10 117

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-10 t^2+24 t-31+24 t^{-1} -10 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 103, 2 }
Jones polynomial [math]\displaystyle{ -q^8+4 q^7-9 q^6+13 q^5-16 q^4+18 q^3-16 q^2+13 q-8+4 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -z^4+2 z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -z^2+ a^{-2} + a^{-4} - a^{-6} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 z^9 a^{-3} +3 z^9 a^{-5} +7 z^8 a^{-2} +15 z^8 a^{-4} +8 z^8 a^{-6} +7 z^7 a^{-1} +9 z^7 a^{-3} +10 z^7 a^{-5} +8 z^7 a^{-7} -8 z^6 a^{-2} -28 z^6 a^{-4} -12 z^6 a^{-6} +4 z^6 a^{-8} +4 z^6+a z^5-11 z^5 a^{-1} -26 z^5 a^{-3} -29 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} +17 z^4 a^{-4} +6 z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-a z^3+5 z^3 a^{-1} +18 z^3 a^{-3} +21 z^3 a^{-5} +8 z^3 a^{-7} -z^3 a^{-9} +2 z^2 a^{-2} -4 z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} +2 z^2-z a^{-1} -3 z a^{-3} -5 z a^{-5} -3 z a^{-7} - a^{-2} + a^{-4} + a^{-6} }[/math]
The A2 invariant [math]\displaystyle{ -q^6+2 q^4-q^2-1+4 q^{-2} -3 q^{-4} +3 q^{-6} +3 q^{-12} -3 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} +2 q^{-22} - q^{-24} }[/math]
The G2 invariant [math]\displaystyle{ q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+15 q^{24}-14 q^{22}+3 q^{20}+21 q^{18}-49 q^{16}+81 q^{14}-100 q^{12}+87 q^{10}-40 q^8-54 q^6+173 q^4-269 q^2+308-240 q^{-2} +63 q^{-4} +178 q^{-6} -398 q^{-8} +501 q^{-10} -428 q^{-12} +190 q^{-14} +119 q^{-16} -379 q^{-18} +476 q^{-20} -352 q^{-22} +81 q^{-24} +223 q^{-26} -405 q^{-28} +367 q^{-30} -133 q^{-32} -203 q^{-34} +486 q^{-36} -585 q^{-38} +465 q^{-40} -143 q^{-42} -252 q^{-44} +583 q^{-46} -727 q^{-48} +629 q^{-50} -331 q^{-52} -66 q^{-54} +415 q^{-56} -593 q^{-58} +555 q^{-60} -307 q^{-62} -27 q^{-64} +311 q^{-66} -430 q^{-68} +315 q^{-70} -41 q^{-72} -263 q^{-74} +450 q^{-76} -428 q^{-78} +211 q^{-80} +108 q^{-82} -391 q^{-84} +522 q^{-86} -465 q^{-88} +250 q^{-90} +19 q^{-92} -247 q^{-94} +349 q^{-96} -321 q^{-98} +213 q^{-100} -69 q^{-102} -46 q^{-104} +107 q^{-106} -122 q^{-108} +94 q^{-110} -52 q^{-112} +18 q^{-114} +7 q^{-116} -16 q^{-118} +16 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_117.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^5+3 q^3-4 q+5 q^{-1} -3 q^{-3} +2 q^{-5} +2 q^{-7} -3 q^{-9} +4 q^{-11} -5 q^{-13} +3 q^{-15} - q^{-17} }[/math]
