6 3

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^2-3 t+5-3 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	{ 13, 0 }
Jones polynomial	[math]\displaystyle{ -q^3+2 q^2-2 q+3-2 q^{-1} +2 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4-a^2 z^2-z^2 a^{-2} +3 z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^5+z^5 a^{-1} +2 a^2 z^4+2 z^4 a^{-2} +4 z^4+a^3 z^3+a z^3+z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-3 z^2 a^{-2} -6 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{10}+2 q^2+1+2 q^{-2} - q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-q^{44}+q^{42}-3 q^{40}+4 q^{38}-4 q^{36}+q^{34}-3 q^{30}+3 q^{28}-3 q^{26}+q^{24}+q^{22}-2 q^{20}+q^{18}+q^{16}-q^{14}+4 q^{12}-3 q^{10}+3 q^8+q^6-q^4+6 q^2-5+6 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +4 q^{-12} - q^{-14} + q^{-16} + q^{-18} -2 q^{-20} + q^{-22} + q^{-24} -3 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-34} -4 q^{-36} +4 q^{-38} -3 q^{-40} + q^{-42} - q^{-44} -2 q^{-46} +2 q^{-48} - q^{-50} + q^{-52} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 6_3.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^7+q^5+q+ q^{-1} + q^{-5} - q^{-7} }[/math]
2	[math]\displaystyle{ q^{20}-q^{18}-2 q^{16}+2 q^{14}-2 q^{10}+2 q^8+q^6-q^4+q^2+1+ q^{-2} - q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-14} -2 q^{-16} - q^{-18} + q^{-20} }[/math]
3	[math]\displaystyle{ -q^{39}+q^{37}+2 q^{35}-3 q^{31}-2 q^{29}+4 q^{27}+2 q^{25}-4 q^{23}-4 q^{21}+3 q^{19}+4 q^{17}-2 q^{15}-4 q^{13}+3 q^{11}+4 q^9-2 q^5+q+ q^{-1} -2 q^{-5} +4 q^{-9} +3 q^{-11} -4 q^{-13} -2 q^{-15} +4 q^{-17} +3 q^{-19} -4 q^{-21} -4 q^{-23} +2 q^{-25} +4 q^{-27} -2 q^{-29} -3 q^{-31} +2 q^{-35} + q^{-37} - q^{-39} }[/math]
4	[math]\displaystyle{ q^{64}-q^{62}-2 q^{60}+q^{56}+5 q^{54}-4 q^{50}-4 q^{48}-2 q^{46}+10 q^{44}+5 q^{42}-4 q^{40}-9 q^{38}-8 q^{36}+9 q^{34}+9 q^{32}+q^{30}-9 q^{28}-11 q^{26}+7 q^{24}+10 q^{22}+4 q^{20}-6 q^{18}-9 q^{16}+3 q^{14}+6 q^{12}+4 q^{10}-2 q^8-5 q^6+2 q^2+3+2 q^{-2} -5 q^{-6} -2 q^{-8} +4 q^{-10} +6 q^{-12} +3 q^{-14} -9 q^{-16} -6 q^{-18} +4 q^{-20} +10 q^{-22} +7 q^{-24} -11 q^{-26} -9 q^{-28} + q^{-30} +9 q^{-32} +9 q^{-34} -8 q^{-36} -9 q^{-38} -4 q^{-40} +5 q^{-42} +10 q^{-44} -2 q^{-46} -4 q^{-48} -4 q^{-50} +5 q^{-54} + q^{-56} -2 q^{-60} - q^{-62} + q^{-64} }[/math]
5	[math]\displaystyle{ -q^{95}+q^{93}+2 q^{91}-q^{87}-3 q^{85}-3 q^{83}+6 q^{79}+6 q^{77}+q^{75}-6 q^{73}-10 q^{71}-6 q^{69}+5 q^{67}+17 q^{65}+13 q^{63}-3 q^{61}-17 q^{59}-20 q^{57}-5 q^{55}+17 q^{53}+25 q^{51}+9 q^{49}-15 q^{47}-27 q^{45}-17 q^{43}+12 q^{41}+27 q^{39}+19 q^{37}-6 q^{35}-25 q^{33}-19 q^{31}+4 q^{29}+21 q^{27}+17 q^{25}-16 q^{21}-16 q^{19}+11 q^{15}+11 q^{13}+3 q^{11}-6 q^9-8 q^7-3 q^5+3 q^3+6 q+6 q^{-1} +3 q^{-3} -3 q^{-5} -8 q^{-7} -6 q^{-9} +3 q^{-11} +11 q^{-13} +11 q^{-15} -16 q^{-19} -16 q^{-21} +17 q^{-25} +21 q^{-27} +4 q^{-29} -19 q^{-31} -25 q^{-33} -6 q^{-35} +19 q^{-37} +27 q^{-39} +12 q^{-41} -17 q^{-43} -27 q^{-45} -15 q^{-47} +9 q^{-49} +25 q^{-51} +17 q^{-53} -5 q^{-55} -20 q^{-57} -17 q^{-59} -3 q^{-61} +13 q^{-63} +17 q^{-65} +5 q^{-67} -6 q^{-69} -10 q^{-71} -6 q^{-73} + q^{-75} +6 q^{-77} +6 q^{-79} -3 q^{-83} -3 q^{-85} - q^{-87} +2 q^{-91} + q^{-93} - q^{-95} }[/math]
