8 6

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+6 t-7+6 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 23, -2 }
Jones polynomial	[math]\displaystyle{ q-1+3 q^{-1} -4 q^{-2} +4 q^{-3} -4 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6+a^6-z^4 a^4-2 z^2 a^4-a^4-z^4 a^2-2 z^2 a^2-a^2+z^2+2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8-2 z^2 a^8+2 z^5 a^7-4 z^3 a^7+z a^7+2 z^6 a^6-4 z^4 a^6+3 z^2 a^6-a^6+z^7 a^5-z^5 a^5+2 z^3 a^5-z a^5+3 z^6 a^4-6 z^4 a^4+6 z^2 a^4-a^4+z^7 a^3-2 z^5 a^3+5 z^3 a^3-3 z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}+q^{16}-q^{14}-q^{10}-q^8-q^4+2 q^2+1+ q^{-2} + q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-3 q^{102}+6 q^{100}-7 q^{98}+7 q^{96}-4 q^{94}-2 q^{92}+7 q^{90}-9 q^{88}+11 q^{86}-6 q^{84}+q^{82}+5 q^{80}-7 q^{78}+7 q^{76}-2 q^{74}-3 q^{72}+6 q^{70}-5 q^{68}+2 q^{66}+4 q^{64}-9 q^{62}+12 q^{60}-10 q^{58}+5 q^{56}+q^{54}-10 q^{52}+13 q^{50}-13 q^{48}+10 q^{46}-5 q^{44}-3 q^{42}+7 q^{40}-10 q^{38}+6 q^{36}-3 q^{34}-4 q^{32}+5 q^{30}-5 q^{28}-q^{26}+6 q^{24}-8 q^{22}+8 q^{20}-6 q^{18}-q^{16}+6 q^{14}-8 q^{12}+10 q^{10}-5 q^8+3 q^6+2 q^4-2 q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_6.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-q^{13}+q^{11}-q^9-q^3+2 q+ q^{-3} }[/math]
2	[math]\displaystyle{ q^{42}-q^{40}-q^{38}+3 q^{36}-q^{34}-4 q^{32}+3 q^{30}+q^{28}-4 q^{26}+3 q^{24}+2 q^{22}-2 q^{20}+q^{16}+q^{14}-3 q^{12}+q^{10}+4 q^8-4 q^6-q^4+4 q^2-2- q^{-2} +2 q^{-4} + q^{-10} }[/math]
3	[math]\displaystyle{ q^{81}-q^{79}-q^{77}+q^{75}+2 q^{73}-q^{71}-5 q^{69}+7 q^{65}+3 q^{63}-7 q^{61}-7 q^{59}+6 q^{57}+10 q^{55}-5 q^{53}-10 q^{51}+3 q^{49}+12 q^{47}-q^{45}-10 q^{43}-q^{41}+7 q^{39}+q^{37}-5 q^{35}-3 q^{33}+q^{31}+5 q^{29}+q^{27}-6 q^{25}-4 q^{23}+8 q^{21}+7 q^{19}-7 q^{17}-10 q^{15}+8 q^{13}+10 q^{11}-3 q^9-10 q^7+q^5+9 q^3+q-5 q^{-1} -2 q^{-3} +3 q^{-5} + q^{-7} - q^{-9} - q^{-11} + q^{-13} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{132}-q^{130}-q^{128}+q^{126}+2 q^{122}-3 q^{120}-3 q^{118}+2 q^{116}+2 q^{114}+9 q^{112}-3 q^{110}-11 q^{108}-6 q^{106}-q^{104}+20 q^{102}+10 q^{100}-7 q^{98}-19 q^{96}-19 q^{94}+20 q^{92}+26 q^{90}+9 q^{88}-21 q^{86}-36 q^{84}+6 q^{82}+28 q^{80}+24 q^{78}-11 q^{76}-39 q^{74}-5 q^{72}+20 q^{70}+25 q^{68}-q^{66}-26 q^{64}-9 q^{62}+8 q^{60}+17 q^{58}+5 q^{56}-10 q^{54}-10 q^{52}-2 q^{50}+9 q^{48}+12 q^{46}+5 q^{44}-16 q^{42}-15 q^{40}+3 q^{38}+19 q^{36}+19 q^{34}-20 q^{32}-28 q^{30}-8 q^{28}+24 q^{26}+36 q^{24}-13 q^{22}-30 q^{20}-19 q^{18}+14 q^{16}+38 q^{14}+4 q^{12}-16 q^{10}-23 q^8-4 q^6+23 q^4+10 q^2-12 q^{-2} -9 q^{-4} +6 q^{-6} +4 q^{-8} +5 q^{-10} -2 q^{-12} -4 q^{-14} + q^{-16} - q^{-18} +2 q^{-20} - q^{-24} + q^{-26} - q^{-28} + q^{-36} }[/math]