2 [math]\displaystyle{ q^{16}-3 q^{14}+10 q^{10}-14 q^8-7 q^6+36 q^4-21 q^2-34+56 q^{-2} -3 q^{-4} -54 q^{-6} +40 q^{-8} +22 q^{-10} -40 q^{-12} + q^{-14} +31 q^{-16} -4 q^{-18} -38 q^{-20} +24 q^{-22} +33 q^{-24} -56 q^{-26} +3 q^{-28} +54 q^{-30} -40 q^{-32} -18 q^{-34} +41 q^{-36} -10 q^{-38} -17 q^{-40} +11 q^{-42} + q^{-44} -3 q^{-46} + q^{-48} }[/math]
3 [math]\displaystyle{ -q^{33}+3 q^{31}-6 q^{27}-q^{25}+14 q^{23}+6 q^{21}-39 q^{19}-16 q^{17}+74 q^{15}+59 q^{13}-115 q^{11}-147 q^9+133 q^7+272 q^5-93 q^3-399 q-20 q^{-1} +493 q^{-3} +173 q^{-5} -500 q^{-7} -332 q^{-9} +428 q^{-11} +451 q^{-13} -298 q^{-15} -495 q^{-17} +144 q^{-19} +475 q^{-21} +13 q^{-23} -414 q^{-25} -146 q^{-27} +323 q^{-29} +263 q^{-31} -219 q^{-33} -374 q^{-35} +107 q^{-37} +458 q^{-39} +28 q^{-41} -518 q^{-43} -170 q^{-45} +512 q^{-47} +319 q^{-49} -441 q^{-51} -431 q^{-53} +306 q^{-55} +478 q^{-57} -141 q^{-59} -437 q^{-61} -11 q^{-63} +335 q^{-65} +101 q^{-67} -210 q^{-69} -117 q^{-71} +93 q^{-73} +95 q^{-75} -28 q^{-77} -54 q^{-79} +5 q^{-81} +20 q^{-83} +2 q^{-85} -7 q^{-87} - q^{-89} +3 q^{-91} - q^{-93} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^6+2 q^4-q^2-1+4 q^{-2} -3 q^{-4} +3 q^{-6} +3 q^{-12} -3 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} +2 q^{-22} - q^{-24} }[/math]
1,1 [math]\displaystyle{ q^{20}-6 q^{18}+20 q^{16}-50 q^{14}+107 q^{12}-206 q^{10}+360 q^8-594 q^6+914 q^4-1306 q^2+1748-2170 q^{-2} +2477 q^{-4} -2562 q^{-6} +2350 q^{-8} -1780 q^{-10} +871 q^{-12} +322 q^{-14} -1636 q^{-16} +2924 q^{-18} -4030 q^{-20} +4806 q^{-22} -5162 q^{-24} +5046 q^{-26} -4476 q^{-28} +3528 q^{-30} -2312 q^{-32} +982 q^{-34} +301 q^{-36} -1396 q^{-38} +2168 q^{-40} -2576 q^{-42} +2636 q^{-44} -2420 q^{-46} +2024 q^{-48} -1544 q^{-50} +1087 q^{-52} -708 q^{-54} +422 q^{-56} -230 q^{-58} +113 q^{-60} -50 q^{-62} +20 q^{-64} -6 q^{-66} + q^{-68} }[/math]
2,0 [math]\displaystyle{ q^{18}-2 q^{16}-2 q^{14}+6 q^{12}+2 q^{10}-11 q^8-6 q^6+17 q^4+10 q^2-22-8 q^{-2} +28 q^{-4} +5 q^{-6} -27 q^{-8} +2 q^{-10} +21 q^{-12} -2 q^{-14} -12 q^{-16} +10 q^{-18} +9 q^{-20} -15 q^{-22} +8 q^{-24} +8 q^{-26} -18 q^{-28} -7 q^{-30} +22 q^{-32} + q^{-34} -26 q^{-36} +3 q^{-38} +23 q^{-40} - q^{-42} -24 q^{-44} +5 q^{-46} +17 q^{-48} -3 q^{-50} -9 q^{-52} -2 q^{-54} +6 q^{-56} -2 q^{-60} + q^{-62} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{14}-3 q^{12}+q^{10}+7 q^8-13 q^6+3 q^4+21 q^2-29+3 q^{-2} +36 q^{-4} -40 q^{-6} +2 q^{-8} +38 q^{-10} -29 q^{-12} -4 q^{-14} +23 q^{-16} -3 q^{-18} -9 q^{-20} -2 q^{-22} +21 q^{-24} -6 q^{-26} -30 q^{-28} +32 q^{-30} +2 q^{-32} -43 q^{-34} +34 q^{-36} +6 q^{-38} -32 q^{-40} +23 q^{-42} +3 q^{-44} -14 q^{-46} +8 q^{-48} + q^{-50} -3 q^{-52} + q^{-54} }[/math]
