6 2

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^2+3 t-3+3 t^{-1} - t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 11, -2 }
Jones polynomial	[math]\displaystyle{ q-1+2 q^{-1} -2 q^{-2} +2 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^2-3 z^2 a^2-2 a^2+z^2+2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6+2 z^3 a^5-z a^5+2 z^4 a^4-2 z^2 a^4+a^4+z^5 a^3-z a^3+3 z^4 a^2-6 z^2 a^2+2 a^2+z^5 a-2 z^3 a+z^4-3 z^2+2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{16}-q^8-q^4+q^2+1+ q^{-2} + q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{86}-q^{84}+q^{82}-q^{80}-q^{78}-q^{74}+3 q^{72}-2 q^{70}+q^{68}+2 q^{62}-q^{60}+q^{58}+q^{56}+q^{52}+3 q^{46}-3 q^{44}+q^{42}-q^{40}-q^{38}+2 q^{36}-4 q^{34}+q^{32}-2 q^{30}-3 q^{24}+q^{22}-q^{20}-q^{14}+q^{12}+q^{10}+2 q^6-q^4+2 q^2+1- q^{-2} +3 q^{-4} - q^{-6} +2 q^{-8} - q^{-12} +2 q^{-14} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 6_2.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{11}-q^9+q+ q^{-3} }[/math]
2	[math]\displaystyle{ q^{30}-q^{28}-q^{26}+2 q^{24}-q^{22}-q^{20}+q^{18}+q^{10}-q^6+q^4+q^2-1+ q^{-2} + q^{-4} - q^{-6} + q^{-10} }[/math]
3	[math]\displaystyle{ q^{57}-q^{55}-q^{53}+q^{51}+q^{49}-q^{47}-3 q^{45}+2 q^{43}+3 q^{41}-q^{39}-2 q^{37}+q^{35}+2 q^{33}-q^{29}-q^{27}-2 q^{19}-q^{17}+2 q^{15}+2 q^{13}-q^{11}+2 q^7+2 q^5-q^3-2 q+ q^{-1} +2 q^{-3} -2 q^{-7} +2 q^{-11} + q^{-13} - q^{-15} - q^{-17} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{92}-q^{90}-q^{88}+q^{86}+q^{82}-3 q^{80}-q^{78}+3 q^{76}+2 q^{74}+3 q^{72}-5 q^{70}-4 q^{68}+3 q^{66}+4 q^{64}+3 q^{62}-5 q^{60}-5 q^{58}+2 q^{54}+3 q^{52}-q^{50}-2 q^{48}+q^{44}+2 q^{42}+2 q^{40}+q^{38}-2 q^{36}-2 q^{34}+2 q^{30}+q^{28}-3 q^{26}-3 q^{24}-q^{22}+4 q^{20}+2 q^{18}-3 q^{16}-2 q^{14}+5 q^{10}+3 q^8-2 q^6-2 q^4-2 q^2+4+4 q^{-2} -2 q^{-6} -4 q^{-8} + q^{-10} +3 q^{-12} +2 q^{-14} -4 q^{-18} - q^{-20} + q^{-22} +2 q^{-24} +2 q^{-26} - q^{-28} - q^{-30} - q^{-32} + q^{-36} }[/math]
5	[math]\displaystyle{ q^{135}-q^{133}-q^{131}+q^{129}-q^{123}-q^{121}+3 q^{117}+4 q^{115}-5 q^{111}-5 q^{109}+4 q^{105}+8 q^{103}+3 q^{101}-8 q^{99}-10 q^{97}-5 q^{95}+5 q^{93}+11 q^{91}+7 q^{89}-2 q^{87}-9 q^{85}-6 q^{83}+q^{81}+6 q^{79}+6 q^{77}+q^{75}-4 q^{73}-5 q^{71}-2 q^{69}+q^{67}+2 q^{65}+3 q^{63}+q^{61}-q^{59}-2 q^{57}-3 q^{55}-q^{53}+3 q^{51}+4 q^{49}+q^{47}-3 q^{45}-5 q^{43}-q^{41}+5 q^{39}+5 q^{37}+q^{35}-4 q^{33}-6 q^{31}-q^{29}+5 q^{27}+6 q^{25}-6 q^{21}-7 q^{19}-2 q^{17}+7 q^{15}+8 q^{13}+3 q^{11}-4 q^9-7 q^7-3 q^5+3 q^3+8 q+6 q^{-1} - q^{-3} -6 q^{-5} -6 q^{-7} -2 q^{-9} +4 q^{-11} +7 q^{-13} +4 q^{-15} - q^{-17} -5 q^{-19} -5 q^{-21} - q^{-23} +3 q^{-25} +5 q^{-27} +3 q^{-29} - q^{-31} -4 q^{-33} -3 q^{-35} - q^{-37} + q^{-39} +3 q^{-41} +2 q^{-43} - q^{-47} - q^{-49} - q^{-51} + q^{-55} }[/math]