5	[math]\displaystyle{ q^{195}-q^{193}-q^{191}+q^{189}-q^{181}-2 q^{179}+2 q^{177}+5 q^{175}+2 q^{173}-q^{171}-6 q^{169}-9 q^{167}-5 q^{165}+8 q^{163}+17 q^{161}+13 q^{159}-20 q^{155}-28 q^{153}-17 q^{151}+16 q^{149}+42 q^{147}+37 q^{145}+3 q^{143}-45 q^{141}-63 q^{139}-29 q^{137}+37 q^{135}+81 q^{133}+58 q^{131}-18 q^{129}-87 q^{127}-86 q^{125}-11 q^{123}+82 q^{121}+106 q^{119}+33 q^{117}-65 q^{115}-108 q^{113}-54 q^{111}+47 q^{109}+105 q^{107}+63 q^{105}-30 q^{103}-86 q^{101}-64 q^{99}+13 q^{97}+69 q^{95}+56 q^{93}-2 q^{91}-50 q^{89}-47 q^{87}-6 q^{85}+30 q^{83}+38 q^{81}+12 q^{79}-18 q^{77}-30 q^{75}-21 q^{73}+4 q^{71}+28 q^{69}+32 q^{67}+8 q^{65}-27 q^{63}-44 q^{61}-23 q^{59}+29 q^{57}+61 q^{55}+40 q^{53}-24 q^{51}-78 q^{49}-61 q^{47}+20 q^{45}+90 q^{43}+80 q^{41}-93 q^{37}-105 q^{35}-14 q^{33}+81 q^{31}+106 q^{29}+41 q^{27}-59 q^{25}-106 q^{23}-60 q^{21}+32 q^{19}+85 q^{17}+68 q^{15}-58 q^{11}-64 q^9-19 q^7+34 q^5+49 q^3+28 q-7 q^{-1} -31 q^{-3} -26 q^{-5} -4 q^{-7} +14 q^{-9} +18 q^{-11} +10 q^{-13} -5 q^{-15} -10 q^{-17} -7 q^{-19} -2 q^{-21} +5 q^{-23} +6 q^{-25} + q^{-27} - q^{-29} - q^{-31} -3 q^{-33} +2 q^{-37} + q^{-43} - q^{-45} - q^{-47} + q^{-55} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}+q^{16}-q^{14}-q^{10}-q^8-q^4+2 q^2+1+ q^{-2} + q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{60}-2 q^{58}+4 q^{56}-8 q^{54}+13 q^{52}-16 q^{50}+22 q^{48}-26 q^{46}+25 q^{44}-24 q^{42}+16 q^{40}-8 q^{38}-5 q^{36}+18 q^{34}-30 q^{32}+40 q^{30}-46 q^{28}+50 q^{26}-44 q^{24}+42 q^{22}-28 q^{20}+18 q^{18}-4 q^{16}-8 q^{14}+15 q^{12}-24 q^{10}+22 q^8-24 q^6+18 q^4-16 q^2+12-6 q^{-2} +8 q^{-4} -2 q^{-6} +4 q^{-8} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{56}+q^{48}-3 q^{44}-q^{42}+q^{40}-2 q^{36}+2 q^{32}-q^{28}+2 q^{26}+2 q^{24}+2 q^{20}+2 q^{18}-q^{16}+q^{12}-q^{10}-4 q^8-2 q^6+q^4-2 q^2-1+2 q^{-2} +3 q^{-4} + q^{-6} + q^{-8} + q^{-10} + q^{-12} }[/math]
