8 8

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 25, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+2 q^4-3 q^3+4 q^2-4 q+5-3 q^{-1} +2 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{-2} +z^4-a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +2 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+2 a z^5+z^5 a^{-1} +z^5 a^{-5} +2 a^2 z^4-9 z^4 a^{-2} -6 z^4 a^{-4} -z^4+a^3 z^3-3 z^3 a^{-1} -5 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+5 z^2 a^{-2} +4 z^2 a^{-4} -z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{10}-q^4+2 q^2+1+2 q^{-2} + q^{-4} + q^{-8} - q^{-10} - q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-3 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-5 q^{32}+q^{30}+3 q^{28}-6 q^{26}+8 q^{24}-9 q^{22}+7 q^{20}-4 q^{18}-3 q^{16}+7 q^{14}-8 q^{12}+9 q^{10}-q^8-3 q^6+6 q^4-3 q^2+1+6 q^{-2} -10 q^{-4} +12 q^{-6} -5 q^{-8} +10 q^{-12} -14 q^{-14} +17 q^{-16} -11 q^{-18} +3 q^{-20} +4 q^{-22} -9 q^{-24} +14 q^{-26} -10 q^{-28} +6 q^{-30} + q^{-32} -5 q^{-34} +7 q^{-36} -5 q^{-38} +4 q^{-42} -8 q^{-44} +7 q^{-46} -3 q^{-48} -4 q^{-50} +10 q^{-52} -13 q^{-54} +10 q^{-56} -5 q^{-58} -4 q^{-60} +7 q^{-62} -10 q^{-64} +9 q^{-66} -5 q^{-68} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_8.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^7+q^5-q^3+2 q+ q^{-1} + q^{-5} - q^{-7} + q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{20}-q^{18}-q^{16}+2 q^{14}-3 q^{12}-q^{10}+5 q^8-3 q^6-2 q^4+5 q^2- q^{-2} + q^{-4} +3 q^{-6} -2 q^{-10} +3 q^{-12} -5 q^{-16} +2 q^{-18} +2 q^{-20} -4 q^{-22} + q^{-24} +3 q^{-26} -2 q^{-28} - q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{39}+q^{37}+q^{35}-q^{31}+q^{29}+q^{27}-4 q^{25}-q^{23}+5 q^{21}+2 q^{19}-9 q^{17}-4 q^{15}+10 q^{13}+7 q^{11}-11 q^9-7 q^7+7 q^5+11 q^3-3 q-7 q^{-1} + q^{-3} +6 q^{-5} +5 q^{-7} -3 q^{-9} -6 q^{-11} + q^{-13} +9 q^{-15} - q^{-17} -9 q^{-19} -2 q^{-21} +10 q^{-23} +3 q^{-25} -10 q^{-27} -6 q^{-29} +9 q^{-31} +7 q^{-33} -6 q^{-35} -10 q^{-37} +3 q^{-39} +10 q^{-41} -8 q^{-45} -3 q^{-47} +6 q^{-49} +5 q^{-51} -3 q^{-53} -4 q^{-55} +2 q^{-59} + q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{64}-q^{62}-q^{60}-q^{56}+3 q^{54}-q^{52}+q^{50}+2 q^{48}-4 q^{46}+q^{44}-4 q^{42}+5 q^{40}+10 q^{38}-6 q^{36}-8 q^{34}-14 q^{32}+9 q^{30}+26 q^{28}+q^{26}-17 q^{24}-34 q^{22}+3 q^{20}+40 q^{18}+18 q^{16}-11 q^{14}-43 q^{12}-12 