1,0,0 [math]\displaystyle{ -q^7+2 q^5-2 q^3+2 q-2 q^{-1} +4 q^{-3} -3 q^{-5} +3 q^{-7} + q^{-11} + q^{-13} +3 q^{-17} -3 q^{-19} +3 q^{-21} -3 q^{-23} + q^{-25} -3 q^{-27} +2 q^{-29} - q^{-31} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{16}-2 q^{14}-q^{12}+5 q^{10}-2 q^8-8 q^6+6 q^4+12 q^2-11-14 q^{-2} +20 q^{-4} +15 q^{-6} -31 q^{-8} -6 q^{-10} +39 q^{-12} -5 q^{-14} -35 q^{-16} +20 q^{-18} +28 q^{-20} -25 q^{-22} -11 q^{-24} +36 q^{-26} + q^{-28} -30 q^{-30} +24 q^{-32} +24 q^{-34} -39 q^{-36} -11 q^{-38} +34 q^{-40} -13 q^{-42} -37 q^{-44} +14 q^{-46} +26 q^{-48} -17 q^{-50} -15 q^{-52} +20 q^{-54} +9 q^{-56} -15 q^{-58} - q^{-60} +8 q^{-62} -2 q^{-64} -2 q^{-66} + q^{-68} }[/math]
1,0,0,0 [math]\displaystyle{ -q^8+2 q^6-2 q^4+q^2+1-2 q^{-2} +4 q^{-4} -3 q^{-6} +3 q^{-8} + q^{-12} + q^{-14} + q^{-16} + q^{-18} +3 q^{-22} -3 q^{-24} +3 q^{-26} -3 q^{-28} -3 q^{-34} +2 q^{-36} - q^{-38} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{14}+3 q^{12}-7 q^{10}+13 q^8-21 q^6+31 q^4-41 q^2+49-51 q^{-2} +48 q^{-4} -36 q^{-6} +20 q^{-8} +4 q^{-10} -29 q^{-12} +56 q^{-14} -77 q^{-16} +93 q^{-18} -99 q^{-20} +96 q^{-22} -83 q^{-24} +62 q^{-26} -36 q^{-28} +10 q^{-30} +14 q^{-32} -33 q^{-34} +46 q^{-36} -52 q^{-38} +50 q^{-40} -45 q^{-42} +35 q^{-44} -24 q^{-46} +14 q^{-48} -7 q^{-50} +3 q^{-52} - q^{-54} }[/math]
1,0 [math]\displaystyle{ q^{24}-3 q^{20}-3 q^{18}+4 q^{16}+10 q^{14}-17 q^{10}-13 q^8+17 q^6+31 q^4-3 q^2-43-21 q^{-2} +39 q^{-4} +45 q^{-6} -19 q^{-8} -55 q^{-10} -6 q^{-12} +52 q^{-14} +26 q^{-16} -36 q^{-18} -34 q^{-20} +23 q^{-22} +37 q^{-24} -10 q^{-26} -36 q^{-28} +3 q^{-30} +36 q^{-32} +6 q^{-34} -34 q^{-36} -14 q^{-38} +33 q^{-40} +25 q^{-42} -31 q^{-44} -39 q^{-46} +20 q^{-48} +50 q^{-50} -2 q^{-52} -55 q^{-54} -25 q^{-56} +43 q^{-58} +44 q^{-60} -20 q^{-62} -46 q^{-64} -5 q^{-66} +34 q^{-68} +19 q^{-70} -15 q^{-72} -19 q^{-74} + q^{-76} +11 q^{-78} +4 q^{-80} -3 q^{-82} -3 q^{-84} + q^{-88} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{18}-3 q^{16}+4 q^{14}-6 q^{12}+11 q^{10}-17 q^8+20 q^6-24 q^4+34 q^2-39+38 q^{-2} -38 q^{-4} +41 q^{-6} -33 q^{-8} +20 q^{-10} -12 q^{-12} +4 q^{-14} +19 q^{-16} -34 q^{-18} +43 q^{-20} -53 q^{-22} +73 q^{-24} -73 q^{-26} +74 q^{-28} -74 q^{-30} +75 q^{-32} -58 q^{-34} +47 