}+19 q^{223}+17 q^{221}-3 q^{219}-23 q^{217}-33 q^{215}-14 q^{213}+25 q^{211}+52 q^{209}+35 q^{207}-14 q^{205}-59 q^{203}-65 q^{201}-15 q^{199}+54 q^{197}+88 q^{195}+53 q^{193}-24 q^{191}-85 q^{189}-85 q^{187}-22 q^{185}+52 q^{183}+90 q^{181}+65 q^{179}-56 q^{175}-80 q^{173}-64 q^{171}-14 q^{169}+60 q^{167}+104 q^{165}+94 q^{163}+16 q^{161}-107 q^{159}-172 q^{157}-106 q^{155}+66 q^{153}+210 q^{151}+200 q^{149}+16 q^{147}-212 q^{145}-274 q^{143}-104 q^{141}+171 q^{139}+310 q^{137}+184 q^{135}-104 q^{133}-311 q^{131}-244 q^{129}+40 q^{127}+281 q^{125}+263 q^{123}+17 q^{121}-230 q^{119}-259 q^{117}-54 q^{115}+179 q^{113}+228 q^{111}+69 q^{109}-131 q^{107}-184 q^{105}-70 q^{103}+92 q^{101}+152 q^{99}+64 q^{97}-71 q^{95}-116 q^{93}-63 q^{91}+38 q^{89}+108 q^{87}+84 q^{85}-18 q^{83}-98 q^{81}-113 q^{79}-46 q^{77}+93 q^{75}+169 q^{73}+110 q^{71}-60 q^{69}-216 q^{67}-205 q^{65}+242 q^{61}+300 q^{59}+89 q^{57}-230 q^{55}-371 q^{53}-187 q^{51}+171 q^{49}+397 q^{47}+276 q^{45}-85 q^{43}-369 q^{41}-326 q^{39}-9 q^{37}+294 q^{35}+329 q^{33}+89 q^{31}-199 q^{29}-294 q^{27}-132 q^{25}+115 q^{23}+225 q^{21}+143 q^{19}-40 q^{17}-163 q^{15}-131 q^{13}+4 q^{11}+105 q^9+103 q^7+23 q^5-63 q^3-80 q-30 q^{-1} +37 q^{-3} +58 q^{-5} +30 q^{-7} -16 q^{-9} -40 q^{-11} -27 q^{-13} +3 q^{-15} +27 q^{-17} +24 q^{-19} + q^{-21} -13 q^{-23} -16 q^{-25} -7 q^{-27} +5 q^{-29} +12 q^{-31} +5 q^{-33} -2 q^{-35} -3 q^{-37} -5 q^{-39} - q^{-41} +3 q^{-43} +2 q^{-45} - q^{-51} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{28}+q^{22}-2 q^{20}-q^{18}-q^{14}+q^{12}+q^8+q^6-q^4+2 q^2+ q^{-4} }[/math]
1,1 [math]\displaystyle{ q^{76}-2 q^{74}+4 q^{72}-8 q^{70}+17 q^{68}-26 q^{66}+36 q^{64}-54 q^{62}+67 q^{60}-78 q^{58}+82 q^{56}-80 q^{54}+68 q^{52}-38 q^{50}+10 q^{48}+26 q^{46}-65 q^{44}+96 q^{42}-122 q^{40}+132 q^{38}-135 q^{36}+134 q^{34}-112 q^{32}+92 q^{30}-66 q^{28}+36 q^{26}-14 q^{24}-10 q^{22}+22 q^{20}-36 q^{18}+40 q^{16}-38 q^{14}+39 q^{12}-42 q^{10}+42 q^8-36 q^6+38 q^4-32 q^2+30-20 q^{-2} +15 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0 [math]\displaystyle{ q^{72}+q^{64}-q^{62}-4 q^{60}-q^{58}+2 q^{56}+q^{54}-3 q^{52}+5 q^{48}+4 q^{46}-3 q^{44}-2 q^{42}+3 q^{40}+q^{38}-3 q^{36}-2 q^{34}+3 q^{32}+q^{30}-2 q^{28}+q^{26}-q^{24}-3 q^{22}+2 q^{20}+q^{18}-4 q^{16}-2 q^{14}+5 q^{12}+2 q^{10}-5 q^8-q^6+7 q^4+2 q^2-2+ q^{-2} +2 q^{-4} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{60}-q^{58}+q^{54}-3 q^{52}+q^{48}-3 q^{46}+4 q^{44}+4 q^{42}-3 q^{40}+5 q^{38}+3 q^{36}-6 q^{34}-q^{32}+q^{30}-5 q^{28}-2 q^{26}+q^{24}+q^{22}-2 q^{20}-q^{18}+6 q^{16}-3 q^{14}-2 q^{12}+7 