q^{10}+30 q^8+27 q^6+10 q^4-27 q^2-21+6 q^{-2} +18 q^{-4} +18 q^{-6} -3 q^{-8} -16 q^{-10} -15 q^{-12} +2 q^{-14} +19 q^{-16} +14 q^{-18} -10 q^{-20} -25 q^{-22} -3 q^{-24} +20 q^{-26} +24 q^{-28} -8 q^{-30} -33 q^{-32} -8 q^{-34} +19 q^{-36} +31 q^{-38} -3 q^{-40} -35 q^{-42} -17 q^{-44} +9 q^{-46} +37 q^{-48} +12 q^{-50} -26 q^{-52} -24 q^{-54} -9 q^{-56} +29 q^{-58} +24 q^{-60} -4 q^{-62} -19 q^{-64} -26 q^{-66} +8 q^{-68} +21 q^{-70} +13 q^{-72} - q^{-74} -22 q^{-76} -8 q^{-78} +4 q^{-80} +12 q^{-82} +11 q^{-84} -7 q^{-86} -7 q^{-88} -5 q^{-90} + q^{-92} +6 q^{-94} + q^{-96} -2 q^{-100} - q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{95}+q^{93}+q^{91}+q^{87}-q^{85}-3 q^{83}-q^{81}+q^{79}+4 q^{75}+3 q^{73}-2 q^{71}-6 q^{69}-6 q^{67}+9 q^{63}+15 q^{61}+7 q^{59}-16 q^{57}-27 q^{55}-13 q^{53}+18 q^{51}+43 q^{49}+34 q^{47}-19 q^{45}-68 q^{43}-55 q^{41}+10 q^{39}+83 q^{37}+87 q^{35}+7 q^{33}-94 q^{31}-118 q^{29}-36 q^{27}+85 q^{25}+139 q^{23}+63 q^{21}-64 q^{19}-131 q^{17}-93 q^{15}+31 q^{13}+120 q^{11}+99 q^9+3 q^7-78 q^5-93 q^3-33 q+47 q^{-1} +77 q^{-3} +48 q^{-5} -9 q^{-7} -50 q^{-9} -53 q^{-11} -20 q^{-13} +30 q^{-15} +53 q^{-17} +35 q^{-19} -10 q^{-21} -54 q^{-23} -52 q^{-25} +3 q^{-27} +59 q^{-29} +57 q^{-31} +2 q^{-33} -62 q^{-35} -71 q^{-37} -3 q^{-39} +74 q^{-41} +77 q^{-43} +9 q^{-45} -78 q^{-47} -92 q^{-49} -16 q^{-51} +80 q^{-53} +103 q^{-55} +31 q^{-57} -73 q^{-59} -114 q^{-61} -50 q^{-63} +57 q^{-65} +111 q^{-67} +73 q^{-69} -30 q^{-71} -106 q^{-73} -89 q^{-75} +82 q^{-79} +96 q^{-81} +36 q^{-83} -52 q^{-85} -89 q^{-87} -58 q^{-89} +15 q^{-91} +68 q^{-93} +67 q^{-95} +21 q^{-97} -38 q^{-99} -63 q^{-101} -41 q^{-103} +7 q^{-105} +42 q^{-107} +45 q^{-109} +19 q^{-111} -19 q^{-113} -37 q^{-115} -28 q^{-117} - q^{-119} +20 q^{-121} +25 q^{-123} +15 q^{-125} -6 q^{-127} -17 q^{-129} -14 q^{-131} -3 q^{-133} +5 q^{-135} +9 q^{-137} +7 q^{-139} - q^{-141} -4 q^{-143} -3 q^{-145} - q^{-147} +2 q^{-151} + q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{10}-q^4+2 q^2+1+2 q^{-2} + q^{-4} + q^{-8} - q^{-10} - q^{-16} }[/math]