6	[math]\displaystyle{ q^{186}-q^{184}-q^{182}+q^{180}-2 q^{174}+q^{172}+5 q^{166}+2 q^{164}-2 q^{162}-6 q^{160}-3 q^{158}-3 q^{156}+3 q^{154}+12 q^{152}+7 q^{150}-3 q^{148}-14 q^{146}-10 q^{144}-8 q^{142}+4 q^{140}+21 q^{138}+17 q^{136}+5 q^{134}-14 q^{132}-18 q^{130}-17 q^{128}-3 q^{126}+18 q^{124}+20 q^{122}+12 q^{120}-5 q^{118}-13 q^{116}-17 q^{114}-9 q^{112}+7 q^{110}+13 q^{108}+12 q^{106}+2 q^{104}-4 q^{102}-10 q^{100}-9 q^{98}-2 q^{96}+3 q^{94}+6 q^{92}+4 q^{90}+3 q^{88}-q^{86}-3 q^{84}-4 q^{82}-3 q^{80}+4 q^{76}+6 q^{74}+4 q^{72}-6 q^{68}-6 q^{66}-3 q^{64}+5 q^{62}+8 q^{60}+5 q^{58}-q^{56}-11 q^{54}-8 q^{52}-q^{50}+8 q^{48}+9 q^{46}+4 q^{44}-4 q^{42}-14 q^{40}-9 q^{38}+q^{36}+12 q^{34}+12 q^{32}+7 q^{30}-4 q^{28}-16 q^{26}-12 q^{24}-q^{22}+12 q^{20}+14 q^{18}+11 q^{16}-15 q^{12}-15 q^{10}-7 q^8+7 q^6+12 q^4+15 q^2+7-8 q^{-2} -13 q^{-4} -12 q^{-6} -2 q^{-8} +4 q^{-10} +14 q^{-12} +13 q^{-14} +2 q^{-16} -5 q^{-18} -11 q^{-20} -9 q^{-22} -6 q^{-24} +5 q^{-26} +10 q^{-28} +8 q^{-30} +4 q^{-32} -2 q^{-34} -6 q^{-36} -10 q^{-38} -3 q^{-40} +2 q^{-42} +5 q^{-44} +6 q^{-46} +4 q^{-48} + q^{-50} -5 q^{-52} -4 q^{-54} -3 q^{-56} - q^{-58} + q^{-60} +3 q^{-62} +3 q^{-64} - q^{-70} - q^{-72} - q^{-74} + q^{-78} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{16}-q^8-q^4+q^2+1+ q^{-2} + q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+2 q^{40}-2 q^{38}+5 q^{36}-4 q^{34}+2 q^{32}-4 q^{30}-2 q^{24}+6 q^{22}-4 q^{20}+8 q^{18}-6 q^{16}+6 q^{14}-6 q^{12}+4 q^{10}-2 q^8+q^4-2 q^2+4-4 q^{-2} +4 q^{-4} -2 q^{-6} +4 q^{-8} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{40}-q^{30}-2 q^{28}+q^{20}+2 q^{18}+q^{16}+q^{14}+q^{12}-q^8-q^6-q^2+ q^{-2} + q^{-4} + q^{-8} + q^{-10} + q^{-12} }[/math]
3,0	[math]\displaystyle{ q^{72}-3 q^{60}-2 q^{58}-q^{56}+2 q^{54}+2 q^{52}+2 q^{46}+4 q^{44}+2 q^{42}-q^{38}-q^{34}-3 q^{32}-4 q^{30}-3 q^{28}-2 q^{26}-q^{24}-q^{22}+2 q^{20}+4 q^{18}+4 q^{16}+3 q^{14}+2 q^{12}+3 q^{10}+q^8-q^6-3 q^4-q^2+1+ q^{-2} - q^{-4} - q^{-6} +2 q^{-10} + q^{-12} - q^{-16} + q^{-20} + q^{-22} + q^{-24} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{36}-q^{34}-q^{32}+q^{30}-q^{28}+2 q^{24}+q^{22}+q^{20}+q^{18}-q^{14}-2 q^{12}-q^{10}-q^8-2 q^6+q^4+2 q^2+1+2 q^{-2} +2 q^{-4} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{21}+q^{17}-q^{11}-q^9-q^7-q^5+q^3+q+2 q^{-1} + q^{-3} + q^{-5} }[/math]
1,0,1	[math]\displaystyle{ q^{58}-2 q^{56}+q^{54}+q^{52}-2 q^{50}+5 q^{48}-2 q^{46}-q^{44}+2 q^{42}-5 q^{40}+3 q^{38}-2 q^{36}-2 q^{34}+3 q^{32}-5 q^{30}+2 q^{28}+q^{26}+4 q^{22}+4 q^{20}+q^{18}+3 q^{16}+2 q^{14}-4 q^{12}+3 q^{10}-7 q^8-q^6-q^4-5 q^2+4-2 q^{-2} +5 q^{-4} +4 q^{-6} +2 q^{-8} +4 q^{-10} + q^{-14} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{50}-q^{48}-q^{44}+q^{42}-q^{40}+q^{34}+2 q^{32}+q^{30}+2 q^{28}+2 q^{24}-q^{22}-3 q^{18}-q^{16}-3 q^{14}-q^{12}-2 q^{10}-q^8+q^6+q^4+2 q^2+2+3 q^{-2} + q^{-4} +2 q^{-6} + q^{-10} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{86}-q^{84}+q^{82}-q^{80}-q^{78}-q^{74}+3 q^{72}-2 q^{70}+q^{68}+2 q^{62}-q^{60}+q^{58}+q^{56}+q^{52}+3 q^{46}-3 q^{44}+q^{42}-q^{40}-q^{38}+2 q^{36}-4 q^{34}+q^{32}-2 q^{30}-3 q^{24}+q^{22}-q^{20}-q^{14}+q^{12}+q^{10}+2 q^6-q^4+2 q^2+1- q^{-2} +3 q^{-4} - q^{-6} +2 q^{-8} - q^{-12} +2 q^{-14} + 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["6 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

{ 11, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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