8 2

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-3 z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 17, -4 }
Jones polynomial	[math]\displaystyle{ 1- q^{-1} +2 q^{-2} -2 q^{-3} +3 q^{-4} -3 q^{-5} +2 q^{-6} -2 q^{-7} + q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^6+3 z^2 a^6+a^6-z^6 a^4-5 z^4 a^4-7 z^2 a^4-3 a^4+z^4 a^2+4 z^2 a^2+3 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^2 a^{10}+2 z^3 a^9-z a^9+2 z^4 a^8-z^2 a^8+2 z^5 a^7-2 z^3 a^7-z a^7+2 z^6 a^6-5 z^4 a^6+3 z^2 a^6-a^6+z^7 a^5-2 z^5 a^5-z^3 a^5+z a^5+3 z^6 a^4-12 z^4 a^4+12 z^2 a^4-3 a^4+z^7 a^3-4 z^5 a^3+3 z^3 a^3+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^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^6+q^4+q^2+1 }[/math]
The G2 invariant	[math]\displaystyle{ q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2 q^{120}-2 q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2 q^{92}-q^{84}+q^{82}-2 q^{78}+q^{76}-q^{74}+q^{70}-4 q^{68}+2 q^{66}-3 q^{64}-3 q^{58}+3 q^{56}-2 q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2 q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2 q^{30}-2 q^{28}+3 q^{26}-q^{24}+q^{22}+3 q^{20}-3 q^{18}+4 q^{16}+q^{12}+q^{10}-q^8+2 q^6+q^2 }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_2.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{17}-q^{15}-q^{11}+q^7+q^3+ q^{-1} }[/math]
2	[math]\displaystyle{ q^{46}-q^{44}-q^{42}+q^{40}-q^{38}+q^{36}+q^{34}-q^{32}+q^{28}-q^{24}-q^{18}-q^{16}+q^{14}+2 q^8+q^6-q^4+q^2+1- q^{-2} + q^{-6} }[/math]
3	[math]\displaystyle{ q^{87}-q^{85}-q^{83}+q^{79}+q^{77}-q^{75}-q^{71}+q^{69}+q^{67}-2 q^{63}+2 q^{59}-q^{55}-q^{53}+q^{49}+2 q^{47}-q^{45}-q^{43}+2 q^{39}-q^{37}-2 q^{35}+q^{31}-q^{29}-q^{27}+q^{23}+q^{21}-q^{19}-q^{17}+q^{15}+2 q^{13}+q^{11}-q^9-q^7+2 q^5+2 q^3-2 q^{-1} +2 q^{-5} + q^{-7} - q^{-9} - q^{-11} + q^{-15} }[/math]
4	[math]\displaystyle{ q^{140}-q^{138}-q^{136}+3 q^{130}-q^{128}-q^{126}-2 q^{124}-q^{122}+5 q^{120}+q^{118}-q^{116}-4 q^{114}-2 q^{112}+5 q^{110}+4 q^{108}-q^{106}-6 q^{104}-4 q^{102}+4 q^{100}+5 q^{98}+2 q^{96}-4 q^{94}-6 q^{92}+3 q^{88}+4 q^{86}-2 q^{82}-q^{80}-2 q^{78}+q^{76}+4 q^{74}+2 q^{72}-q^{70}-3 q^{68}-q^{66}+3 q^{64}+2 q^{62}-2 q^{60}-3 q^{58}-q^{56}+3 q^{54}+2 q^{52}-3 q^{50}-3 q^{48}+4 q^{44}+3 q^{42}-3 q^{40}-4 q^{38}-2 q^{36}+2 q^{34}+3 q^{32}-q^{30}-q^{28}-2 q^{26}+3 q^{22}+2 q^{20}+2 q^{18}-q^{16}-2 q^{14}+4 q^8+2 q^6-q^4-2 q^2-3+2 q^{-2} +3 q^{-4} +2 q^{-6} -4 q^{-10} - q^{-12} + q^{-14} +2 q^{-16} +2 q^{-18} - q^{-20} - q^{-22} - q^{-24} + q^{-28} }[/math]
