10 113

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-11 t^2+26 t-33+26 t^{-1} -11 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 111, 2 }
Jones polynomial [math]\displaystyle{ -q^8+5 q^7-10 q^6+14 q^5-18 q^4+19 q^3-17 q^2+14 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} +z^4 a^{-4} -z^4 a^{-6} -z^4+3 z^2 a^{-2} -2 z^2 a^{-4} -z^2+3 a^{-2} -3 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} +16 z^8 a^{-4} +9 z^8 a^{-6} +7 z^7 a^{-1} +12 z^7 a^{-3} +15 z^7 a^{-5} +10 z^7 a^{-7} -5 z^6 a^{-2} -23 z^6 a^{-4} -9 z^6 a^{-6} +5 z^6 a^{-8} +4 z^6+a z^5-10 z^5 a^{-1} -30 z^5 a^{-3} -36 z^5 a^{-5} -16 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} +z^4 a^{-4} -4 z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-a z^3+5 z^3 a^{-1} +17 z^3 a^{-3} +16 z^3 a^{-5} +5 z^3 a^{-7} +8 z^2 a^{-2} +8 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2-z a^{-1} -z a^{-3} +z a^{-5} +z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} }[/math]
The A2 invariant [math]\displaystyle{ -q^6+2 q^4-q^2-1+5 q^{-2} -2 q^{-4} +4 q^{-6} -2 q^{-10} + q^{-12} -5 q^{-14} +3 q^{-16} - q^{-18} - q^{-20} +3 q^{-22} - q^{-24} }[/math]
The G2 invariant [math]\displaystyle{ q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+15 q^{24}-15 q^{22}+4 q^{20}+20 q^{18}-50 q^{16}+86 q^{14}-108 q^{12}+100 q^{10}-52 q^8-47 q^6+182 q^4-301 q^2+359-303 q^{-2} +110 q^{-4} +168 q^{-6} -443 q^{-8} +605 q^{-10} -562 q^{-12} +315 q^{-14} +63 q^{-16} -415 q^{-18} +597 q^{-20} -516 q^{-22} +220 q^{-24} +165 q^{-26} -445 q^{-28} +484 q^{-30} -265 q^{-32} -118 q^{-34} +506 q^{-36} -702 q^{-38} +623 q^{-40} -277 q^{-42} -216 q^{-44} +661 q^{-46} -905 q^{-48} +841 q^{-50} -511 q^{-52} +19 q^{-54} +454 q^{-56} -748 q^{-58} +762 q^{-60} -503 q^{-62} +81 q^{-64} +313 q^{-66} -530 q^{-68} +465 q^{-70} -168 q^{-72} -207 q^{-74} +498 q^{-76} -548 q^{-78} +346 q^{-80} +29 q^{-82} -415 q^{-84} +644 q^{-86} -631 q^{-88} +398 q^{-90} -50 q^{-92} -270 q^{-94} +454 q^{-96} -460 q^{-98} +334 q^{-100} -139 q^{-102} -42 q^{-104} +147 q^{-106} -183 q^{-108} +147 q^{-110} -84 q^{-112} +31 q^{-114} +10 q^{-116} -25 q^{-118} +26 q^{-120} -20 q^{-122} +10 q^{-124} -4 q^{-126} + q^{-128} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_113.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^5+3 q^3-4 q+6 q^{-1} -3 q^{-3} +2 q^{-5} + q^{-7} -4 q^{-9} +4 q^{-11} -5 q^{-13} +4 q^{-15} - q^{-17} }[/math]
