7 6

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^2- t^{-2} +5 t+5 t^{-1} -7 }[/math]
Conway polynomial	[math]\displaystyle{ -z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 19, -2 }
Jones polynomial	[math]\displaystyle{ q-2+3 q^{-1} -3 q^{-2} +4 q^{-3} -3 q^{-4} +2 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^6+2 a^4 z^2+2 a^4-a^2 z^4-2 a^2 z^2-a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^7 z^3-a^7 z+2 a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^5-a^5 z^3+a^4 z^6+2 a^4 z^4-4 a^4 z^2+2 a^4+4 a^3 z^5-6 a^3 z^3+2 a^3 z+a^2 z^6+a^2 z^4-4 a^2 z^2+a^2+2 a z^5-4 a z^3+a z+z^4-2 z^2+1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{20}-q^{18}+q^{16}+q^{12}+q^{10}+q^6-q^4+q^2+ q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-q^{98}+2 q^{96}-2 q^{94}-3 q^{88}+5 q^{86}-6 q^{84}+4 q^{82}-4 q^{80}-q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-5 q^{70}+q^{68}+2 q^{66}-5 q^{64}+4 q^{62}-2 q^{58}+6 q^{56}-5 q^{54}+2 q^{52}+6 q^{50}-9 q^{48}+11 q^{46}-9 q^{44}+5 q^{42}+2 q^{40}-6 q^{38}+10 q^{36}-9 q^{34}+8 q^{32}-3 q^{30}-3 q^{28}+5 q^{26}-5 q^{24}+3 q^{22}-3 q^{18}+5 q^{16}-3 q^{14}-q^{12}+5 q^{10}-8 q^8+8 q^6-5 q^4-q^2+5-6 q^{-2} +8 q^{-4} -4 q^{-6} +2 q^{-8} + q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 7_6.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+q^{11}-q^9+q^7+q^5+q- q^{-1} + q^{-3} }[/math]
2	[math]\displaystyle{ q^{36}-q^{34}-q^{32}+3 q^{30}-2 q^{28}-3 q^{26}+4 q^{24}-q^{22}-3 q^{20}+2 q^{18}+q^{16}-q^{12}+2 q^{10}+q^8-3 q^6+2 q^4+3 q^2-3+ q^{-2} +3 q^{-4} -2 q^{-6} - q^{-8} + q^{-10} }[/math]
3	[math]\displaystyle{ -q^{69}+q^{67}+q^{65}-q^{63}-2 q^{61}+2 q^{59}+5 q^{57}-3 q^{55}-7 q^{53}+2 q^{51}+9 q^{49}-q^{47}-11 q^{45}-q^{43}+11 q^{41}+2 q^{39}-7 q^{37}-4 q^{35}+5 q^{33}+3 q^{31}-2 q^{29}-4 q^{27}-2 q^{25}+3 q^{23}+4 q^{21}-3 q^{19}-6 q^{17}+5 q^{15}+8 q^{13}-2 q^{11}-8 q^9+2 q^7+10 q^5+q^3-9 q-2 q^{-1} +7 q^{-3} +4 q^{-5} -5 q^{-7} -5 q^{-9} +3 q^{-11} +4 q^{-13} -2 q^{-17} - q^{-19} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{112}-q^{110}-q^{108}+q^{106}+2 q^{102}-4 q^{100}-3 q^{98}+4 q^{96}+4 q^{94}+7 q^{92}-10 q^{90}-12 q^{88}+4 q^{86}+12 q^{84}+18 q^{82}-13 q^{80}-25 q^{78}-6 q^{76}+15 q^{74}+32 q^{72}-5 q^{70}-28 q^{68}-15 q^{66}+8 q^{64}+31 q^{62}+5 q^{60}-16 q^{58}-16 q^{56}-3 