5	[math]\displaystyle{ q^{205}-q^{203}-q^{201}+2 q^{195}+q^{193}-q^{191}-3 q^{189}-q^{187}+q^{185}+4 q^{183}+3 q^{181}-q^{179}-5 q^{177}-5 q^{175}+3 q^{173}+6 q^{171}+4 q^{169}-2 q^{167}-9 q^{165}-7 q^{163}+5 q^{161}+11 q^{159}+5 q^{157}-5 q^{155}-11 q^{153}-9 q^{151}+5 q^{149}+14 q^{147}+10 q^{145}-2 q^{143}-12 q^{141}-10 q^{139}-q^{137}+9 q^{135}+12 q^{133}+5 q^{131}-5 q^{129}-9 q^{127}-8 q^{125}-2 q^{123}+4 q^{121}+5 q^{119}+4 q^{117}-5 q^{113}-5 q^{111}-3 q^{109}+2 q^{107}+6 q^{105}+6 q^{103}-q^{101}-5 q^{99}-3 q^{97}+q^{95}+5 q^{93}+4 q^{91}-3 q^{89}-5 q^{87}-q^{85}+3 q^{83}+6 q^{81}+3 q^{79}-4 q^{77}-8 q^{75}-3 q^{73}+4 q^{71}+8 q^{69}+5 q^{67}-3 q^{65}-10 q^{63}-7 q^{61}+q^{59}+8 q^{57}+7 q^{55}-8 q^{51}-9 q^{49}-3 q^{47}+7 q^{45}+9 q^{43}+4 q^{41}-3 q^{39}-8 q^{37}-5 q^{35}+5 q^{31}+6 q^{29}+2 q^{27}-q^{25}-3 q^{23}-3 q^{21}-q^{19}+2 q^{17}+4 q^{15}+3 q^{13}+3 q^{11}-3 q^7-4 q^5-2 q^3+q+4 q^{-1} +5 q^{-3} +2 q^{-5} -2 q^{-7} -5 q^{-9} -4 q^{-11} +3 q^{-15} +5 q^{-17} +3 q^{-19} - q^{-21} -4 q^{-23} -3 q^{-25} - q^{-27} + q^{-29} +3 q^{-31} +2 q^{-33} - q^{-37} - q^{-39} - q^{-41} + q^{-45} }[/math]
6	[math]\displaystyle{ q^{282}-q^{280}-q^{278}+2 q^{272}+q^{268}-3 q^{266}-2 q^{264}+q^{262}+q^{260}+4 q^{258}+2 q^{256}+q^{254}-7 q^{252}-4 q^{250}+2 q^{246}+6 q^{244}+3 q^{242}-q^{240}-9 q^{238}-3 q^{236}+2 q^{234}+4 q^{232}+6 q^{230}-6 q^{226}-10 q^{224}+2 q^{222}+11 q^{220}+10 q^{218}+6 q^{216}-8 q^{214}-17 q^{212}-14 q^{210}+4 q^{208}+19 q^{206}+19 q^{204}+8 q^{202}-10 q^{200}-24 q^{198}-24 q^{196}-5 q^{194}+14 q^{192}+24 q^{190}+20 q^{188}+4 q^{186}-14 q^{184}-24 q^{182}-15 q^{180}-q^{178}+11 q^{176}+18 q^{174}+15 q^{172}+5 q^{170}-7 q^{168}-11 q^{166}-11 q^{164}-7 q^{162}+q^{160}+8 q^{158}+12 q^{156}+8 q^{154}-8 q^{150}-12 q^{148}-8 q^{146}+9 q^{142}+9 q^{140}+q^{138}-6 q^{136}-8 q^{134}-4 q^{132}+3 q^{130}+10 q^{128}+5 q^{126}-5 q^{124}-9 q^{122}-7 q^{120}+q^{118}+9 q^{116}+14 q^{114}+8 q^{112}-8 q^{110}-14 q^{108}-10 q^{106}+11 q^{102}+16 q^{100}+11 q^{98}-5 q^{96}-15 q^{94}-14 q^{92}-7 q^{90}+6 q^{88}+16 q^{86}+16 q^{84}+2 q^{82}-12 q^{80}-17 q^{78}-15 q^{76}-2 q^{74}+13 q^{72}+21 q^{70}+13 q^{68}-2 q^{66}-14 q^{64}-21 q^{62}-14 q^{60}+2 q^{58}+17 q^{56}+19 q^{54}+10 q^{52}-3 q^{50}-17 q^{48}-19 q^{46}-10 q^{44}+6 q^{42}+14 q^{40}+15 q^{38}+10 q^{36}-3 q^{34}-10 q^{32}-12 q^{30}-5 q^{28}+q^{26}+6 q^{24}+10 q^{22}+6 q^{20}+3 q^{18}-3 q^{16}-3 q^{14}-5 q^{12}-5 q^{10}-q^8+q^6+5 q^4+4 q^2+7+3 q^{-2} -3 q^{-4} -5 q^{-6} -7 q^{-8} -4 q^{-10} -2 q^{-12} +6 q^{-14} +8 q^{-16} +6 q^{-18} +2 q^{-20} -3 q^{-22} -6 q^{-24} -9 q^{-26} -2 q^{-28} +2 q^{-30} +5 q^{-32} +6 q^{-34} +4 q^{-36} + q^{-38} -5 q^{-40} -4 q^{-42} -3 q^{-44} - q^{-46} + q^{-48} +3 q^{-50} +3 q^{-52} - q^{-58} - q^{-60} - q^{-62} + q^{-66} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^6+q^4+q^2+1 }[/math]
1,1	[math]\displaystyle{ q^{68}-2 q^{66}+2 q^{64}-2 q^{62}+3 q^{60}-4 q^{58}+2 q^{56}+2 q^{52}-2 q^{50}-2 q^{46}+q^{44}-2 q^{42}+2 q^{40}+2 q^{38}+6 q^{34}-6 q^{32}+6 q^{30}-11 q^{28}+8 q^{26}-10 q^{24}+6 q^{22}-6 q^{20}+4 q^{18}-2 q^{14}+5 q^{12}-4 q^{10}+8 q^8-6 q^6+6 q^4-2 q^2+4+ q^{-4} }[/math]