2 [math]\displaystyle{ q^{16}-3 q^{14}+10 q^{10}-15 q^8-6 q^6+39 q^4-26 q^2-34+67 q^{-2} -10 q^{-4} -61 q^{-6} +51 q^{-8} +19 q^{-10} -50 q^{-12} +6 q^{-14} +35 q^{-16} -10 q^{-18} -40 q^{-20} +32 q^{-22} +34 q^{-24} -65 q^{-26} +10 q^{-28} +61 q^{-30} -52 q^{-32} -17 q^{-34} +50 q^{-36} -17 q^{-38} -20 q^{-40} +16 q^{-42} + q^{-44} -4 q^{-46} + q^{-48} }[/math]
3 [math]\displaystyle{ -q^{33}+3 q^{31}-6 q^{27}-q^{25}+15 q^{23}+5 q^{21}-43 q^{19}-14 q^{17}+84 q^{15}+58 q^{13}-140 q^{11}-150 q^9+181 q^7+296 q^5-167 q^3-458 q+67 q^{-1} +608 q^{-3} +93 q^{-5} -663 q^{-7} -290 q^{-9} +618 q^{-11} +460 q^{-13} -492 q^{-15} -556 q^{-17} +304 q^{-19} +579 q^{-21} -105 q^{-23} -535 q^{-25} -86 q^{-27} +459 q^{-29} +252 q^{-31} -346 q^{-33} -413 q^{-35} +228 q^{-37} +541 q^{-39} -78 q^{-41} -643 q^{-43} -92 q^{-45} +678 q^{-47} +275 q^{-49} -626 q^{-51} -442 q^{-53} +498 q^{-55} +535 q^{-57} -304 q^{-59} -540 q^{-61} +107 q^{-63} +454 q^{-65} +41 q^{-67} -318 q^{-69} -104 q^{-71} +171 q^{-73} +108 q^{-75} -69 q^{-77} -75 q^{-79} +22 q^{-81} +31 q^{-83} -12 q^{-87} - q^{-89} +4 q^{-91} - q^{-93} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^6+2 q^4-q^2-1+5 q^{-2} -2 q^{-4} +4 q^{-6} -2 q^{-10} + q^{-12} -5 q^{-14} +3 q^{-16} - q^{-18} - q^{-20} +3 q^{-22} - q^{-24} }[/math]
1,1 [math]\displaystyle{ q^{20}-6 q^{18}+20 q^{16}-50 q^{14}+109 q^{12}-218 q^{10}+392 q^8-662 q^6+1048 q^4-1548 q^2+2134-2722 q^{-2} +3230 q^{-4} -3482 q^{-6} +3384 q^{-8} -2820 q^{-10} +1769 q^{-12} -308 q^{-14} -1436 q^{-16} +3228 q^{-18} -4886 q^{-20} +6174 q^{-22} -6924 q^{-24} +7056 q^{-26} -6543 q^{-28} +5466 q^{-30} -3938 q^{-32} +2154 q^{-34} -340 q^{-36} -1288 q^{-38} +2552 q^{-40} -3344 q^{-42} +3642 q^{-44} -3508 q^{-46} +3056 q^{-48} -2430 q^{-50} +1773 q^{-52} -1192 q^{-54} +730 q^{-56} -400 q^{-58} +198 q^{-60} -88 q^{-62} +32 q^{-64} -8 q^{-66} + q^{-68} }[/math]
2,0 [math]\displaystyle{ q^{18}-2 q^{16}-2 q^{14}+6 q^{12}+2 q^{10}-12 q^8-6 q^6+19 q^4+10 q^2-24-7 q^{-2} +35 q^{-4} +7 q^{-6} -29 q^{-8} +4 q^{-10} +24 q^{-12} -7 q^{-14} -22 q^{-16} +7 q^{-18} +6 q^{-20} -20 q^{-22} +8 q^{-24} +13 q^{-26} -15 q^{-28} -4 q^{-30} +28 q^{-32} +2 q^{-34} -30 q^{-36} +4 q^{-38} +27 q^{-40} -3 q^{-42} -29 q^{-44} +7 q^{-46} +19 q^{-48} -6 q^{-50} -11 q^{-52} +9 q^{-56} - q^{-58} -3 q^{-60} + q^{-62} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{14}-3 q^{12}+q^{10}+7 q^8-14 q^6+3 q^4+24 q^2-31+2 q^{-2} +44 q^{-4} -44 q^{-6} + q^{-8} +48 q^{-10} -34 q^{-12} -7 q^{-14} +27 q^{-16} -10 q^{-18} -17 q^{-20} -5 q^{-22} +20 q^{-24} -6 q^{-26} -31 q^{-28} +41 q^{-30} +9 q^{-32} -48 q^{-34} +39 q^{-36} +9 q^{-38} -42 q^{-40} +26 q^{-42} +5 q^{-44} -19 q^{-46} +11 q^{-48} +2 q^{-50} -4 q^{-52} + q^{-54} }[/math]