q^{54}+17 q^{52}+9 q^{50}-3 q^{48}-10 q^{46}-8 q^{44}+9 q^{40}+9 q^{38}-5 q^{36}-13 q^{34}-10 q^{32}+12 q^{30}+19 q^{28}-q^{26}-16 q^{24}-18 q^{22}+14 q^{20}+26 q^{18}+4 q^{16}-16 q^{14}-26 q^{12}+7 q^{10}+25 q^8+13 q^6-7 q^4-29 q^2-4+16 q^{-2} +17 q^{-4} +6 q^{-6} -19 q^{-8} -11 q^{-10} + q^{-12} +11 q^{-14} +12 q^{-16} -6 q^{-18} -7 q^{-20} -5 q^{-22} + q^{-24} +6 q^{-26} + q^{-28} -2 q^{-32} - q^{-34} + q^{-36} }[/math]
5	[math]\displaystyle{ -q^{165}+q^{163}+q^{161}-q^{159}+2 q^{151}+2 q^{149}-4 q^{147}-6 q^{145}-2 q^{143}+5 q^{141}+11 q^{139}+7 q^{137}-6 q^{135}-21 q^{133}-17 q^{131}+9 q^{129}+31 q^{127}+31 q^{125}-q^{123}-41 q^{121}-52 q^{119}-11 q^{117}+47 q^{115}+68 q^{113}+30 q^{111}-43 q^{109}-81 q^{107}-49 q^{105}+31 q^{103}+85 q^{101}+61 q^{99}-16 q^{97}-78 q^{95}-65 q^{93}-q^{91}+60 q^{89}+68 q^{87}+12 q^{85}-42 q^{83}-52 q^{81}-21 q^{79}+21 q^{77}+43 q^{75}+25 q^{73}-8 q^{71}-27 q^{69}-26 q^{67}-8 q^{65}+15 q^{63}+26 q^{61}+18 q^{59}-7 q^{57}-29 q^{55}-29 q^{53}-2 q^{51}+34 q^{49}+39 q^{47}+4 q^{45}-39 q^{43}-48 q^{41}-10 q^{39}+44 q^{37}+61 q^{35}+15 q^{33}-47 q^{31}-67 q^{29}-26 q^{27}+46 q^{25}+75 q^{23}+36 q^{21}-35 q^{19}-77 q^{17}-47 q^{15}+23 q^{13}+73 q^{11}+58 q^9-3 q^7-58 q^5-64 q^3-14 q+42 q^{-1} +60 q^{-3} +28 q^{-5} -19 q^{-7} -48 q^{-9} -38 q^{-11} + q^{-13} +32 q^{-15} +35 q^{-17} +12 q^{-19} -14 q^{-21} -26 q^{-23} -19 q^{-25} +3 q^{-27} +16 q^{-29} +15 q^{-31} +4 q^{-33} -5 q^{-35} -9 q^{-37} -7 q^{-39} + q^{-41} +4 q^{-43} +3 q^{-45} + q^{-47} -2 q^{-51} - q^{-53} + q^{-55} }[/math]
6	[math]\displaystyle{ q^{228}-q^{226}-q^{224}+q^{222}-2 q^{216}+2 q^{214}-q^{212}-2 q^{210}+6 q^{208}+4 q^{206}-8 q^{202}-4 q^{200}-8 q^{198}-3 q^{196}+19 q^{194}+21 q^{192}+9 q^{190}-19 q^{188}-24 q^{186}-38 q^{184}-19 q^{182}+41 q^{180}+67 q^{178}+57 q^{176}-9 q^{174}-58 q^{172}-110 q^{170}-86 q^{168}+32 q^{166}+126 q^{164}+155 q^{162}+67 q^{160}-49 q^{158}-188 q^{156}-200 q^{154}-51 q^{152}+124 q^{150}+237 q^{148}+178 q^{146}+35 q^{144}-185 q^{142}-273 q^{140}-156 q^{138}+43 q^{136}+222 q^{134}+233 q^{132}+128 q^{130}-98 q^{128}-235 q^{126}-194 q^{124}-46 q^{122}+123 q^{120}+186 q^{118}+156 q^{116}-4 q^{114}-128 q^{112}-149 q^{110}-83 q^{108}+25 q^{106}+93 q^{104}+116 q^{102}+47 q^{100}-29 q^{98}-77 q^{96}-75 q^{94}-33 q^{92}+20 q^{90}+65 q^{88}+65 q^{86}+35 q^{84}-26 q^{82}-66 