2,0	[math]\displaystyle{ q^{60}-q^{54}-q^{52}-q^{44}+q^{42}+q^{40}+q^{38}+2 q^{34}+q^{32}-q^{30}-q^{28}-2 q^{26}-2 q^{24}-2 q^{22}+q^{16}+2 q^{14}+2 q^{12}+q^6+q^4+1+ q^{-2} + q^{-4} }[/math]
3,0	[math]\displaystyle{ q^{108}-q^{102}-q^{100}+q^{94}-2 q^{92}-q^{90}+3 q^{86}+q^{84}-q^{82}+2 q^{78}+4 q^{76}-3 q^{72}-3 q^{70}+q^{66}-3 q^{62}-q^{60}+2 q^{56}+q^{50}+q^{48}-q^{44}+q^{42}-q^{38}-2 q^{36}+2 q^{32}+q^{30}-2 q^{28}-3 q^{26}-q^{24}+2 q^{22}+q^{20}-q^{16}+2 q^{14}+4 q^{12}+3 q^{10}-q^6+2 q^2+1- q^{-4} + q^{-8} + q^{-10} + q^{-12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{56}-q^{54}-q^{52}+q^{50}-q^{48}-q^{46}+q^{44}+q^{42}+2 q^{38}+q^{36}-2 q^{26}-q^{24}-q^{22}-3 q^{20}-2 q^{18}+2 q^{12}+3 q^{10}+2 q^8+2 q^6+2 q^4+1 }[/math]
1,0,0	[math]\displaystyle{ q^{31}+q^{27}-q^{25}-q^{21}-q^{19}-q^{17}-q^{15}+2 q^9+q^7+2 q^5+q^3+q }[/math]
1,0,1	[math]\displaystyle{ q^{90}-2 q^{88}+q^{86}+q^{84}-2 q^{82}+3 q^{80}-2 q^{78}-q^{76}+q^{74}+q^{70}-q^{68}+q^{62}-2 q^{60}+2 q^{58}-3 q^{56}-2 q^{54}+3 q^{52}-4 q^{50}+5 q^{48}-q^{46}+4 q^{44}+q^{40}+3 q^{38}-2 q^{36}+4 q^{34}-5 q^{32}+2 q^{30}-5 q^{28}-q^{26}-5 q^{24}-4 q^{22}-q^{20}-3 q^{18}+4 q^{16}-2 q^{14}+6 q^{12}+2 q^{10}+5 q^8+6 q^6+2 q^4+4 q^2+ q^{-2} }[/math]

A4 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{78}-q^{76}-q^{72}+q^{70}-q^{68}-q^{64}+q^{62}+q^{58}+q^{54}+q^{52}+2 q^{48}-q^{46}+2 q^{44}-q^{42}+2 q^{40}-2 q^{38}+q^{36}-3 q^{34}-q^{32}-4 q^{30}-2 q^{28}-3 q^{26}-2 q^{24}-q^{22}+3 q^{18}+2 q^{16}+4 q^{14}+2 q^{12}+4 q^{10}+q^8+2 q^6+q^2 }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2 q^{120}-2 q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2 q^{92}-q^{84}+q^{82}-2 q^{78}+q^{76}-q^{74}+q^{70}-4 q^{68}+2 q^{66}-3 q^{64}-3 q^{58}+3 q^{56}-2 q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2 q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2 q^{30}-2 q^{28}+3 q^{26}-q^{24}+q^{22}+3 q^{20}-3 q^{18}+4 q^{16}+q^{12}+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["8 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-3 z^4+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]=

{ 17, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 1- q^{-1} +2 q^{-2} -2 q^{-3} +3 q^{-4} -3 q^{-5} +2 q^{-6} -2 q^{-7} + q^{-8} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^4 a^6+3 z^2 a^6+a^6-z^6 a^4-5 z^4 a^4-7 z^2 a^4-3 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^2 a^{10}+2 z^3 a^9-z a^9+2 z^4 a^8-z^2 a^8+2 z^5 a^7-2 z^3 a^7-z a^7+2 z^6 a^6-5 z^4 a^6+3 z^2 a^6-a^6+z^7 a^5-2 z^5 a^5-z^3 a^5+z a^5+3 z^6 a^4-12 z^4 a^4+12 z^2 a^4-3 a^4+z^7 a^3-4 z^5 a^3+3 z^3 a^3+z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 a^2 }[/math]