10 151

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+2 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 43, 2 }
Jones polynomial [math]\displaystyle{ -2 q^6+4 q^5-6 q^4+8 q^3-7 q^2+7 q-5+3 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+6 z^2 a^{-2} -z^2 a^{-4} -2 z^2+3 a^{-2} - a^{-6} -1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +5 z^7 a^{-3} +2 z^7 a^{-5} +5 z^6 a^{-2} +3 z^6 a^{-4} +z^6 a^{-6} +3 z^6+a z^5-4 z^5 a^{-1} -7 z^5 a^{-3} -2 z^5 a^{-5} -15 z^4 a^{-2} -6 z^4 a^{-4} +2 z^4 a^{-6} -7 z^4-2 a z^3-3 z^3 a^{-1} +z^3 a^{-3} +5 z^3 a^{-5} +3 z^3 a^{-7} +10 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} +4 z^2+a z+2 z a^{-1} +z a^{-3} -3 z a^{-5} -3 z a^{-7} -3 a^{-2} + a^{-6} -1 }[/math]
The A2 invariant [math]\displaystyle{ -q^6+q^4-q^2+2 q^{-2} - q^{-4} +3 q^{-6} +2 q^{-10} + q^{-12} - q^{-14} + q^{-16} -2 q^{-18} - q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+7 q^{24}-4 q^{22}-6 q^{20}+19 q^{18}-30 q^{16}+34 q^{14}-27 q^{12}+q^{10}+27 q^8-52 q^6+62 q^4-46 q^2+15+23 q^{-2} -51 q^{-4} +55 q^{-6} -34 q^{-8} +33 q^{-12} -46 q^{-14} +35 q^{-16} -35 q^{-20} +62 q^{-22} -62 q^{-24} +40 q^{-26} - q^{-28} -43 q^{-30} +75 q^{-32} -80 q^{-34} +64 q^{-36} -24 q^{-38} -18 q^{-40} +56 q^{-42} -68 q^{-44} +57 q^{-46} -25 q^{-48} -11 q^{-50} +42 q^{-52} -45 q^{-54} +24 q^{-56} +13 q^{-58} -42 q^{-60} +55 q^{-62} -42 q^{-64} +5 q^{-66} +29 q^{-68} -56 q^{-70} +60 q^{-72} -45 q^{-74} +15 q^{-76} +13 q^{-78} -34 q^{-80} +34 q^{-82} -28 q^{-84} +15 q^{-86} -2 q^{-88} -7 q^{-90} +7 q^{-92} -8 q^{-94} +6 q^{-96} -2 q^{-98} + q^{-100} + q^{-102} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_151.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^5+2 q^3-2 q+2 q^{-1} + q^{-5} +2 q^{-7} -2 q^{-9} +2 q^{-11} -2 q^{-13} }[/math]
2 [math]\displaystyle{ q^{16}-2 q^{14}-2 q^{12}+7 q^{10}-2 q^8-10 q^6+10 q^4+5 q^2-14+5 q^{-2} +10 q^{-4} -9 q^{-6} - q^{-8} +9 q^{-10} -6 q^{-14} +2 q^{-16} +10 q^{-18} -10 q^{-20} -6 q^{-22} +15 q^{-24} -6 q^{-26} -10 q^{-28} +9 q^{-30} -5 q^{-34} +2 q^{-36} + q^{-38} }[/math]