q^{80}-71 q^{78}-24 q^{76}+39 q^{74}+88 q^{72}+80 q^{70}-6 q^{68}-83 q^{66}-118 q^{64}-54 q^{62}+43 q^{60}+133 q^{58}+132 q^{56}+7 q^{54}-115 q^{52}-173 q^{50}-92 q^{48}+42 q^{46}+178 q^{44}+191 q^{42}+43 q^{40}-123 q^{38}-220 q^{36}-149 q^{34}-q^{32}+184 q^{30}+239 q^{28}+111 q^{26}-69 q^{24}-213 q^{22}-200 q^{20}-88 q^{18}+116 q^{16}+230 q^{14}+175 q^{12}+32 q^{10}-130 q^8-192 q^6-166 q^4-q^2+137+170 q^{-2} +114 q^{-4} -6 q^{-6} -101 q^{-8} -160 q^{-10} -84 q^{-12} +16 q^{-14} +86 q^{-16} +108 q^{-18} +69 q^{-20} +3 q^{-22} -78 q^{-24} -76 q^{-26} -47 q^{-28} +39 q^{-32} +56 q^{-34} +43 q^{-36} -8 q^{-38} -23 q^{-40} -32 q^{-42} -23 q^{-44} -6 q^{-46} +14 q^{-48} +23 q^{-50} +9 q^{-52} +4 q^{-54} -5 q^{-56} -8 q^{-58} -9 q^{-60} - q^{-62} +4 q^{-64} + q^{-66} +3 q^{-68} + q^{-70} -2 q^{-74} - q^{-76} + q^{-78} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{20}-q^{18}+q^{16}+q^{12}+q^{10}+q^6-q^4+q^2+ q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{52}-2 q^{50}+4 q^{48}-6 q^{46}+11 q^{44}-14 q^{42}+16 q^{40}-22 q^{38}+19 q^{36}-18 q^{34}+10 q^{32}-2 q^{30}-7 q^{28}+22 q^{26}-28 q^{24}+36 q^{22}-39 q^{20}+36 q^{18}-34 q^{16}+26 q^{14}-16 q^{12}+8 q^{10}+6 q^8-10 q^6+19 q^4-20 q^2+22-18 q^{-2} +13 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{50}+q^{48}-2 q^{44}-q^{42}+q^{40}-q^{38}-2 q^{36}+2 q^{32}-2 q^{28}+q^{26}+q^{24}-q^{22}+q^{20}+q^{18}+2 q^{12}-q^8+q^6+3 q^4-1+2 q^{-2} + q^{-4} - q^{-6} - q^{-8} + q^{-12} }[/math]
3,0	[math]\displaystyle{ -q^{90}-q^{88}+q^{84}+3 q^{82}-q^{78}-q^{76}+2 q^{74}+5 q^{72}-2 q^{70}-5 q^{68}-5 q^{66}+3 q^{64}+7 q^{62}-q^{60}-7 q^{58}-6 q^{56}+4 q^{54}+8 q^{52}-5 q^{48}-2 q^{46}+5 q^{44}+3 q^{42}-3 q^{40}-4 q^{38}-q^{36}-4 q^{32}-2 q^{30}+q^{28}+4 q^{26}+q^{24}-2 q^{22}+3 q^{20}+8 q^{18}+5 q^{16}-3 q^{14}-6 q^{12}+3 q^{10}+7 q^8+2 q^6-6 q^4-6 q^2+4+6 q^{-2} +2 q^{-4} -4 q^{-6} -4 q^{-8} +2 q^{-10} +3 q^{-12} +3 q^{-14} - q^{-16} -2 q^{-18} - q^{-20} + q^{-24} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{42}-q^{40}+2 q^{36}-3 q^{34}-2 q^{32}+2 q^{30}-3 q^{28}-q^{26}+3 q^{24}+q^{20}+q^{18}+2 q^{16}-q^{12}+2 q^{10}+q^8-3 q^6+2 q^4+2 q^2-2+2 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ -q^{27}-q^{25}-q^{23}+q^{21}+2 q^{17}+q^{15}+q^{13}-q^5+q^3+ q^{-1} + q^{-5} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{56}+q^{54}-q^{50}-3 