6 1

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t+5-2 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 1-2 z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 9, 0 }
Jones polynomial	[math]\displaystyle{ q^2-q+2-2 q^{-1} + q^{-2} - q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^4-z^2 a^2-a^2-z^2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^5+a z^5+a^4 z^4+2 a^2 z^4+z^4-3 a^3 z^3-2 a z^3+z^3 a^{-1} -3 a^4 z^2-4 a^2 z^2+z^2 a^{-2} +2 a^3 z+2 a z+a^4+a^2- a^{-2} }[/math]
The A2 invariant	[math]\displaystyle{ q^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2 q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2 q^{28}+q^{26}+q^{24}-2 q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}-2 q^{10}-q^8+2 q^6-q^4-1+ q^{-4} + q^{-10} +2 q^{-14} - q^{-18} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 6_1.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-q^3+ q^{-1} + q^{-5} }[/math]
2	[math]\displaystyle{ q^{26}-q^{22}-q^{16}+q^{12}+q^8+q^6+ q^{-2} - q^{-4} - q^{-6} + q^{-8} + q^{-14} }[/math]
3	[math]\displaystyle{ q^{51}-q^{47}-q^{45}+q^{41}-q^{37}+q^{33}+q^{31}-q^{27}+q^{25}+q^{23}-q^{19}-q^{13}-q^{11}+q^7+q^3+ q^{-1} +2 q^{-3} + q^{-5} - q^{-7} - q^{-9} + q^{-11} + q^{-13} - q^{-15} - q^{-17} + q^{-27} }[/math]
4	[math]\displaystyle{ q^{84}-q^{80}-q^{78}-q^{76}+q^{74}+q^{72}+q^{70}-2 q^{66}+q^{62}+q^{60}+q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+2 q^{48}-q^{44}-2 q^{42}+q^{38}-q^{34}-2 q^{32}+2 q^{28}+q^{26}+q^{24}+q^{20}+2 q^{18}-q^{14}-q^{12}-1- q^{-2} +2 q^{-4} + q^{-6} + q^{-8} - q^{-10} -2 q^{-12} + q^{-14} +2 q^{-16} +3 q^{-18} -2 q^{-22} + q^{-28} - q^{-32} - q^{-36} + q^{-44} }[/math]
5	[math]\displaystyle{ q^{125}-q^{121}-q^{119}-q^{117}+q^{113}+2 q^{111}+q^{109}-q^{105}-2 q^{103}-q^{101}+q^{99}+2 q^{97}+q^{95}-q^{91}-2 q^{89}-q^{87}+2 q^{83}+2 q^{81}+q^{79}-q^{77}-3 q^{75}-2 q^{73}+2 q^{69}+2 q^{67}-2 q^{63}-2 q^{61}-q^{59}+2 q^{57}+4 q^{55}+2 q^{53}-q^{51}-q^{49}-q^{47}+q^{45}+2 q^{43}+q^{41}-2 q^{39}-3 q^{37}-2 q^{35}+q^{31}-q^{27}-q^{25}+q^{23}+2 q^{21}+q^{19}-q^{13}+q^{11}+q^9+2 q^5+2 q^3- q^{-1} - q^{-3} + q^{-7} +2 q^{-9} + q^{-11} - q^{-13} -4 q^{-15} -3 q^{-17} +3 q^{-21} +4 q^{-23} + q^{-25} -2 q^{-27} -4 q^{-29} - q^{-31} +2 q^{-33} +3 q^{-35} +2 q^{-37} - q^{-41} - q^{-43} + q^{-47} - q^{-55} - q^{-57} + q^{-65} }[/math]
6	[math]\displaystyle{ q^{174}-q^{170}-q^{168}-q^{166}+2 q^{160}+2 q^{158}+q^{156}-q^{152}-2 q^{150}-3 q^{148}+q^{144}+2 q^{142}+2 q^{140}+q^{138}-3 q^{134}-2 q^{132}-q^{130}+q^{126}+3 q^{124}+3 q^{122}-q^{118}-3 q^{116}-3 q^{114}-3 q^{112}+q^{110}+4 q^{108}+3 q^{106}+2 q^{104}-q^{102}-3 q^{100}-4 q^{98}-q^{96}+2 q^{94}+4 q^{92}+5 q^{90}+2 q^{88}-2 q^{86}-5 q^{84}-3 q^{82}-q^{80}+2 q^{78}+4 q^{76}+2 q^{74}-q^{72}-5 q^{70}-4 q^{68}-2 q^{66}+q^{64}+4 q^{62}+3 q^{60}-3 q^{56}-2 q^{54}+2 q^{50}+4 q^{48}+3 q^{46}-3 q^{42}-q^{40}+q^{38}+q^{36}+q^{34}-q^{30}-2 q^{28}+q^{24}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}+q^6+2 q^4+2 q^2+2- q^{-2} -2 q^{-4} - q^{-8} +2 q^{-10} +4 q^{-12} +5 q^{-14} + q^{-16} -3 q^{-18} -4 q^{-20} -5 q^{-22} - q^{-24} +4 q^{-26} +8 q^{-28} +4 q^{-30} - q^{-32} -5 q^{-34} -8 q^{-36} -4 q^{-38} + q^{-40} +6 q^{-42} +4 q^{-44} + q^{-46} - q^{-48} -4 q^{-50} -3 q^{-52} +3 q^{-56} +2 q^{-58} + q^{-60} + q^{-62} - q^{-64} - q^{-66} + q^{-70} + q^{-76} - q^{-78} - q^{-80} - q^{-82} + q^{-90} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}+q^{24}-q^{20}+q^{12}+2 q^{10}-q^4-q^2-1- q^{-2} + q^{-4} + q^{-8} +2 q^{-10} + q^{-12} + q^{-16} }[/math]
1,0,0	[math]\displaystyle{ q^{19}+q^{17}+q^{15}-q^9-q^7-q^5+ q^{-3} + q^{-7} + q^{-9} + q^{-11} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{38}+q^{34}+q^{30}+q^{16}+q^{12}-q^{10}-q^6-2 q^2- q^{-2} + q^{-6} + q^{-10} + q^{-12} +2 q^{-14} + q^{-16} + q^{-18} + q^{-22} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2 q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2 q^{28}+q^{26}+q^{24}-2 q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}-2 q^{10}-q^8+2 q^6-q^4-1+ q^{-4} + q^{-10} +2 q^{-14} - q^{-18} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} }[/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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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