8 4

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+5 t-5+5 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 19, -2 }
Jones polynomial	[math]\displaystyle{ q^3-q^2+2 q-3+3 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^4+a^4-z^4 a^2-2 z^2 a^2-z^4-3 z^2-2+z^2 a^{-2} +2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+z^6 a^{-2} +3 z^6+3 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^4 z^4-3 a^2 z^4-5 z^4 a^{-2} -11 z^4+2 a^5 z^3-5 a^3 z^3-3 a z^3+4 z^3 a^{-1} +a^6 z^2-3 a^4 z^2-a^2 z^2+7 z^2 a^{-2} +10 z^2+2 a^3 z+a z-z a^{-1} +a^4-2 a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{16}+q^{10}+q^6-q^4-q^2-1- q^{-2} + q^{-4} + q^{-6} + q^{-8} + q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3 q^{72}-2 q^{70}+2 q^{68}-2 q^{66}+q^{62}-2 q^{60}+3 q^{58}-2 q^{56}+q^{54}+q^{50}+2 q^{48}-q^{46}+2 q^{44}+2 q^{38}-q^{36}+q^{34}+2 q^{32}-2 q^{30}+3 q^{28}-3 q^{26}+2 q^{22}-6 q^{20}+5 q^{18}-5 q^{16}+2 q^{12}-4 q^{10}+3 q^8-4 q^6+q^4-q^2-2+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} + q^{-16} +3 q^{-18} -4 q^{-20} +4 q^{-22} -2 q^{-24} + q^{-26} +4 q^{-28} -4 q^{-30} +4 q^{-32} - q^{-34} + q^{-36} + q^{-38} -2 q^{-40} +2 q^{-42} + q^{-46} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_4.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{11}-q^9+q^7- q^{-1} + q^{-3} + q^{-7} }[/math]
2	[math]\displaystyle{ q^{30}-q^{28}+2 q^{24}-2 q^{22}+q^{18}-2 q^{16}+q^{12}-q^8+q^6+2 q^4+2 q^{-2} -2 q^{-6} + q^{-8} -2 q^{-12} + q^{-14} + q^{-16} - q^{-18} + q^{-22} }[/math]
3	[math]\displaystyle{ q^{57}-q^{55}+q^{51}-q^{47}-q^{45}+q^{43}-q^{41}-q^{39}+2 q^{37}+q^{35}-2 q^{33}-q^{31}+3 q^{29}+2 q^{27}-q^{25}-2 q^{23}-q^{21}+2 q^{19}+2 q^{17}-3 q^{13}+3 q^9-3 q^5-q^3+2 q+ q^{-1} - q^{-3} - q^{-5} +2 q^{-7} +2 q^{-9} -2 q^{-13} + q^{-15} +3 q^{-17} + q^{-19} -3 q^{-21} -2 q^{-23} +2 q^{-25} +2 q^{-27} - q^{-29} -3 q^{-31} +2 q^{-35} + q^{-37} - q^{-39} - q^{-41} + q^{-45} }[/math]
4	[math]\displaystyle{ q^{92}-q^{90}+q^{86}-q^{84}+q^{82}-2 q^{80}-2 q^{74}+4 q^{72}-2 q^{66}-4 q^{64}+5 q^{62}+4 q^{60}+2 q^{58}-5 q^{56}-8 q^{54}+3 q^{52}+7 q^{50}+6 q^{48}-2 q^{46}-9 q^{44}-3 q^{42}+2 q^{40}+5 q^{38}+3 q^{36}-2 q^{34}-3 q^{32}-5 q^{30}+5 q^{26}+4 q^{24}-q^{22}-6 q^{20}-3 q^{18}+3 q^{16}+6 q^{14}+q^{12}-4 q^{10}-2 q^8+4 q^6+6 q^4-q^2-3-2 q^{-2} +3 q^{-4} +5 q^{-6} -2 q^{-8} -4 q^{-10} -5 q^{-12} +6 q^{-16} + q^{-18} -5 q^{-22} -4 q^{-24} +4 q^{-26} +4 q^{-28} +5 q^{-30} -2 q^{-32} -6 q^{-34} -2 q^{-36} + q^{-38} +7 q^{-40} +3 q^{-42} -3 q^{-44} -4 q^{-46} -4 q^{-48} +3 q^{-50} +4 q^{-52} +2 q^{-54} - q^{-56} -5 q^{-58} - q^{-60} + q^{-62} +2 q^{-64} +2 q^{-66} - q^{-68} - q^{-70} - q^{-72} + q^{-76} }[/math]