3 [math]\displaystyle{ -q^{33}+2 q^{31}+2 q^{29}-3 q^{27}-7 q^{25}+2 q^{23}+17 q^{21}+2 q^{19}-25 q^{17}-17 q^{15}+28 q^{13}+38 q^{11}-22 q^9-55 q^7+5 q^5+64 q^3+19 q-67 q^{-1} -37 q^{-3} +56 q^{-5} +52 q^{-7} -40 q^{-9} -55 q^{-11} +26 q^{-13} +57 q^{-15} -10 q^{-17} -47 q^{-19} -5 q^{-21} +40 q^{-23} +21 q^{-25} -31 q^{-27} -39 q^{-29} +17 q^{-31} +56 q^{-33} -3 q^{-35} -65 q^{-37} -20 q^{-39} +71 q^{-41} +35 q^{-43} -61 q^{-45} -50 q^{-47} +41 q^{-49} +52 q^{-51} -20 q^{-53} -45 q^{-55} +5 q^{-57} +28 q^{-59} +6 q^{-61} -14 q^{-63} -6 q^{-65} +7 q^{-67} +2 q^{-69} -2 q^{-73} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^6+q^4-q^2+2 q^{-2} - q^{-4} +3 q^{-6} +2 q^{-10} + q^{-12} - q^{-14} + q^{-16} -2 q^{-18} - q^{-20} }[/math]
1,1 [math]\displaystyle{ q^{20}-4 q^{18}+12 q^{16}-28 q^{14}+52 q^{12}-86 q^{10}+128 q^8-170 q^6+196 q^4-206 q^2+188-134 q^{-2} +52 q^{-4} +48 q^{-6} -152 q^{-8} +254 q^{-10} -324 q^{-12} +378 q^{-14} -380 q^{-16} +360 q^{-18} -295 q^{-20} +208 q^{-22} -108 q^{-24} +92 q^{-28} -166 q^{-30} +200 q^{-32} -208 q^{-34} +188 q^{-36} -154 q^{-38} +108 q^{-40} -70 q^{-42} +40 q^{-44} -18 q^{-46} +6 q^{-48} -2 q^{-50} +2 q^{-54} }[/math]
2,0 [math]\displaystyle{ q^{18}-q^{16}-2 q^{14}+2 q^{12}+2 q^{10}-2 q^8-4 q^6+2 q^4+5 q^2-5-3 q^{-2} +6 q^{-4} + q^{-6} -3 q^{-8} +5 q^{-12} + q^{-16} +6 q^{-18} +3 q^{-20} -2 q^{-22} +4 q^{-24} +4 q^{-26} -7 q^{-28} -3 q^{-30} +3 q^{-32} - q^{-34} -6 q^{-36} -4 q^{-38} +3 q^{-40} - q^{-42} -3 q^{-44} + q^{-46} +2 q^{-48} +2 q^{-50} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{14}-2 q^{12}+q^{10}+3 q^8-7 q^6+3 q^4+3 q^2-11+6 q^{-2} +5 q^{-4} -8 q^{-6} +6 q^{-8} +7 q^{-10} - q^{-12} + q^{-14} +4 q^{-16} +4 q^{-18} - q^{-20} -3 q^{-22} +9 q^{-24} -6 q^{-26} -9 q^{-28} +8 q^{-30} -7 q^{-32} -7 q^{-34} +7 q^{-36} - q^{-38} -2 q^{-40} +3 q^{-42} }[/math]
1,0,0 [math]\displaystyle{ -q^7+q^5-2 q^3+q- q^{-1} +2 q^{-3} +2 q^{-7} +2 q^{-9} + q^{-11} +2 q^{-13} +2 q^{-17} -2 q^{-19} + q^{-21} -2 q^{-23} - q^{-25} - q^{-27} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{14}+2 q^{12}-5 q^{10}+7 q^8-9 q^6+11 q^4-11 q^2+9-6 q^{-2} +3 q^{-4} +4 q^{-6} -8 q^{-8} +15 q^{-10} -17 q^{-12} +21 q^{-14} -20 q^{-16} +18 q^{-18} -13 q^{-20} +9 q^{-22} -3 q^{-24} -2 q^{-26} +7 q^{-28} -10 q^{-30} +11 q^{-32} -11 q^{-34} +9 q^{-36} -7 q^{-38} +4 q^{-40} -3 q^{-42} }[/math]