q^{-36} -42 q^{-38} +21 q^{-40} -4 q^{-42} -10 q^{-44} +13 q^{-46} -31 q^{-48} +39 q^{-50} -39 q^{-52} +40 q^{-54} -41 q^{-56} +37 q^{-58} -27 q^{-60} +23 q^{-62} -19 q^{-64} +12 q^{-66} -6 q^{-68} +4 q^{-70} -3 q^{-72} + q^{-74} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+15 q^{24}-14 q^{22}+3 q^{20}+21 q^{18}-49 q^{16}+81 q^{14}-100 q^{12}+87 q^{10}-40 q^8-54 q^6+173 q^4-269 q^2+308-240 q^{-2} +63 q^{-4} +178 q^{-6} -398 q^{-8} +501 q^{-10} -428 q^{-12} +190 q^{-14} +119 q^{-16} -379 q^{-18} +476 q^{-20} -352 q^{-22} +81 q^{-24} +223 q^{-26} -405 q^{-28} +367 q^{-30} -133 q^{-32} -203 q^{-34} +486 q^{-36} -585 q^{-38} +465 q^{-40} -143 q^{-42} -252 q^{-44} +583 q^{-46} -727 q^{-48} +629 q^{-50} -331 q^{-52} -66 q^{-54} +415 q^{-56} -593 q^{-58} +555 q^{-60} -307 q^{-62} -27 q^{-64} +311 q^{-66} -430 q^{-68} +315 q^{-70} -41 q^{-72} -263 q^{-74} +450 q^{-76} -428 q^{-78} +211 q^{-80} +108 q^{-82} -391 q^{-84} +522 q^{-86} -465 q^{-88} +250 q^{-90} +19 q^{-92} -247 q^{-94} +349 q^{-96} -321 q^{-98} +213 q^{-100} -69 q^{-102} -46 q^{-104} +107 q^{-106} -122 q^{-108} +94 q^{-110} -52 q^{-112} +18 q^{-114} +7 q^{-116} -16 q^{-118} +16 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }[/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["10 117"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-10 t^2+24 t-31+24 t^{-1} -10 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+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]= { 103, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^8+4 q^7-9 q^6+13 q^5-16 q^4+18 q^3-16 q^2+13 q-8+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} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -z^4+2 z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -z^2+ a^{-2} + a^{-4} - a^{-6} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 z^9 a^{-3} +3 z^9 a^{-5} +7 z^8 a^{-2} +15 z^8 a^{-4} +8 z^8 a^{-6} +7 z^7 a^{-1} +9 z^7 a^{-3} +10 z^7 a^{-5} +8 z^7 a^{-7} -8 z^6 a^{-2} -28 z^6 a^{-4} -12 z^6 a^{-6} +4 z^6 a^{-8} +4 z^6+a z^5-11 z^5 a^{-1} -26 z^5 a^{-3} -29 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} +17 z^4 a^{-4} +6 z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-a z^3+5 z^3 a^{-1} +18 z^3 a^{-3} +21 z^3 a^{-5} +8 z^3 a^{-7} -z^3 a^{-9} +2 z^2 a^{-2} -4 z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} +2 z^2-z a^{-1} -3 z a^{-3} -5 z a^{-5} -3 z a^{-7} - a^{-2} + a^{-4} + a^{-6} }[/math]
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