10 101

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 7 t^2-21 t+29-21 t^{-1} +7 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 7 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 85, 4 }
Jones polynomial [math]\displaystyle{ q^{12}-4 q^{11}+7 q^{10}-11 q^9+13 q^8-14 q^7+14 q^6-10 q^5+7 q^4-3 q^3+q^2 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^{-4} +3 z^4 a^{-6} +3 z^4 a^{-8} +z^2 a^{-4} +5 z^2 a^{-6} +5 z^2 a^{-8} -4 z^2 a^{-10} +2 a^{-6} +2 a^{-8} -4 a^{-10} + a^{-12} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^9 a^{-9} +2 z^9 a^{-11} +6 z^8 a^{-8} +11 z^8 a^{-10} +5 z^8 a^{-12} +7 z^7 a^{-7} +10 z^7 a^{-9} +7 z^7 a^{-11} +4 z^7 a^{-13} +6 z^6 a^{-6} -6 z^6 a^{-8} -24 z^6 a^{-10} -11 z^6 a^{-12} +z^6 a^{-14} +3 z^5 a^{-5} -8 z^5 a^{-7} -31 z^5 a^{-9} -31 z^5 a^{-11} -11 z^5 a^{-13} +z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +15 z^4 a^{-10} +3 z^4 a^{-12} -2 z^4 a^{-14} -2 z^3 a^{-5} +4 z^3 a^{-7} +26 z^3 a^{-9} +28 z^3 a^{-11} +8 z^3 a^{-13} -z^2 a^{-4} +7 z^2 a^{-6} -z^2 a^{-8} -9 z^2 a^{-10} +z^2 a^{-12} +z^2 a^{-14} -8 z a^{-9} -9 z a^{-11} -z a^{-13} -2 a^{-6} +2 a^{-8} +4 a^{-10} + a^{-12} }[/math]
The A2 invariant [math]\displaystyle{ q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +4 q^{-16} +2 q^{-20} +2 q^{-22} - q^{-24} +2 q^{-26} -4 q^{-28} - q^{-30} -3 q^{-34} + q^{-36} + q^{-38} }[/math]
The G2 invariant [math]\displaystyle{ q^{-30} -2 q^{-32} +4 q^{-34} -6 q^{-36} +6 q^{-38} -5 q^{-40} +11 q^{-44} -21 q^{-46} +33 q^{-48} -38 q^{-50} +31 q^{-52} -14 q^{-54} -15 q^{-56} +52 q^{-58} -84 q^{-60} +107 q^{-62} -105 q^{-64} +68 q^{-66} +3 q^{-68} -87 q^{-70} +169 q^{-72} -206 q^{-74} +183 q^{-76} -94 q^{-78} -40 q^{-80} +166 q^{-82} -226 q^{-84} +203 q^{-86} -85 q^{-88} -58 q^{-90} +169 q^{-92} -188 q^{-94} +107 q^{-96} +45 q^{-98} -192 q^{-100} +265 q^{-102} -220 q^{-104} +73 q^{-106} +125 q^{-108} -289 q^{-110} +359 q^{-112} -307 q^{-114} +147 q^{-116} +50 q^{-118} -229 q^{-120} +322 q^{-122} -304 q^{-124} +183 q^{-126} -14 q^{-128} -146 q^{-130} +224 q^{-132} -204 q^{-134} +85 q^{-136} +65 q^{-138} -188 q^{-140} +214 q^{-142} -138 q^{-144} -16 q^{-146} +177 q^{-148} -270 q^{-150} +259 q^{-152} -149 q^{-154} -14 q^{-156} +158 q^{-158} -232 q^{-160} +224 q^{-162} -139 q^{-164} +32 q^{-166} +59 q^{-168} -106 q^{-170} +105 q^{-172} -71 q^{-174} +33 q^{-176} + q^{-178} -19 q^{-180} +20 q^{-182} -16 q^{-184} +8 q^{-186} -3 q^{-188} + q^{-190} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_101.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-3} -2 q^{-5} +4 q^{-7} -3 q^{-9} +4 q^{-11} - q^{-15} +2 q^{-17} -4 q^{-19} +3 q^{-21} -3 q^{-23} + q^{-25} }[/math]
2 [math]\displaystyle{ q^{-6} -2 q^{-8} + q^{-10} +6 q^{-12} -9 q^{-14} +3 q^{-16} +17 q^{-18} -24 q^{-20} +35 q^{-24} -26 q^{-26} -15 q^{-28} +34 q^{-30} -7 q^{-32} -22 q^{-34} +11 q^{-36} +16 q^{-38} -16 q^{-40} -15 q^{-42} +27 q^{-44} -2 q^{-46} -33 q^{-48} +25 q^{-50} +16 q^{-52} -35 q^{-54} +9 q^{-56} +23 q^{-58} -18 q^{-60} -5 q^{-62} +12 q^{-64} -2 q^{-66} -3 q^{-68} + q^{-70} }[/math]
