10 163

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature { 51, 2 }
Jones polynomial [math]\displaystyle{ -2 q^6+5 q^5-7 q^4+9 q^3-9 q^2+8 q-6+4 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +3 z^4 a^{-2} -z^4 a^{-4} -z^4+2 z^2 a^{-2} -z^2- a^{-2} +2 a^{-4} - a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^8 a^{-2} +2 z^8 a^{-4} +5 z^7 a^{-1} +8 z^7 a^{-3} +3 z^7 a^{-5} +3 z^6 a^{-2} +z^6 a^{-6} +4 z^6+a z^5-10 z^5 a^{-1} -15 z^5 a^{-3} -4 z^5 a^{-5} -11 z^4 a^{-2} +z^4 a^{-4} +4 z^4 a^{-6} -8 z^4-a z^3+3 z^3 a^{-1} +8 z^3 a^{-3} +7 z^3 a^{-5} +3 z^3 a^{-7} +2 z^2 a^{-2} -4 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2-z a^{-3} -3 z a^{-5} -2 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1 }[/math]
The A2 invariant [math]\displaystyle{ -q^6+2 q^4+1+2 q^{-2} -3 q^{-4} + q^{-6} -2 q^{-8} +2 q^{-10} +2 q^{-12} +2 q^{-16} -2 q^{-18} - q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+13 q^{24}-8 q^{22}-7 q^{20}+31 q^{18}-49 q^{16}+61 q^{14}-48 q^{12}+5 q^{10}+46 q^8-91 q^6+109 q^4-79 q^2+21+50 q^{-2} -92 q^{-4} +96 q^{-6} -53 q^{-8} -13 q^{-10} +68 q^{-12} -90 q^{-14} +59 q^{-16} +6 q^{-18} -74 q^{-20} +115 q^{-22} -108 q^{-24} +61 q^{-26} +6 q^{-28} -81 q^{-30} +123 q^{-32} -134 q^{-34} +101 q^{-36} -34 q^{-38} -40 q^{-40} +99 q^{-42} -113 q^{-44} +90 q^{-46} -30 q^{-48} -35 q^{-50} +79 q^{-52} -79 q^{-54} +39 q^{-56} +32 q^{-58} -85 q^{-60} +106 q^{-62} -70 q^{-64} +3 q^{-66} +62 q^{-68} -104 q^{-70} +100 q^{-72} -65 q^{-74} +13 q^{-76} +32 q^{-78} -59 q^{-80} +55 q^{-82} -36 q^{-84} +14 q^{-86} +5 q^{-88} -14 q^{-90} +9 q^{-92} -8 q^{-94} +5 q^{-96} -2 q^{-98} + q^{-100} + q^{-102} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_163.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^5+3 q^3-2 q+2 q^{-1} - q^{-3} +2 q^{-7} -2 q^{-9} +3 q^{-11} -2 q^{-13} }[/math]
2 [math]\displaystyle{ q^{16}-3 q^{14}-2 q^{12}+11 q^{10}-3 q^8-15 q^6+14 q^4+8 q^2-20+6 q^{-2} +15 q^{-4} -12 q^{-6} -3 q^{-8} +12 q^{-10} -11 q^{-14} +2 q^{-16} +14 q^{-18} -14 q^{-20} -9 q^{-22} +22 q^{-24} -6 q^{-26} -14 q^{-28} +13 q^{-30} +2 q^{-32} -7 q^{-34} + q^{-36} + q^{-38} }[/math]