q^{44}-4 q^{42}-2 q^{36}+q^{34}+4 q^{32}+2 q^{30}+2 q^{26}+q^{24}-2 q^{22}-q^{20}+3 q^{18}-q^{16}+4 q^{12}+2 q^{10}-2 q^8+2 q^4-1+ q^{-2} +2 q^{-4} + q^{-10} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+2 q^{22}+2 q^{20}+q^{18}+q^{16}-q^{10}-q^6+q^4+1+ q^{-2} + q^{-6} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{42}+q^{40}-2 q^{38}+2 q^{36}-3 q^{34}+2 q^{32}-2 q^{30}+q^{28}+q^{26}-q^{24}+4 q^{22}-3 q^{20}+5 q^{18}-4 q^{16}+4 q^{14}-3 q^{12}+2 q^{10}-q^8-q^6+2 q^4-2 q^2+2-2 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{68}-q^{64}-q^{62}+q^{60}+2 q^{58}-3 q^{54}-2 q^{52}+q^{50}+2 q^{48}-q^{46}-3 q^{44}+2 q^{40}+q^{38}-2 q^{36}+2 q^{32}+2 q^{30}-q^{28}-q^{26}+q^{24}+2 q^{22}-q^{18}+2 q^{14}+q^{12}-2 q^{10}-2 q^8+2 q^6+3 q^4-2- q^{-2} +2 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{58}-q^{56}+q^{54}-q^{52}+2 q^{50}-3 q^{48}-3 q^{44}+q^{42}-2 q^{40}-q^{38}+4 q^{32}-q^{30}+4 q^{28}-2 q^{26}+5 q^{24}-2 q^{22}+3 q^{20}-3 q^{18}+2 q^{16}-q^{14}+q^{12}-q^{10}-q^8+2 q^6-q^4+2 q^2-1+3 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} + q^{-10} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-q^{98}+2 q^{96}-2 q^{94}-3 q^{88}+5 q^{86}-6 q^{84}+4 q^{82}-4 q^{80}-q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-5 q^{70}+q^{68}+2 q^{66}-5 q^{64}+4 q^{62}-2 q^{58}+6 q^{56}-5 q^{54}+2 q^{52}+6 q^{50}-9 q^{48}+11 q^{46}-9 q^{44}+5 q^{42}+2 q^{40}-6 q^{38}+10 q^{36}-9 q^{34}+8 q^{32}-3 q^{30}-3 q^{28}+5 q^{26}-5 q^{24}+3 q^{22}-3 q^{18}+5 q^{16}-3 q^{14}-q^{12}+5 q^{10}-8 q^8+8 q^6-5 q^4-q^2+5-6 q^{-2} +8 q^{-4} -4 q^{-6} +2 q^{-8} + 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["7 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^2- t^{-2} +5 t+5 t^{-1} -7 }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -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]=

{ 19, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q-2+3 q^{-1} -3 q^{-2} +4 q^{-3} -3 q^{-4} +2 q^{-5} - q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -a^6+2 a^4 z^2+2 a^4-a^2 z^4-2 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{ a^7 z^3-a^7 z+2 a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^5-a^5 z^3+a^4 z^6+2 a^4 z^4-4 a^4 z^2+2 a^4+4 a^3 z^5-6 a^3 z^3+2 a^3 z+a^2 z^6+a^2 z^4-4 a^2 z^2+a^2+2 a z^5-4 a z^3+a z+z^4-2 z^2+1 }[/math]