6	[math]\displaystyle{ q^{132}-q^{130}-2 q^{128}+q^{124}+3 q^{122}+q^{120}+3 q^{118}-2 q^{116}-8 q^{114}-5 q^{112}-q^{110}+6 q^{108}+8 q^{106}+14 q^{104}+q^{102}-14 q^{100}-19 q^{98}-15 q^{96}+13 q^{92}+38 q^{90}+25 q^{88}-4 q^{86}-29 q^{84}-41 q^{82}-28 q^{80}-q^{78}+49 q^{76}+53 q^{74}+26 q^{72}-18 q^{70}-53 q^{68}-58 q^{66}-30 q^{64}+39 q^{62}+64 q^{60}+53 q^{58}+7 q^{56}-43 q^{54}-66 q^{52}-49 q^{50}+18 q^{48}+53 q^{46}+57 q^{44}+22 q^{42}-24 q^{40}-53 q^{38}-47 q^{36}+4 q^{34}+33 q^{32}+43 q^{30}+21 q^{28}-9 q^{26}-32 q^{24}-33 q^{22}-3 q^{20}+14 q^{18}+23 q^{16}+15 q^{14}+q^{12}-13 q^{10}-17 q^8-7 q^6+2 q^4+11 q^2+13+11 q^{-2} +2 q^{-4} -7 q^{-6} -17 q^{-8} -13 q^{-10} + q^{-12} +15 q^{-14} +23 q^{-16} +14 q^{-18} -3 q^{-20} -33 q^{-22} -32 q^{-24} -9 q^{-26} +21 q^{-28} +43 q^{-30} +33 q^{-32} +4 q^{-34} -47 q^{-36} -53 q^{-38} -24 q^{-40} +22 q^{-42} +57 q^{-44} +53 q^{-46} +18 q^{-48} -49 q^{-50} -66 q^{-52} -43 q^{-54} +7 q^{-56} +53 q^{-58} +64 q^{-60} +39 q^{-62} -30 q^{-64} -58 q^{-66} -53 q^{-68} -18 q^{-70} +26 q^{-72} +53 q^{-74} +49 q^{-76} - q^{-78} -28 q^{-80} -41 q^{-82} -29 q^{-84} -4 q^{-86} +25 q^{-88} +38 q^{-90} +13 q^{-92} -15 q^{-96} -19 q^{-98} -14 q^{-100} + q^{-102} +14 q^{-104} +8 q^{-106} +6 q^{-108} - q^{-110} -5 q^{-112} -8 q^{-114} -2 q^{-116} +3 q^{-118} + q^{-120} +3 q^{-122} + q^{-124} -2 q^{-128} - q^{-130} + q^{-132} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{22}-q^{20}+q^{16}-3 q^{14}-q^{12}-2 q^8+3 q^4+3 q^2+4+3 q^{-2} +3 q^{-4} -2 q^{-8} - q^{-12} -3 q^{-14} + q^{-16} - q^{-20} + q^{-22} }[/math]
1,0,0	[math]\displaystyle{ -q^{13}-q^9+2 q^3+2 q+2 q^{-1} +2 q^{-3} - q^{-9} - q^{-13} }[/math]
1,0,1	[math]\displaystyle{ q^{36}-2 q^{34}+3 q^{32}-q^{30}-3 q^{28}+7 q^{26}-9 q^{24}+6 q^{22}-q^{20}-9 q^{18}+10 q^{16}-15 q^{14}+5 q^{12}-q^{10}-7 q^8+11 q^6+8 q^2+9+8 q^{-2} +11 q^{-6} -7 q^{-8} - q^{-10} +5 q^{-12} -15 q^{-14} +10 q^{-16} -9 q^{-18} - q^{-20} +6 q^{-22} -9 q^{-24} +7 q^{-26} -3 q^{-28} - q^{-30} +3 q^{-32} -2 q^{-34} + q^{-36} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{28}+q^{22}-3 q^{18}-3 q^{16}-2 q^{14}-3 q^{12}-3 q^{10}+4 q^6+3 q^4+6 q^2+8+6 q^{-2} +3 q^{-4} +4 q^{-6} -3 q^{-10} -3 q^{-12} -2 q^{-14} -3 q^{-16} -3 q^{-18} + q^{-22} + q^{-28} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{16}-q^{12}-q^{10}+2 q^4+2 q^2+3+2 q^{-2} +2 q^{-4} - q^{-10} - q^{-12} - q^{-16} }[/math]

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-q^{44}+q^{42}-3 q^{40}+4 q^{38}-4 q^{36}+q^{34}-3 q^{30}+3 q^{28}-3 q^{26}+q^{24}+q^{22}-2 q^{20}+q^{18}+q^{16}-q^{14}+4 q^{12}-3 q^{10}+3 q^8+q^6-q^4+6 q^2-5+6 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +4 q^{-12} - q^{-14} + q^{-16} + q^{-18} -2 q^{-20} + q^{-22} + q^{-24} -3 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-34} -4 q^{-36} +4 q^{-38} -3 q^{-40} + q^{-42} - q^{-44} -2 q^{-46} +2 q^{-48} - q^{-50} + q^{-52} }[/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 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^2-3 t+5-3 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]=

{ 13, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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