1,0 [math]\displaystyle{ q^{24}-2 q^{20}-2 q^{18}+3 q^{16}+5 q^{14}-2 q^{12}-8 q^{10}-3 q^8+9 q^6+7 q^4-8 q^2-11+2 q^{-2} +12 q^{-4} +4 q^{-6} -10 q^{-8} -5 q^{-10} +7 q^{-12} +8 q^{-14} -3 q^{-16} -5 q^{-18} +3 q^{-20} +9 q^{-22} + q^{-24} -7 q^{-26} - q^{-28} +9 q^{-30} +5 q^{-32} -6 q^{-34} -7 q^{-36} +6 q^{-38} +9 q^{-40} -4 q^{-42} -13 q^{-44} -2 q^{-46} +10 q^{-48} +4 q^{-50} -9 q^{-52} -10 q^{-54} +2 q^{-56} +8 q^{-58} +2 q^{-60} -4 q^{-62} -3 q^{-64} + q^{-66} +3 q^{-68} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{18}-2 q^{16}+3 q^{14}-4 q^{12}+6 q^{10}-8 q^8+7 q^6-10 q^4+8 q^2-9+4 q^{-2} -4 q^{-4} +3 q^{-6} +3 q^{-8} -4 q^{-10} +12 q^{-12} -6 q^{-14} +16 q^{-16} -13 q^{-18} +17 q^{-20} -13 q^{-22} +17 q^{-24} -12 q^{-26} +10 q^{-28} -7 q^{-30} +7 q^{-32} - q^{-34} -4 q^{-36} + q^{-38} -9 q^{-40} +7 q^{-42} -11 q^{-44} +5 q^{-46} -10 q^{-48} +8 q^{-50} -4 q^{-52} +4 q^{-54} -3 q^{-56} +3 q^{-58} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+7 q^{24}-4 q^{22}-6 q^{20}+19 q^{18}-30 q^{16}+34 q^{14}-27 q^{12}+q^{10}+27 q^8-52 q^6+62 q^4-46 q^2+15+23 q^{-2} -51 q^{-4} +55 q^{-6} -34 q^{-8} +33 q^{-12} -46 q^{-14} +35 q^{-16} -35 q^{-20} +62 q^{-22} -62 q^{-24} +40 q^{-26} - q^{-28} -43 q^{-30} +75 q^{-32} -80 q^{-34} +64 q^{-36} -24 q^{-38} -18 q^{-40} +56 q^{-42} -68 q^{-44} +57 q^{-46} -25 q^{-48} -11 q^{-50} +42 q^{-52} -45 q^{-54} +24 q^{-56} +13 q^{-58} -42 q^{-60} +55 q^{-62} -42 q^{-64} +5 q^{-66} +29 q^{-68} -56 q^{-70} +60 q^{-72} -45 q^{-74} +15 q^{-76} +13 q^{-78} -34 q^{-80} +34 q^{-82} -28 q^{-84} +15 q^{-86} -2 q^{-88} -7 q^{-90} +7 q^{-92} -8 q^{-94} +6 q^{-96} -2 q^{-98} + q^{-100} + q^{-102} }[/math]

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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["10 151"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+2 z^4+3 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]= { 43, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -2 q^6+4 q^5-6 q^4+8 q^3-7 q^2+7 q-5+3 q^{-1} - q^{-2} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+6 z^2 a^{-2} -z^2 a^{-4} -2 z^2+3 a^{-2} - a^{-6} -1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +5 z^7 a^{-3} +2 z^7 a^{-5} +5 z^6 a^{-2} +3 z^6 a^{-4} +z^6 a^{-6} +3 z^6+a z^5-4 z^5 a^{-1} -7 z^5 a^{-3} -2 z^5 a^{-5} -15 z^4 a^{-2} -6 z^4 a^{-4} +2 z^4 a^{-6} -7 z^4-2 a z^3-3 z^3 a^{-1} +z^3 a^{-3} +5 z^3 a^{-5} +3 z^3 a^{-7} +10 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} +4 z^2+a z+2 z a^{-1} +z a^{-3} -3 z a^{-5} -3 z a^{-7} -3 a^{-2} + a^{-6} -1 }[/math]
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