1,0,0 [math]\displaystyle{ -q^7+2 q^5-2 q^3+2 q-2 q^{-1} +5 q^{-3} -2 q^{-5} +5 q^{-7} + q^{-9} + q^{-11} - q^{-13} -3 q^{-15} -5 q^{-19} +3 q^{-21} -2 q^{-23} +3 q^{-25} -2 q^{-27} +3 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^8-9 q^6+5 q^4+14 q^2-10-16 q^{-2} +22 q^{-4} +22 q^{-6} -31 q^{-8} -8 q^{-10} +47 q^{-12} +2 q^{-14} -40 q^{-16} +18 q^{-18} +33 q^{-20} -32 q^{-22} -27 q^{-24} +30 q^{-26} -4 q^{-28} -45 q^{-30} +20 q^{-32} +34 q^{-34} -35 q^{-36} -8 q^{-38} +51 q^{-40} -2 q^{-42} -40 q^{-44} +13 q^{-46} +30 q^{-48} -22 q^{-50} -24 q^{-52} +21 q^{-54} +12 q^{-56} -17 q^{-58} -2 q^{-60} +11 q^{-62} -2 q^{-64} -3 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} +5 q^{-4} -2 q^{-6} +5 q^{-8} +2 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} -2 q^{-18} -4 q^{-20} -5 q^{-24} +3 q^{-26} -2 q^{-28} +2 q^{-30} +2 q^{-32} -2 q^{-34} +3 q^{-36} - q^{-38} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{14}+3 q^{12}-7 q^{10}+13 q^8-22 q^6+33 q^4-44 q^2+55-58 q^{-2} +58 q^{-4} -46 q^{-6} +29 q^{-8} -2 q^{-10} -26 q^{-12} +57 q^{-14} -83 q^{-16} +104 q^{-18} -115 q^{-20} +113 q^{-22} -102 q^{-24} +78 q^{-26} -51 q^{-28} +19 q^{-30} +9 q^{-32} -34 q^{-34} +51 q^{-36} -59 q^{-38} +60 q^{-40} -54 q^{-42} +45 q^{-44} -31 q^{-46} +19 q^{-48} -10 q^{-50} +4 q^{-52} - q^{-54} }[/math]
1,0 [math]\displaystyle{ q^{24}-3 q^{20}-3 q^{18}+4 q^{16}+10 q^{14}-18 q^{10}-14 q^8+18 q^6+34 q^4-q^2-47-25 q^{-2} +42 q^{-4} +54 q^{-6} -17 q^{-8} -64 q^{-10} -11 q^{-12} +60 q^{-14} +35 q^{-16} -39 q^{-18} -44 q^{-20} +22 q^{-22} +45 q^{-24} -8 q^{-26} -45 q^{-28} -4 q^{-30} +39 q^{-32} +9 q^{-34} -39 q^{-36} -22 q^{-38} +35 q^{-40} +32 q^{-42} -30 q^{-44} -45 q^{-46} +21 q^{-48} +60 q^{-50} +5 q^{-52} -61 q^{-54} -32 q^{-56} +48 q^{-58} +54 q^{-60} -21 q^{-62} -57 q^{-64} -10 q^{-66} +40 q^{-68} +26 q^{-70} -19 q^{-72} -25 q^{-74} + q^{-76} +15 q^{-78} +6 q^{-80} -4 q^{-82} -4 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}-18 q^8+21 q^6-25 q^4+37 q^2-43+43 q^{-2} -43 q^{-4} +50 q^{-6} -40 q^{-8} +29 q^{-10} -17 q^{-12} +12 q^{-14} +18 q^{-16} -32 q^{-18} +43 q^{-20} -59 