5	[math]\displaystyle{ q^{135}-q^{133}+q^{129}-q^{127}-q^{121}-q^{119}+q^{115}+2 q^{113}+2 q^{111}-q^{109}-5 q^{107}-4 q^{105}+4 q^{103}+8 q^{101}+5 q^{99}-3 q^{97}-12 q^{95}-8 q^{93}+5 q^{91}+16 q^{89}+11 q^{87}-4 q^{85}-17 q^{83}-17 q^{81}-q^{79}+18 q^{77}+20 q^{75}+4 q^{73}-14 q^{71}-21 q^{69}-11 q^{67}+7 q^{65}+19 q^{63}+14 q^{61}-q^{59}-11 q^{57}-13 q^{55}-7 q^{53}+4 q^{51}+11 q^{49}+12 q^{47}+4 q^{45}-3 q^{43}-11 q^{41}-11 q^{39}+10 q^{35}+11 q^{33}+3 q^{31}-9 q^{29}-11 q^{27}-3 q^{25}+7 q^{23}+9 q^{21}+2 q^{19}-7 q^{17}-7 q^{15}+8 q^{11}+8 q^9-2 q^7-10 q^5-10 q^3+q+12 q^{-1} +12 q^{-3} -11 q^{-7} -13 q^{-9} -2 q^{-11} +11 q^{-13} +16 q^{-15} +7 q^{-17} -5 q^{-19} -13 q^{-21} -11 q^{-23} +2 q^{-25} +12 q^{-27} +11 q^{-29} +4 q^{-31} -7 q^{-33} -13 q^{-35} -10 q^{-37} + q^{-39} +9 q^{-41} +10 q^{-43} +6 q^{-45} -4 q^{-47} -11 q^{-49} -10 q^{-51} - q^{-53} +7 q^{-55} +11 q^{-57} +8 q^{-59} - q^{-61} -9 q^{-63} -10 q^{-65} -4 q^{-67} +4 q^{-69} +9 q^{-71} +8 q^{-73} + q^{-75} -5 q^{-77} -8 q^{-79} -5 q^{-81} + q^{-83} +5 q^{-85} +6 q^{-87} +3 q^{-89} -2 q^{-91} -5 q^{-93} -3 q^{-95} - q^{-97} + q^{-99} +3 q^{-101} +2 q^{-103} - q^{-107} - q^{-109} - q^{-111} + q^{-115} }[/math]
6	[math]\displaystyle{ q^{186}-q^{184}+q^{180}-q^{178}-q^{174}+q^{172}-2 q^{170}-q^{168}+3 q^{166}+2 q^{162}+q^{160}-q^{158}-7 q^{156}-3 q^{154}+6 q^{152}+5 q^{150}+7 q^{148}+q^{146}-7 q^{144}-16 q^{142}-5 q^{140}+13 q^{138}+15 q^{136}+13 q^{134}-4 q^{132}-21 q^{130}-28 q^{128}-6 q^{126}+24 q^{124}+30 q^{122}+22 q^{120}-6 q^{118}-35 q^{116}-42 q^{114}-15 q^{112}+26 q^{110}+43 q^{108}+38 q^{106}+8 q^{104}-32 q^{102}-52 q^{100}-35 q^{98}+5 q^{96}+37 q^{94}+49 q^{92}+31 q^{90}-5 q^{88}-38 q^{86}-44 q^{84}-23 q^{82}+3 q^{80}+28 q^{78}+35 q^{76}+24 q^{74}+q^{72}-19 q^{70}-25 q^{68}-24 q^{66}-10 q^{64}+9 q^{62}+26 q^{60}+25 q^{58}+12 q^{56}-8 q^{54}-25 q^{52}-25 q^{50}-11 q^{48}+13 q^{46}+22 q^{44}+18 q^{42}-16 q^{38}-17 q^{36}-7 q^{34}+12 q^{32}+16 q^{30}+7 q^{28}-9 q^{26}-16 q^{24}-9 q^{22}+5 q^{20}+22 q^{18}+20 q^{16}+2 q^{14}-19 q^{12}-24 q^{10}-15 q^8+6 q^6+28 q^4+30 q^2+10-18 q^{-2} -30 q^{-4} -28 q^{-6} -6 q^{-8} +24 q^{-10} +38 q^{-12} +24 q^{-14} -6 q^{-16} -27 q^{-18} -37 q^{-20} -23 q^{-22} +7 q^{-24} +34 q^{-26} +35 q^{-28} +13 q^{-30} -9 q^{-32} -31 q^{-34} -33 q^{-36} -15 q^{-38} +15 q^{-40} +32 q^{-42} +28 q^{-44} +17 q^{-46} -9 q^{-48} -27 q^{-50} -30 q^{-52} -12 q^{-54} +7 q^{-56} +19 q^{-58} +28 q^{-60} +16 q^{-62} - q^{-64} -20 q^{-66} -23 q^{-68} -18 q^{-70} -7 q^{-72} +14 q^{-74} +21 q^{-76} +21 q^{-78} +7 q^{-80} -5 q^{-82} -18 q^{-84} -23 q^{-86} -11 q^{-88} + q^{-90} +15 q^{-92} +17 q^{-94} +17 q^{-96} +4 q^{-98} -11 q^{-100} -15 q^{-102} -15 q^{-104} -6 q^{-106} +2 q^{-108} +14 q^{-110} +14 q^{-112} +7 q^{-114} -7 q^{-118} -10 q^{-120} -11 q^{-122} - q^{-124} +4 q^{-126} +7 q^{-128} +7 q^{-130} +4 q^{-132} -6 q^{-136} -4 q^{-138} -3 q^{-140} - q^{-142} + q^{-144} +3 q^{-146} +3 