3 [math]\displaystyle{ -q^{33}+3 q^{31}+2 q^{29}-7 q^{27}-10 q^{25}+7 q^{23}+31 q^{21}-47 q^{17}-30 q^{15}+53 q^{13}+70 q^{11}-39 q^9-103 q^7+3 q^5+120 q^3+44 q-115 q^{-1} -80 q^{-3} +92 q^{-5} +101 q^{-7} -58 q^{-9} -107 q^{-11} +31 q^{-13} +100 q^{-15} -4 q^{-17} -85 q^{-19} -21 q^{-21} +70 q^{-23} +45 q^{-25} -52 q^{-27} -75 q^{-29} +30 q^{-31} +101 q^{-33} +2 q^{-35} -119 q^{-37} -42 q^{-39} +122 q^{-41} +77 q^{-43} -100 q^{-45} -101 q^{-47} +60 q^{-49} +104 q^{-51} -18 q^{-53} -85 q^{-55} -8 q^{-57} +48 q^{-59} +23 q^{-61} -21 q^{-63} -17 q^{-65} +5 q^{-67} +5 q^{-69} +2 q^{-71} -2 q^{-73} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^6+2 q^4+1+2 q^{-2} -3 q^{-4} + q^{-6} -2 q^{-8} +2 q^{-10} +2 q^{-12} +2 q^{-16} -2 q^{-18} - q^{-20} }[/math]
1,1 [math]\displaystyle{ q^{20}-6 q^{18}+20 q^{16}-50 q^{14}+95 q^{12}-160 q^{10}+244 q^8-318 q^6+372 q^4-384 q^2+336-224 q^{-2} +62 q^{-4} +132 q^{-6} -332 q^{-8} +510 q^{-10} -629 q^{-12} +698 q^{-14} -694 q^{-16} +628 q^{-18} -496 q^{-20} +318 q^{-22} -126 q^{-24} -70 q^{-26} +232 q^{-28} -346 q^{-30} +396 q^{-32} -382 q^{-34} +326 q^{-36} -242 q^{-38} +158 q^{-40} -92 q^{-42} +41 q^{-44} -10 q^{-46} -2 q^{-52} +2 q^{-54} }[/math]
2,0 [math]\displaystyle{ q^{18}-2 q^{16}-3 q^{14}+4 q^{12}+4 q^{10}-2 q^8-5 q^6+4 q^4+8 q^2-7-6 q^{-2} +6 q^{-4} -3 q^{-8} +2 q^{-10} +9 q^{-12} +2 q^{-14} +4 q^{-18} -4 q^{-20} -9 q^{-22} +4 q^{-26} -9 q^{-28} +10 q^{-32} +5 q^{-34} -3 q^{-36} -4 q^{-38} +5 q^{-40} -2 q^{-42} -5 q^{-44} - q^{-46} + q^{-48} +2 q^{-50} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{14}-3 q^{12}+q^{10}+5 q^8-9 q^6+7 q^4+7 q^2-13+10 q^{-2} +5 q^{-4} -13 q^{-6} +4 q^{-8} +5 q^{-10} -5 q^{-12} - q^{-14} +3 q^{-16} +5 q^{-18} -2 q^{-20} -4 q^{-22} +14 q^{-24} -6 q^{-26} -9 q^{-28} +14 q^{-30} -8 q^{-32} -8 q^{-34} +9 q^{-36} -3 q^{-38} -3 q^{-40} +3 q^{-42} }[/math]
1,0,0 [math]\displaystyle{ -q^7+2 q^5-q^3+3 q+2 q^{-3} -2 q^{-5} - q^{-7} - q^{-9} - q^{-11} +2 q^{-13} + q^{-15} +4 q^{-17} - q^{-19} +2 q^{-21} -2 q^{-23} - q^{-25} - q^{-27} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{14}+3 q^{12}-7 q^{10}+11 q^8-13 q^6+17 q^4-15 q^2+13-8 q^{-2} +3 q^{-4} +5 q^{-6} -14 q^{-8} +21 q^{-10} -27 q^{-12} +29 q^{-14} -29 q^{-16} +25 q^{-18} -18 q^{-20} +12 q^{-22} -2 q^{-24} -4 q^{-26} +11 q^{-28} -14 q^{-30} +16 q^{-32} -16 q^{-34} +13 q^{-36} -9 q^{-38} +5 q^{-40} -3 q^{-42} }[/math]