q^{10}-q^8-4 q^6+5 q^4+q^2-2+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0 [math]\displaystyle{ q^{37}+q^{33}+q^{29}-2 q^{27}-q^{25}-2 q^{23}-q^{19}+q^{17}+q^{15}+q^{11}+q^7-q^5+2 q^3+ q^{-1} + q^{-5} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{60}-q^{58}+2 q^{56}-3 q^{54}+5 q^{52}-6 q^{50}+7 q^{48}-7 q^{46}+6 q^{44}-6 q^{42}+3 q^{40}-q^{38}-3 q^{36}+6 q^{34}-9 q^{32}+11 q^{30}-13 q^{28}+14 q^{26}-13 q^{24}+11 q^{22}-8 q^{20}+5 q^{18}-2 q^{16}-q^{14}+4 q^{12}-5 q^{10}+7 q^8-6 q^6+7 q^4-5 q^2+4-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }[/math]
1,0 [math]\displaystyle{ q^{98}-q^{94}-q^{92}+q^{90}+2 q^{88}-q^{86}-4 q^{84}-q^{82}+4 q^{80}+3 q^{78}-4 q^{76}-6 q^{74}+2 q^{72}+8 q^{70}+4 q^{68}-6 q^{66}-5 q^{64}+4 q^{62}+8 q^{60}+q^{58}-6 q^{56}-3 q^{54}+3 q^{52}+2 q^{50}-3 q^{48}-4 q^{46}+q^{44}+3 q^{42}-2 q^{40}-5 q^{38}+5 q^{34}+q^{32}-5 q^{30}-3 q^{28}+5 q^{26}+5 q^{24}-3 q^{22}-6 q^{20}+q^{18}+7 q^{16}+4 q^{14}-4 q^{12}-5 q^{10}+6 q^6+3 q^4-2 q^2-3+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-3 q^{132}-q^{130}+6 q^{128}-10 q^{126}+12 q^{124}-12 q^{122}+8 q^{120}+2 q^{118}-12 q^{116}+22 q^{114}-24 q^{112}+21 q^{110}-8 q^{108}-7 q^{106}+20 q^{104}-23 q^{102}+22 q^{100}-9 q^{98}-4 q^{96}+14 q^{94}-16 q^{92}+8 q^{90}+2 q^{88}-14 q^{86}+19 q^{84}-16 q^{82}+q^{80}+10 q^{78}-23 q^{76}+27 q^{74}-25 q^{72}+11 q^{70}+2 q^{68}-19 q^{66}+29 q^{64}-30 q^{62}+21 q^{60}-6 q^{58}-8 q^{56}+18 q^{54}-20 q^{52}+15 q^{50}-3 q^{48}-6 q^{46}+11 q^{44}-8 q^{42}+q^{40}+10 q^{38}-14 q^{36}+14 q^{34}-7 q^{32}-2 q^{30}+9 q^{28}-14 q^{26}+16 q^{24}-13 q^{22}+9 q^{20}-2 q^{18}-5 q^{16}+10 q^{14}-12 q^{12}+14 q^{10}-10 q^8+6 q^6-6 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/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 7"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -3 t^2+11 t-15+11 t^{-1} -3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -3 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]= { 43, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q-2+4 q^{-1} -5 q^{-2} +7 q^{-3} -7 q^{-4} +6 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4+a^4-z^4 a^2-z^2 a^2+z^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+8 z^3 a^9-3 z a^9+2 z^8 a^8-7 z^6 a^8+6 z^4 a^8-3 z^2 a^8+a^8+z^9 a^7-z^7 a^7-6 z^5 a^7+10 z^3 a^7-5 z a^7+4 z^8 a^6-15 z^6 a^6+20 z^4 a^6-10 z^2 a^6+2 a^6+z^9 a^5-z^7 a^5-2 z^5 a^5+6 z^3 a^5-2 z a^5+2 z^8 a^4-5 z^6 a^4+8 z^4 a^4-4 z^2 a^4+a^4+2 z^7 a^3-2 z^5 a^3+z^3 a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1 }[/math]
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