10 122

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^3+11 t^2-24 t+31-24 t^{-1} +11 t^{-2} -2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -2 z^6-z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \left\{3,t^2+1\right\} }[/math]
Determinant and Signature { 105, 0 }
Jones polynomial [math]\displaystyle{ q^6-5 q^5+9 q^4-13 q^3+17 q^2-17 q+17-13 q^{-1} +8 q^{-2} -4 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -2 z^4+a^2 z^2+3 z^2 a^{-2} -2 z^2+4 a^{-2} -2 a^{-4} -1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 4 z^9 a^{-1} +4 z^9 a^{-3} +18 z^8 a^{-2} +8 z^8 a^{-4} +10 z^8+11 a z^7+9 z^7 a^{-1} +3 z^7 a^{-3} +5 z^7 a^{-5} +8 a^2 z^6-42 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -13 z^6+4 a^3 z^5-14 a z^5-32 z^5 a^{-1} -25 z^5 a^{-3} -11 z^5 a^{-5} +a^4 z^4-7 a^2 z^4+24 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +3 z^4-2 a^3 z^3+6 a z^3+18 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +2 a^2 z^2+2 z^2+2 z a^{-3} +2 z a^{-5} -4 a^{-2} -2 a^{-4} -1 }[/math]
The A2 invariant [math]\displaystyle{ q^{12}-2 q^{10}+q^8+q^6-4 q^4+3 q^2-2+2 q^{-2} +3 q^{-4} +5 q^{-8} -3 q^{-10} -3 q^{-16} + q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+10 q^{58}-8 q^{56}+q^{54}+15 q^{52}-31 q^{50}+51 q^{48}-63 q^{46}+56 q^{44}-32 q^{42}-14 q^{40}+73 q^{38}-133 q^{36}+183 q^{34}-198 q^{32}+149 q^{30}-34 q^{28}-132 q^{26}+304 q^{24}-402 q^{22}+375 q^{20}-211 q^{18}-57 q^{16}+321 q^{14}-469 q^{12}+426 q^{10}-200 q^8-114 q^6+360 q^4-422 q^2+253+68 q^{-2} -380 q^{-4} +538 q^{-6} -459 q^{-8} +166 q^{-10} +226 q^{-12} -552 q^{-14} +696 q^{-16} -597 q^{-18} +301 q^{-20} +97 q^{-22} -438 q^{-24} +626 q^{-26} -590 q^{-28} +366 q^{-30} -28 q^{-32} -290 q^{-34} +464 q^{-36} -423 q^{-38} +194 q^{-40} +131 q^{-42} -392 q^{-44} +458 q^{-46} -299 q^{-48} -24 q^{-50} +353 q^{-52} -547 q^{-54} +517 q^{-56} -287 q^{-58} -43 q^{-60} +323 q^{-62} -459 q^{-64} +419 q^{-66} -246 q^{-68} +30 q^{-70} +130 q^{-72} -204 q^{-74} +186 q^{-76} -115 q^{-78} +45 q^{-80} +12 q^{-82} -35 q^{-84} +34 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_122.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^9-3 q^7+4 q^5-5 q^3+4 q+4 q^{-5} -4 q^{-7} +4 q^{-9} -4 q^{-11} + q^{-13} }[/math]
2 [math]\displaystyle{ q^{26}-3 q^{24}+q^{22}+7 q^{20}-14 q^{18}+7 q^{16}+23 q^{14}-40 q^{12}+q^{10}+52 q^8-43 q^6-24 q^4+54 q^2-9-35 q^{-2} +20 q^{-4} +27 q^{-6} -25 q^{-8} -20 q^{-10} +45 q^{-12} -3 q^{-14} -51 q^{-16} +40 q^{-18} +24 q^{-20} -56 q^{-22} +12 q^{-24} +36 q^{-26} -28 q^{-28} -10 q^{-30} +18 q^{-32} - q^{-34} -4 q^{-36} + q^{-38} }[/math]
3 [math]\displaystyle{ q^{51}-3 q^{49}+q^{47}+4 q^{45}-2 q^{43}-5 q^{41}+6 q^{39}+13 q^{37}-32 q^{35}-24 q^{33}+72 q^{31}+68 q^{29}-125 q^{27}-165 q^{25}+162 q^{23}+308 q^{21}-132 q^{19}-453 q^{17}+19 q^{15}+544 q^{13}+147 q^{11}-543 q^9-308 q^7+419 q^5+433 q^3-247 q-463 q^{-1} +53 q^{-3} +437 q^{-5} +119 q^{-7} -367 q^{-9} -253 q^{-11} +290 q^{-13} +360 q^{-15} -208 q^{-17} -441 q^{-19} +111 q^{-21} +511 q^{-23} +5 q^{-25} -542 q^{-27} -156 q^{-29} +525 q^{-31} +300 q^{-33} -420 q^{-35} -431 q^{-37} +258 q^{-39} +476 q^{-41} -67 q^{-43} -426 q^{-45} -97 q^{-47} +300 q^{-49} +184 q^{-51} -153 q^{-53} -177 q^{-55} +35 q^{-57} +118 q^{-59} +26 q^{-61} -57 q^{-63} -25 q^{-65} +12 q^{-67} +13 q^{-69} - q^{-71} -4 q^{-73} + q^{-75} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{12}-2 q^{10}+q^8+q^6-4 q^4+3 q^2-2+2 q^{-2} +3 q^{-4} +5 q^{-8} -3 q^{-10} -3 q^{-16} + q^{-18} }[/math]
