8 20

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^2-2 t+3-2 t^{-1} + t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 9, 0 }
Jones polynomial	[math]\displaystyle{ -q+2- q^{-1} +2 q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+4 a^2-z^2-1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^4 z^6+a^2 z^6+a^5 z^5+2 a^3 z^5+a z^5-4 a^4 z^4-4 a^2 z^4-4 a^5 z^3-7 a^3 z^3-3 a z^3+4 a^4 z^2+6 a^2 z^2+2 z^2+3 a^5 z+5 a^3 z+3 a z+z a^{-1} -2 a^4-4 a^2-1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}-q^{14}-q^{12}+2 q^8+2 q^6+2 q^4+q^2- q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}+q^{76}-q^{74}-2 q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2 q^{58}-3 q^{52}-q^{50}-q^{48}+q^{44}-2 q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+2 q^{30}+2 q^{28}+3 q^{26}+2 q^{22}+3 q^{20}+q^{18}+q^{16}+q^{14}+3 q^{10}-2 q^6+q^4+1- q^{-2} -2 q^{-4} - q^{-12} - q^{-14} - q^{-20} + q^{-24} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_20.

A1 Invariants.

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

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}+q^{30}-q^{26}-2 q^{24}-4 q^{22}-4 q^{20}-3 q^{18}+2 q^{14}+6 q^{12}+6 q^{10}+6 q^8+3 q^6+q^4-q^2-2-2 q^{-2} - q^{-4} - q^{-6} + q^{-10} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{19}-2 q^{17}-q^{15}+2 q^{11}+3 q^9+3 q^7+2 q^5+q^3- q^{-1} - q^{-5} }[/math]
1,0,1	[math]\displaystyle{ q^{56}+2 q^{52}+q^{48}-q^{44}+q^{42}-2 q^{40}+q^{38}-4 q^{36}-2 q^{34}-6 q^{32}-5 q^{30}-5 q^{28}-6 q^{26}-2 q^{22}+6 q^{20}+5 q^{18}+10 q^{16}+9 q^{14}+9 q^{12}+8 q^{10}+2 q^8+3 q^6-2 q^4-3- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} - q^{-10} + q^{-16} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}+q^{42}+2 q^{40}+2 q^{38}+q^{36}-q^{34}-3 q^{32}-6 q^{30}-9 q^{28}-9 q^{26}-7 q^{24}-3 q^{22}+q^{20}+8 q^{18}+12 q^{16}+13 q^{14}+12 q^{12}+9 q^{10}+3 q^8-q^6-4 q^4-5 q^2-5-3 q^{-2} - q^{-4} + q^{-10} + q^{-12} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{24}-2 q^{22}-2 q^{20}-q^{18}+2 q^{14}+3 q^{12}+4 q^{10}+3 q^8+2 q^6+q^4-1- q^{-2} - q^{-6} }[/math]

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{80}+q^{76}-q^{74}-2 q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2 q^{58}-3 q^{52}-q^{50}-q^{48}+q^{44}-2 q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+2 q^{30}+2 q^{28}+3 q^{26}+2 q^{22}+3 q^{20}+q^{18}+q^{16}+q^{14}+3 q^{10}-2 q^6+q^4+1- q^{-2} -2 q^{-4} - q^{-12} - q^{-14} - q^{-20} + q^{-24} }[/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 20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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