q^{-22} +77 q^{-24} -88 q^{-26} +79 q^{-28} -93 q^{-30} +85 q^{-32} -74 q^{-34} +60 q^{-36} -52 q^{-38} +35 q^{-40} -6 q^{-42} - q^{-44} +14 q^{-46} -30 q^{-48} +44 q^{-50} -45 q^{-52} +45 q^{-54} -50 q^{-56} +44 q^{-58} -35 q^{-60} +30 q^{-62} -25 q^{-64} +16 q^{-66} -8 q^{-68} +6 q^{-70} -4 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}-15 q^{22}+4 q^{20}+20 q^{18}-50 q^{16}+86 q^{14}-108 q^{12}+100 q^{10}-52 q^8-47 q^6+182 q^4-301 q^2+359-303 q^{-2} +110 q^{-4} +168 q^{-6} -443 q^{-8} +605 q^{-10} -562 q^{-12} +315 q^{-14} +63 q^{-16} -415 q^{-18} +597 q^{-20} -516 q^{-22} +220 q^{-24} +165 q^{-26} -445 q^{-28} +484 q^{-30} -265 q^{-32} -118 q^{-34} +506 q^{-36} -702 q^{-38} +623 q^{-40} -277 q^{-42} -216 q^{-44} +661 q^{-46} -905 q^{-48} +841 q^{-50} -511 q^{-52} +19 q^{-54} +454 q^{-56} -748 q^{-58} +762 q^{-60} -503 q^{-62} +81 q^{-64} +313 q^{-66} -530 q^{-68} +465 q^{-70} -168 q^{-72} -207 q^{-74} +498 q^{-76} -548 q^{-78} +346 q^{-80} +29 q^{-82} -415 q^{-84} +644 q^{-86} -631 q^{-88} +398 q^{-90} -50 q^{-92} -270 q^{-94} +454 q^{-96} -460 q^{-98} +334 q^{-100} -139 q^{-102} -42 q^{-104} +147 q^{-106} -183 q^{-108} +147 q^{-110} -84 q^{-112} +31 q^{-114} +10 q^{-116} -25 q^{-118} +26 q^{-120} -20 q^{-122} +10 q^{-124} -4 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["10 113"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-11 t^2+26 t-33+26 t^{-1} -11 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+z^4+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]= { 111, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^8+5 q^7-10 q^6+14 q^5-18 q^4+19 q^3-17 q^2+14 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} +z^4 a^{-4} -z^4 a^{-6} -z^4+3 z^2 a^{-2} -2 z^2 a^{-4} -z^2+3 a^{-2} -3 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} +16 z^8 a^{-4} +9 z^8 a^{-6} +7 z^7 a^{-1} +12 z^7 a^{-3} +15 z^7 a^{-5} +10 z^7 a^{-7} -5 z^6 a^{-2} -23 z^6 a^{-4} -9 z^6 a^{-6} +5 z^6 a^{-8} +4 z^6+a z^5-10 z^5 a^{-1} -30 z^5 a^{-3} -36 z^5 a^{-5} -16 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} +z^4 a^{-4} -4 z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-a z^3+5 z^3 a^{-1} +17 z^3 a^{-3} +16 z^3 a^{-5} +5 z^3 a^{-7} +8 z^2 a^{-2} +8 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2-z a^{-1} -z a^{-3} +z a^{-5} +z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} }[/math]
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