1,1 [math]\displaystyle{ q^{36}-6 q^{34}+18 q^{32}-38 q^{30}+71 q^{28}-128 q^{26}+212 q^{24}-320 q^{22}+460 q^{20}-650 q^{18}+896 q^{16}-1186 q^{14}+1517 q^{12}-1834 q^{10}+2064 q^8-2134 q^6+1924 q^4-1414 q^2+586+492 q^{-2} -1670 q^{-4} +2834 q^{-6} -3788 q^{-8} +4442 q^{-10} -4690 q^{-12} +4518 q^{-14} -3944 q^{-16} +3034 q^{-18} -1890 q^{-20} +652 q^{-22} +538 q^{-24} -1540 q^{-26} +2230 q^{-28} -2566 q^{-30} +2544 q^{-32} -2260 q^{-34} +1804 q^{-36} -1300 q^{-38} +842 q^{-40} -480 q^{-42} +244 q^{-44} -104 q^{-46} +34 q^{-48} -8 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+5 q^{26}-2 q^{24}-8 q^{22}+6 q^{20}+14 q^{18}-8 q^{16}-20 q^{14}+10 q^{12}+26 q^{10}-20 q^8-20 q^6+18 q^4+10 q^2-13-11 q^{-2} +17 q^{-4} -2 q^{-6} -6 q^{-8} +13 q^{-10} +5 q^{-12} -10 q^{-14} +12 q^{-16} +21 q^{-18} -17 q^{-20} -14 q^{-22} +11 q^{-24} +10 q^{-26} -21 q^{-28} -18 q^{-30} +19 q^{-32} +9 q^{-34} -12 q^{-36} -7 q^{-38} +7 q^{-40} +9 q^{-42} -3 q^{-44} -3 q^{-46} + q^{-48} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{28}-3 q^{26}+9 q^{22}-10 q^{20}-6 q^{18}+24 q^{16}-15 q^{14}-20 q^{12}+40 q^{10}-15 q^8-29 q^6+40 q^4-13 q^2-24+18 q^{-2} +4 q^{-4} -4 q^{-6} -4 q^{-8} +22 q^{-10} +17 q^{-12} -28 q^{-14} +11 q^{-16} +23 q^{-18} -43 q^{-20} +3 q^{-22} +26 q^{-24} -34 q^{-26} +10 q^{-28} +16 q^{-30} -17 q^{-32} +7 q^{-34} +3 q^{-36} -4 q^{-38} + q^{-40} }[/math]
1,0,0 [math]\displaystyle{ q^{15}-2 q^{13}+2 q^{11}-2 q^9+2 q^7-4 q^5+3 q^3-3 q+ q^{-1} + q^{-3} +2 q^{-5} +4 q^{-7} + q^{-9} +5 q^{-11} -3 q^{-13} + q^{-15} -4 q^{-17} + q^{-19} -3 q^{-21} + q^{-23} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{34}-2 q^{32}-2 q^{30}+6 q^{28}+q^{26}-9 q^{24}-q^{22}+14 q^{20}+q^{18}-22 q^{16}+4 q^{14}+30 q^{12}-13 q^{10}-29 q^8+26 q^6+17 q^4-36 q^2-12+27 q^{-2} -8 q^{-4} -30 q^{-6} +25 q^{-8} +31 q^{-10} -24 q^{-12} +10 q^{-14} +53 q^{-16} -11 q^{-18} -32 q^{-20} +28 q^{-22} +11 q^{-24} -42 q^{-26} -19 q^{-28} +23 q^{-30} -2 q^{-32} -25 q^{-34} +9 q^{-36} +19 q^{-38} -8 q^{-40} -5 q^{-42} +9 q^{-44} - q^{-46} -3 q^{-48} + q^{-50} }[/math]
1,0,0,0 [math]\displaystyle{ q^{18}-2 q^{16}+2 q^{14}-q^{12}-q^{10}+2 q^8-4 q^6+3 q^4-3 q^2+ q^{-4} +2 q^{-6} +3 q^{-8} +5 q^{-10} + q^{-12} +5 q^{-14} -3 q^{-16} + q^{-18} -3 q^{-20} -3 q^{-22} + q^{-24} -3 q^{-26} + q^{-28} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{28}-3 q^{26}+6 q^{24}-11 q^{22}+18 q^{20}-26 q^{18}+36 q^{16}-45 q^{14}+50 q^{12}-52 q^{10}+43 q^8-29 q^6+8 q^4+17 q^2-42+68 q^{-2} -86 q^{-4} +100 q^{-6} -100 q^{-8} +94 q^{-10} -75 q^{-12} +54 q^{-14} -25 q^{-16} - q^{-18} +25 q^{-20} -41 q^{-22} +50 q^{-24} -54 q^{-26} +50 q^{-28} -44 q^{-30} +31 q^{-32} -21 q^{-34} +11 q^{-36} -4 q^{-38} + q^{-40} }[/math]