3 [math]\displaystyle{ q^{-9} -2 q^{-11} + q^{-13} +3 q^{-15} -5 q^{-19} +3 q^{-21} +11 q^{-23} -11 q^{-25} -19 q^{-27} +30 q^{-29} +42 q^{-31} -44 q^{-33} -90 q^{-35} +59 q^{-37} +150 q^{-39} -41 q^{-41} -214 q^{-43} -7 q^{-45} +255 q^{-47} +78 q^{-49} -255 q^{-51} -147 q^{-53} +201 q^{-55} +200 q^{-57} -121 q^{-59} -219 q^{-61} +32 q^{-63} +201 q^{-65} +58 q^{-67} -176 q^{-69} -125 q^{-71} +130 q^{-73} +181 q^{-75} -95 q^{-77} -218 q^{-79} +44 q^{-81} +250 q^{-83} +12 q^{-85} -260 q^{-87} -79 q^{-89} +248 q^{-91} +146 q^{-93} -198 q^{-95} -201 q^{-97} +124 q^{-99} +226 q^{-101} -41 q^{-103} -202 q^{-105} -36 q^{-107} +151 q^{-109} +78 q^{-111} -86 q^{-113} -82 q^{-115} +29 q^{-117} +59 q^{-119} +4 q^{-121} -34 q^{-123} -9 q^{-125} +11 q^{-127} +7 q^{-129} -2 q^{-131} -3 q^{-133} + q^{-135} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +4 q^{-16} +2 q^{-20} +2 q^{-22} - q^{-24} +2 q^{-26} -4 q^{-28} - q^{-30} -3 q^{-34} + q^{-36} + q^{-38} }[/math]
2,0 [math]\displaystyle{ q^{-12} -2 q^{-14} - q^{-16} +7 q^{-18} - q^{-20} -11 q^{-22} +6 q^{-24} +16 q^{-26} -5 q^{-28} -18 q^{-30} +9 q^{-32} +23 q^{-34} -10 q^{-36} -16 q^{-38} +14 q^{-40} +13 q^{-42} -5 q^{-44} -5 q^{-46} +8 q^{-48} -2 q^{-50} -3 q^{-52} +4 q^{-54} -6 q^{-56} -14 q^{-58} +2 q^{-60} +11 q^{-62} -15 q^{-64} -15 q^{-66} +9 q^{-68} +15 q^{-70} -9 q^{-72} -14 q^{-74} +13 q^{-76} +14 q^{-78} -3 q^{-80} -8 q^{-82} +3 q^{-84} +8 q^{-86} -2 q^{-88} -5 q^{-90} -2 q^{-92} + q^{-94} + q^{-96} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-12} -2 q^{-14} +6 q^{-18} -7 q^{-20} -3 q^{-22} +18 q^{-24} -10 q^{-26} -11 q^{-28} +28 q^{-30} -8 q^{-32} -16 q^{-34} +31 q^{-36} -3 q^{-38} -11 q^{-40} +14 q^{-42} +2 q^{-44} -9 q^{-46} -13 q^{-48} +4 q^{-50} + q^{-52} -25 q^{-54} +7 q^{-56} +19 q^{-58} -23 q^{-60} +8 q^{-62} +20 q^{-64} -20 q^{-66} +6 q^{-68} +10 q^{-70} -12 q^{-72} +4 q^{-74} +2 q^{-76} -3 q^{-78} + q^{-80} }[/math]
1,0,0 [math]\displaystyle{ q^{-9} -2 q^{-11} +2 q^{-13} - q^{-15} +2 q^{-17} -2 q^{-19} +4 q^{-21} +2 q^{-25} +2 q^{-27} +2 q^{-29} +2 q^{-31} - q^{-33} +2 q^{-35} -4 q^{-37} - q^{-39} -4 q^{-41} -3 q^{-45} + q^{-47} + q^{-49} + q^{-51} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +13 q^{-20} -17 q^{-22} +24 q^{-24} -28 q^{-26} +33 q^{-28} -32 q^{-30} +28 q^{-32} -16 q^{-34} +5 q^{-36} +13 q^{-38} -27 q^{-40} +44 q^{-42} -56 q^{-44} +63 q^{-46} -65 q^{-48} +58 q^{-50} -49 q^{-52} +33 q^{-54} -17 q^{-56} - q^{-58} +15 q^{-60} -26 q^{-62} +32 q^{-64} -34 q^{-66} +32 q^{-68} -28 q^{-70} +20 q^{-72} -14 q^{-74} +8 q^{-76} -3 q^{-78} + q^{-80} }[/math]