3,0	[math]\displaystyle{ q^{102}+q^{92}-2 q^{90}-2 q^{88}-3 q^{86}+q^{84}+5 q^{82}+2 q^{80}-2 q^{78}-7 q^{76}-q^{74}+6 q^{72}+5 q^{70}-2 q^{68}-8 q^{66}+10 q^{62}+8 q^{60}-q^{58}-7 q^{56}+6 q^{52}+q^{50}-7 q^{48}-7 q^{46}+q^{42}-3 q^{40}-4 q^{38}+q^{36}+3 q^{34}-2 q^{32}-2 q^{30}+2 q^{28}+7 q^{26}+3 q^{24}-6 q^{22}-q^{20}+7 q^{18}+12 q^{16}+q^{14}-8 q^{12}-2 q^{10}+6 q^8+7 q^6-5 q^4-10 q^2-5+2 q^{-2} +3 q^{-4} - q^{-6} -3 q^{-8} +2 q^{-12} +3 q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} + q^{-24} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-q^{46}+q^{42}-3 q^{40}+q^{38}+3 q^{36}-3 q^{34}+q^{32}+3 q^{30}-2 q^{28}+2 q^{24}+q^{22}+q^{16}-2 q^{14}-5 q^{12}+q^{10}-2 q^8-4 q^6+4 q^4+2 q^2+1+3 q^{-2} +2 q^{-4} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{29}+q^{25}+q^{21}-q^{19}-q^{15}-q^{13}-q^{11}-q^9-q^5+2 q^3+q+2 q^{-1} + q^{-3} + q^{-5} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{62}-q^{58}+q^{56}+q^{54}-2 q^{52}-q^{50}+2 q^{48}+q^{46}-3 q^{44}-q^{42}+2 q^{40}-q^{38}-2 q^{36}+3 q^{34}+3 q^{32}+4 q^{28}+4 q^{26}+q^{24}-q^{22}+q^{20}-2 q^{18}-7 q^{16}-5 q^{14}-2 q^{12}-4 q^{10}-4 q^8+2 q^6+3 q^4+3 q^2+3+4 q^{-2} +3 q^{-4} +2 q^{-6} + q^{-8} + q^{-10} }[/math]
1,0,0,0	[math]\displaystyle{ q^{36}+q^{32}+q^{30}+q^{26}-q^{24}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}-q^6+2 q^4+q^2+2+2 q^{-2} + q^{-4} + q^{-6} }[/math]

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{66}-q^{64}+q^{62}-2 q^{60}+2 q^{58}-3 q^{56}+2 q^{54}-2 q^{52}+3 q^{50}-q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}-3 q^{38}+3 q^{36}-3 q^{34}+5 q^{32}-4 q^{30}+4 q^{28}-4 q^{26}+4 q^{24}-3 q^{22}-4 q^{18}-2 q^{16}-q^{14}-3 q^{12}-3 q^8+4 q^6+4 q^2+1+4 q^{-2} + q^{-4} +2 q^{-6} + q^{-10} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-3 q^{102}+6 q^{100}-7 q^{98}+7 q^{96}-4 q^{94}-2 q^{92}+7 q^{90}-9 q^{88}+11 q^{86}-6 q^{84}+q^{82}+5 q^{80}-7 q^{78}+7 q^{76}-2 q^{74}-3 q^{72}+6 q^{70}-5 q^{68}+2 q^{66}+4 q^{64}-9 q^{62}+12 q^{60}-10 q^{58}+5 q^{56}+q^{54}-10 q^{52}+13 q^{50}-13 q^{48}+10 q^{46}-5 q^{44}-3 q^{42}+7 q^{40}-10 q^{38}+6 q^{36}-3 q^{34}-4 q^{32}+5 q^{30}-5 q^{28}-q^{26}+6 q^{24}-8 q^{22}+8 q^{20}-6 q^{18}-q^{16}+6 q^{14}-8 q^{12}+10 q^{10}-5 q^8+3 q^6+2 q^4-2 q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18} }[/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["8 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^2+6 t-7+6 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^4-2 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]=

{ 23, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

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