1,0 [math]\displaystyle{ q^{46}-3 q^{42}-3 q^{40}+3 q^{38}+10 q^{36}+3 q^{34}-14 q^{32}-16 q^{30}+7 q^{28}+30 q^{26}+12 q^{24}-31 q^{22}-35 q^{20}+14 q^{18}+52 q^{16}+10 q^{14}-50 q^{12}-33 q^{10}+34 q^8+45 q^6-18 q^4-49 q^2-1+44 q^{-2} +11 q^{-4} -36 q^{-6} -18 q^{-8} +31 q^{-10} +27 q^{-12} -19 q^{-14} -26 q^{-16} +22 q^{-18} +38 q^{-20} -8 q^{-22} -45 q^{-24} -6 q^{-26} +48 q^{-28} +27 q^{-30} -44 q^{-32} -52 q^{-34} +18 q^{-36} +55 q^{-38} +6 q^{-40} -48 q^{-42} -28 q^{-44} +29 q^{-46} +34 q^{-48} -9 q^{-50} -24 q^{-52} -4 q^{-54} +14 q^{-56} +7 q^{-58} -4 q^{-60} -4 q^{-62} + q^{-66} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{38}-3 q^{36}+3 q^{34}-4 q^{32}+10 q^{30}-14 q^{28}+14 q^{26}-19 q^{24}+30 q^{22}-33 q^{20}+31 q^{18}-39 q^{16}+45 q^{14}-37 q^{12}+29 q^{10}-26 q^8+13 q^6+4 q^4-18 q^2+26-50 q^{-2} +61 q^{-4} -66 q^{-6} +76 q^{-8} -76 q^{-10} +84 q^{-12} -62 q^{-14} +70 q^{-16} -49 q^{-18} +39 q^{-20} -21 q^{-22} +6 q^{-24} -26 q^{-28} +29 q^{-30} -40 q^{-32} +38 q^{-34} -44 q^{-36} +44 q^{-38} -37 q^{-40} +33 q^{-42} -25 q^{-44} +18 q^{-46} -11 q^{-48} +7 q^{-50} -4 q^{-52} + q^{-54} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+10 q^{58}-8 q^{56}+q^{54}+15 q^{52}-31 q^{50}+51 q^{48}-63 q^{46}+56 q^{44}-32 q^{42}-14 q^{40}+73 q^{38}-133 q^{36}+183 q^{34}-198 q^{32}+149 q^{30}-34 q^{28}-132 q^{26}+304 q^{24}-402 q^{22}+375 q^{20}-211 q^{18}-57 q^{16}+321 q^{14}-469 q^{12}+426 q^{10}-200 q^8-114 q^6+360 q^4-422 q^2+253+68 q^{-2} -380 q^{-4} +538 q^{-6} -459 q^{-8} +166 q^{-10} +226 q^{-12} -552 q^{-14} +696 q^{-16} -597 q^{-18} +301 q^{-20} +97 q^{-22} -438 q^{-24} +626 q^{-26} -590 q^{-28} +366 q^{-30} -28 q^{-32} -290 q^{-34} +464 q^{-36} -423 q^{-38} +194 q^{-40} +131 q^{-42} -392 q^{-44} +458 q^{-46} -299 q^{-48} -24 q^{-50} +353 q^{-52} -547 q^{-54} +517 q^{-56} -287 q^{-58} -43 q^{-60} +323 q^{-62} -459 q^{-64} +419 q^{-66} -246 q^{-68} +30 q^{-70} +130 q^{-72} -204 q^{-74} +186 q^{-76} -115 q^{-78} +45 q^{-80} +12 q^{-82} -35 q^{-84} +34 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94} }[/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["10 122"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^3+11 t^2-24 t+31-24 t^{-1} +11 t^{-2} -2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -2 z^6-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{ \left\{3,t^2+1\right\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 105, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^6-5 q^5+9 q^4-13 q^3+17 q^2-17 q+17-13 q^{-1} +8 q^{-2} -4 q^{-3} + q^{-4} }[/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^2 z^4-z^4 a^{-2} +z^4 a^{-4} -2 z^4+a^2 z^2+3 z^2 a^{-2} -2 z^2+4 a^{-2} -2 a^{-4} -1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 4 z^9 a^{-1} +4 z^9 a^{-3} +18 z^8 a^{-2} +8 z^8 a^{-4} +10 z^8+11 a z^7+9 z^7 a^{-1} +3 z^7 a^{-3} +5 z^7 a^{-5} +8 a^2 z^6-42 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -13 z^6+4 a^3 z^5-14 a z^5-32 z^5 a^{-1} -25 z^5 a^{-3} -11 z^5 a^{-5} +a^4 z^4-7 a^2 z^4+24 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +3 z^4-2 a^3 z^3+6 a z^3+18 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +2 a^2 z^2+2 z^2+2 z a^{-3} +2 z a^{-5} -4 a^{-2} -2 a^{-4} -1 }[/math]
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