1,1	[math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-6 q^{22}+9 q^{20}-12 q^{18}+14 q^{16}-20 q^{14}+19 q^{12}-24 q^{10}+22 q^8-22 q^6+18 q^4-4 q^2+20 q^{-2} -27 q^{-4} +44 q^{-6} -48 q^{-8} +56 q^{-10} -52 q^{-12} +48 q^{-14} -40 q^{-16} +24 q^{-18} -15 q^{-20} -4 q^{-22} +14 q^{-24} -24 q^{-26} +31 q^{-28} -32 q^{-30} +30 q^{-32} -24 q^{-34} +17 q^{-36} -12 q^{-38} +6 q^{-40} -2 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{26}-q^{22}+q^{18}-q^{16}-3 q^{14}+2 q^{10}-3 q^8-3 q^6+2 q^4+2 q^2+2+2 q^{-2} +5 q^{-4} +2 q^{-6} +2 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} -4 q^{-20} -3 q^{-22} + q^{-26} - q^{-28} - q^{-30} +2 q^{-32} + q^{-34} - q^{-36} - q^{-38} + q^{-42} }[/math]
3,0	[math]\displaystyle{ -q^{48}+q^{44}+q^{42}-2 q^{38}+2 q^{36}+3 q^{34}+q^{32}-4 q^{30}-5 q^{28}+4 q^{26}+6 q^{24}-q^{22}-10 q^{20}-8 q^{18}+8 q^{16}+9 q^{14}-3 q^{12}-13 q^{10}-8 q^8+8 q^6+6 q^4-4+4 q^{-2} +10 q^{-4} +8 q^{-6} +3 q^{-8} +3 q^{-10} +7 q^{-12} +2 q^{-14} -3 q^{-16} -2 q^{-18} +3 q^{-20} +2 q^{-22} -8 q^{-24} -9 q^{-26} +6 q^{-30} + q^{-32} -10 q^{-34} -7 q^{-36} +4 q^{-38} +8 q^{-40} +2 q^{-42} -8 q^{-44} -5 q^{-46} +3 q^{-48} +7 q^{-50} +4 q^{-52} -5 q^{-54} -4 q^{-56} +5 q^{-60} +4 q^{-62} -2 q^{-64} -3 q^{-66} -3 q^{-68} + q^{-70} +2 q^{-72} + q^{-74} - q^{-78} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{22}-q^{20}+2 q^{16}-3 q^{14}-3 q^{12}+2 q^{10}-3 q^8-3 q^6+5 q^4+q^2+2+4 q^{-2} +3 q^{-4} + q^{-6} +3 q^{-10} +2 q^{-12} -3 q^{-14} + q^{-16} + q^{-18} -5 q^{-20} - q^{-22} + q^{-24} -3 q^{-26} + q^{-28} + q^{-30} - q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{13}-q^9-q^5+2 q^3+q+2 q^{-1} +2 q^{-3} + q^{-5} + q^{-7} + q^{-11} - q^{-13} - q^{-17} - q^{-21} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-3 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-5 q^{32}+q^{30}+3 q^{28}-6 q^{26}+8 q^{24}-9 q^{22}+7 q^{20}-4 q^{18}-3 q^{16}+7 q^{14}-8 q^{12}+9 q^{10}-q^8-3 q^6+6 q^4-3 q^2+1+6 q^{-2} -10 q^{-4} +12 q^{-6} -5 q^{-8} +10 q^{-12} -14 q^{-14} +17 q^{-16} -11 q^{-18} +3 q^{-20} +4 q^{-22} -9 q^{-24} +14 q^{-26} -10 q^{-28} +6 q^{-30} + q^{-32} -5 q^{-34} +7 q^{-36} -5 q^{-38} +4 q^{-42} -8 q^{-44} +7 q^{-46} -3 q^{-48} -4 q^{-50} +10 q^{-52} -13 q^{-54} +10 q^{-56} -5 q^{-58} -4 q^{-60} +7 q^{-62} -10 q^{-64} +9 q^{-66} -5 q^{-68} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/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 8"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+2 a z^5+z^5 a^{-1} +z^5 a^{-5} +2 a^2 z^4-9 z^4 a^{-2} -6 z^4 a^{-4} -z^4+a^3 z^3-3 z^3 a^{-1} -5 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+5 z^2 a^{-2} +4 z^2 a^{-4} -z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]