q^{-148} - q^{-154} - q^{-156} - q^{-158} + q^{-162} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{16}+q^{10}+q^6-q^4-q^2-1- q^{-2} + q^{-4} + q^{-6} + q^{-8} + q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+2 q^{40}-2 q^{38}+5 q^{36}-4 q^{34}+4 q^{32}-4 q^{30}+5 q^{28}-4 q^{26}+2 q^{24}-4 q^{22}+2 q^{20}-2 q^{18}-2 q^{16}+2 q^{14}-6 q^{12}+10 q^{10}-10 q^8+16 q^6-10 q^4+16 q^2-8+8 q^{-2} -7 q^{-4} -6 q^{-10} +5 q^{-12} -8 q^{-14} +10 q^{-16} -8 q^{-18} +6 q^{-20} -4 q^{-22} +4 q^{-24} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{40}+q^{34}+q^{32}-q^{28}-q^{24}-3 q^{22}+q^{18}-q^{14}+2 q^6+3 q^4+3 q^2+2+3 q^{-2} -2 q^{-6} -2 q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} + q^{-18} + q^{-20} + q^{-24} + q^{-26} + q^{-28} }[/math]
3,0	[math]\displaystyle{ q^{72}+q^{66}+q^{64}+q^{62}-2 q^{60}-q^{58}-q^{54}-2 q^{52}-3 q^{50}+q^{48}+3 q^{46}-4 q^{42}-4 q^{40}+3 q^{38}+6 q^{36}+4 q^{34}-q^{32}+3 q^{28}+3 q^{26}+q^{24}-2 q^{22}+q^{18}-q^{14}-2 q^8-5 q^6-3 q^4-q^2-1-3 q^{-2} -3 q^{-4} +2 q^{-6} +5 q^{-8} +6 q^{-10} +2 q^{-12} +3 q^{-14} +5 q^{-16} +6 q^{-18} +3 q^{-20} -2 q^{-22} -4 q^{-24} -3 q^{-26} - q^{-28} -2 q^{-32} -3 q^{-34} -2 q^{-36} +2 q^{-40} + q^{-42} - q^{-46} + q^{-50} + q^{-52} + q^{-54} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{36}-q^{34}-q^{32}+2 q^{30}-q^{26}+2 q^{24}-q^{20}+q^{18}+q^{16}+q^{12}+q^{10}-2 q^6-q^4+q^2-2+ q^{-4} - q^{-6} + q^{-10} + q^{-14} +2 q^{-16} + q^{-20} }[/math]
1,0,0	[math]\displaystyle{ q^{21}+q^{17}+q^{13}+q^9-q^5-q^3-2 q- q^{-1} - q^{-3} + q^{-5} + q^{-7} +2 q^{-9} + q^{-11} + q^{-13} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{46}-q^{42}+q^{38}+q^{36}+q^{32}+2 q^{30}-q^{26}-2 q^{20}+q^{10}+2 q^8+2 q^4+3 q^2+1-2 q^{-2} - q^{-4} -2 q^{-6} -4 q^{-8} -3 q^{-10} - q^{-12} + q^{-14} + q^{-16} +3 q^{-18} +3 q^{-20} +2 q^{-22} + q^{-24} + q^{-26} }[/math]
1,0,0,0	[math]\displaystyle{ q^{26}+q^{22}+q^{20}+q^{16}+q^{12}-q^6-q^4-2 q^2-2- q^{-2} - q^{-4} + q^{-6} + q^{-8} +2 q^{-10} +2 q^{-12} + q^{-14} + q^{-16} }[/math]

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3 q^{72}-2 q^{70}+2 q^{68}-2 q^{66}+q^{62}-2 q^{60}+3 q^{58}-2 q^{56}+q^{54}+q^{50}+2 q^{48}-q^{46}+2 q^{44}+2 q^{38}-q^{36}+q^{34}+2 q^{32}-2 q^{30}+3 q^{28}-3 q^{26}+2 q^{22}-6 q^{20}+5 q^{18}-5 q^{16}+2 q^{12}-4 q^{10}+3 q^8-4 q^6+q^4-q^2-2+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} + q^{-16} +3 q^{-18} -4 q^{-20} +4 q^{-22} -2 q^{-24} + q^{-26} +4 q^{-28} -4 q^{-30} +4 q^{-32} - q^{-34} + q^{-36} + q^{-38} -2 q^{-40} +2 q^{-42} + q^{-46} }[/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 4"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+z^6 a^{-2} +3 z^6+3 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^4 z^4-3 a^2 z^4-5 z^4 a^{-2} -11 z^4+2 a^5 z^3-5 a^3 z^3-3 a z^3+4 z^3 a^{-1} +a^6 z^2-3 a^4 z^2-a^2 z^2+7 z^2 a^{-2} +10 z^2+2 a^3 z+a z-z a^{-1} +a^4-2 a^{-2} -2 }[/math]