1,0 [math]\displaystyle{ q^{-18} -2 q^{-22} -2 q^{-24} +2 q^{-26} +7 q^{-28} +2 q^{-30} -10 q^{-32} -10 q^{-34} +6 q^{-36} +21 q^{-38} +7 q^{-40} -20 q^{-42} -22 q^{-44} +11 q^{-46} +34 q^{-48} +7 q^{-50} -31 q^{-52} -18 q^{-54} +23 q^{-56} +31 q^{-58} -9 q^{-60} -28 q^{-62} + q^{-64} +29 q^{-66} +7 q^{-68} -24 q^{-70} -13 q^{-72} +16 q^{-74} +13 q^{-76} -17 q^{-78} -20 q^{-80} +9 q^{-82} +20 q^{-84} -9 q^{-86} -30 q^{-88} -4 q^{-90} +32 q^{-92} +18 q^{-94} -26 q^{-96} -31 q^{-98} +14 q^{-100} +37 q^{-102} +6 q^{-104} -29 q^{-106} -18 q^{-108} +18 q^{-110} +22 q^{-112} -5 q^{-114} -16 q^{-116} -4 q^{-118} +9 q^{-120} +5 q^{-122} -3 q^{-124} -3 q^{-126} + q^{-130} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-30} -2 q^{-32} +4 q^{-34} -6 q^{-36} +6 q^{-38} -5 q^{-40} +11 q^{-44} -21 q^{-46} +33 q^{-48} -38 q^{-50} +31 q^{-52} -14 q^{-54} -15 q^{-56} +52 q^{-58} -84 q^{-60} +107 q^{-62} -105 q^{-64} +68 q^{-66} +3 q^{-68} -87 q^{-70} +169 q^{-72} -206 q^{-74} +183 q^{-76} -94 q^{-78} -40 q^{-80} +166 q^{-82} -226 q^{-84} +203 q^{-86} -85 q^{-88} -58 q^{-90} +169 q^{-92} -188 q^{-94} +107 q^{-96} +45 q^{-98} -192 q^{-100} +265 q^{-102} -220 q^{-104} +73 q^{-106} +125 q^{-108} -289 q^{-110} +359 q^{-112} -307 q^{-114} +147 q^{-116} +50 q^{-118} -229 q^{-120} +322 q^{-122} -304 q^{-124} +183 q^{-126} -14 q^{-128} -146 q^{-130} +224 q^{-132} -204 q^{-134} +85 q^{-136} +65 q^{-138} -188 q^{-140} +214 q^{-142} -138 q^{-144} -16 q^{-146} +177 q^{-148} -270 q^{-150} +259 q^{-152} -149 q^{-154} -14 q^{-156} +158 q^{-158} -232 q^{-160} +224 q^{-162} -139 q^{-164} +32 q^{-166} +59 q^{-168} -106 q^{-170} +105 q^{-172} -71 q^{-174} +33 q^{-176} + q^{-178} -19 q^{-180} +20 q^{-182} -16 q^{-184} +8 q^{-186} -3 q^{-188} + q^{-190} }[/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["10 101"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 7 t^2-21 t+29-21 t^{-1} +7 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 7 z^4+7 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]= { 85, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{12}-4 q^{11}+7 q^{10}-11 q^9+13 q^8-14 q^7+14 q^6-10 q^5+7 q^4-3 q^3+q^2 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4 a^{-4} +3 z^4 a^{-6} +3 z^4 a^{-8} +z^2 a^{-4} +5 z^2 a^{-6} +5 z^2 a^{-8} -4 z^2 a^{-10} +2 a^{-6} +2 a^{-8} -4 a^{-10} + a^{-12} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^9 a^{-9} +2 z^9 a^{-11} +6 z^8 a^{-8} +11 z^8 a^{-10} +5 z^8 a^{-12} +7 z^7 a^{-7} +10 z^7 a^{-9} +7 z^7 a^{-11} +4 z^7 a^{-13} +6 z^6 a^{-6} -6 z^6 a^{-8} -24 z^6 a^{-10} -11 z^6 a^{-12} +z^6 a^{-14} +3 z^5 a^{-5} -8 z^5 a^{-7} -31 z^5 a^{-9} -31 z^5 a^{-11} -11 z^5 a^{-13} +z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +15 z^4 a^{-10} +3 z^4 a^{-12} -2 z^4 a^{-14} -2 z^3 a^{-5} +4 z^3 a^{-7} +26 z^3 a^{-9} +28 z^3 a^{-11} +8 z^3 a^{-13} -z^2 a^{-4} +7 z^2 a^{-6} -z^2 a^{-8} -9 z^2 a^{-10} +z^2 a^{-12} +z^2 a^{-14} -8 z a^{-9} -9 z a^{-11} -z a^{-13} -2 a^{-6} +2 a^{-8} +4 a^{-10} + a^{-12} }[/math]
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