1,0 [math]\displaystyle{ q^{24}-3 q^{20}-3 q^{18}+4 q^{16}+8 q^{14}-2 q^{12}-11 q^{10}-4 q^8+15 q^6+11 q^4-11 q^2-15+4 q^{-2} +17 q^{-4} +4 q^{-6} -15 q^{-8} -8 q^{-10} +10 q^{-12} +9 q^{-14} -6 q^{-16} -9 q^{-18} +4 q^{-20} +11 q^{-22} -12 q^{-26} -2 q^{-28} +12 q^{-30} +6 q^{-32} -10 q^{-34} -10 q^{-36} +10 q^{-38} +14 q^{-40} -5 q^{-42} -17 q^{-44} +16 q^{-48} +7 q^{-50} -12 q^{-52} -13 q^{-54} +4 q^{-56} +11 q^{-58} + q^{-60} -6 q^{-62} -4 q^{-64} + q^{-66} +3 q^{-68} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{18}-3 q^{16}+4 q^{14}-6 q^{12}+9 q^{10}-11 q^8+13 q^6-12 q^4+15 q^2-9+8 q^{-2} -4 q^{-4} +3 q^{-6} +2 q^{-8} -11 q^{-10} +12 q^{-12} -16 q^{-14} +18 q^{-16} -23 q^{-18} +22 q^{-20} -20 q^{-22} +24 q^{-24} -16 q^{-26} +15 q^{-28} -8 q^{-30} +10 q^{-32} + q^{-34} -5 q^{-36} +5 q^{-38} -11 q^{-40} +12 q^{-42} -14 q^{-44} +9 q^{-46} -14 q^{-48} +11 q^{-50} -6 q^{-52} +4 q^{-54} -4 q^{-56} +3 q^{-58} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+13 q^{24}-8 q^{22}-7 q^{20}+31 q^{18}-49 q^{16}+61 q^{14}-48 q^{12}+5 q^{10}+46 q^8-91 q^6+109 q^4-79 q^2+21+50 q^{-2} -92 q^{-4} +96 q^{-6} -53 q^{-8} -13 q^{-10} +68 q^{-12} -90 q^{-14} +59 q^{-16} +6 q^{-18} -74 q^{-20} +115 q^{-22} -108 q^{-24} +61 q^{-26} +6 q^{-28} -81 q^{-30} +123 q^{-32} -134 q^{-34} +101 q^{-36} -34 q^{-38} -40 q^{-40} +99 q^{-42} -113 q^{-44} +90 q^{-46} -30 q^{-48} -35 q^{-50} +79 q^{-52} -79 q^{-54} +39 q^{-56} +32 q^{-58} -85 q^{-60} +106 q^{-62} -70 q^{-64} +3 q^{-66} +62 q^{-68} -104 q^{-70} +100 q^{-72} -65 q^{-74} +13 q^{-76} +32 q^{-78} -59 q^{-80} +55 q^{-82} -36 q^{-84} +14 q^{-86} +5 q^{-88} -14 q^{-90} +9 q^{-92} -8 q^{-94} +5 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["10 163"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+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{ \left\{2,t^2+t+1\right\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 51, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -2 q^6+5 q^5-7 q^4+9 q^3-9 q^2+8 q-6+4 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} +3 z^4 a^{-2} -z^4 a^{-4} -z^4+2 z^2 a^{-2} -z^2- a^{-2} +2 a^{-4} - a^{-6} +1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^8 a^{-2} +2 z^8 a^{-4} +5 z^7 a^{-1} +8 z^7 a^{-3} +3 z^7 a^{-5} +3 z^6 a^{-2} +z^6 a^{-6} +4 z^6+a z^5-10 z^5 a^{-1} -15 z^5 a^{-3} -4 z^5 a^{-5} -11 z^4 a^{-2} +z^4 a^{-4} +4 z^4 a^{-6} -8 z^4-a z^3+3 z^3 a^{-1} +8 z^3 a^{-3} +7 z^3 a^{-5} +3 z^3 a^{-7} +2 z^2 a^{-2} -4 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2-z a^{-3} -3 z a^{-5} -2 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1 }[/math]
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