10 108
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ 2 t^3-8 t^2+14 t-15+14 t^{-1} -8 t^{-2} +2 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ 2 z^6+4 z^4+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 63, 2 } |
| Jones polynomial | [math]\displaystyle{ -q^6+3 q^5-5 q^4+8 q^3-10 q^2+10 q-9+8 q^{-1} -5 q^{-2} +3 q^{-3} - q^{-4} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +3 z^4-2 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2+1 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+6 z^8 a^{-2} +9 z^8+a^3 z^7-3 a z^7+4 z^7 a^{-1} +8 z^7 a^{-3} -13 a^2 z^6-13 z^6 a^{-2} +7 z^6 a^{-4} -33 z^6-4 a^3 z^5-11 a z^5-29 z^5 a^{-1} -17 z^5 a^{-3} +5 z^5 a^{-5} +17 a^2 z^4+4 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} +33 z^4+5 a^3 z^3+19 a z^3+28 z^3 a^{-1} +10 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+2 z^2 a^{-4} -z^2 a^{-6} -10 z^2-2 a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} +1 }[/math] |
| The A2 invariant | [math]\displaystyle{ -q^{12}+q^{10}+2 q^4-q^2+2- q^{-4} + q^{-6} -2 q^{-8} +2 q^{-10} + q^{-16} - q^{-18} }[/math] |
| The G2 invariant | [math]\displaystyle{ q^{60}-2 q^{58}+6 q^{56}-11 q^{54}+13 q^{52}-12 q^{50}-2 q^{48}+25 q^{46}-46 q^{44}+58 q^{42}-47 q^{40}+11 q^{38}+36 q^{36}-81 q^{34}+98 q^{32}-77 q^{30}+24 q^{28}+37 q^{26}-81 q^{24}+89 q^{22}-54 q^{20}+3 q^{18}+48 q^{16}-68 q^{14}+53 q^{12}-11 q^{10}-39 q^8+74 q^6-77 q^4+59 q^2-12-42 q^{-2} +88 q^{-4} -108 q^{-6} +93 q^{-8} -49 q^{-10} -18 q^{-12} +74 q^{-14} -102 q^{-16} +93 q^{-18} -49 q^{-20} -10 q^{-22} +61 q^{-24} -75 q^{-26} +47 q^{-28} - q^{-30} -45 q^{-32} +67 q^{-34} -50 q^{-36} +12 q^{-38} +31 q^{-40} -57 q^{-42} +63 q^{-44} -46 q^{-46} +16 q^{-48} +13 q^{-50} -36 q^{-52} +41 q^{-54} -36 q^{-56} +27 q^{-58} -12 q^{-60} +2 q^{-62} +9 q^{-64} -21 q^{-66} +24 q^{-68} -23 q^{-70} +16 q^{-72} -7 q^{-74} +7 q^{-78} -12 q^{-80} +13 q^{-82} -10 q^{-84} +7 q^{-86} -2 q^{-88} - q^{-90} +2 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 10_108.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | [math]\displaystyle{ -q^9+2 q^7-2 q^5+3 q^3-q+ q^{-1} -2 q^{-5} +3 q^{-7} -2 q^{-9} +2 q^{-11} - q^{-13} }[/math] |
| 2 | [math]\displaystyle{ q^{28}-2 q^{26}-3 q^{24}+7 q^{22}+q^{20}-12 q^{18}+7 q^{16}+11 q^{14}-14 q^{12}-q^{10}+15 q^8-8 q^6-9 q^4+12 q^2+1-11 q^{-2} +4 q^{-4} +10 q^{-6} -7 q^{-8} -7 q^{-10} +13 q^{-12} +2 q^{-14} -15 q^{-16} +7 q^{-18} +8 q^{-20} -11 q^{-22} +3 q^{-24} +4 q^{-26} -5 q^{-28} +3 q^{-30} -2 q^{-34} + q^{-36} }[/math] |
| 3 | [math]\displaystyle{ -q^{57}+2 q^{55}+3 q^{53}-2 q^{51}-10 q^{49}-4 q^{47}+18 q^{45}+17 q^{43}-16 q^{41}-37 q^{39}+q^{37}+50 q^{35}+26 q^{33}-46 q^{31}-54 q^{29}+24 q^{27}+74 q^{25}+7 q^{23}-71 q^{21}-42 q^{19}+56 q^{17}+64 q^{15}-33 q^{13}-74 q^{11}+8 q^9+74 q^7+13 q^5-70 q^3-26 q+68 q^{-1} +38 q^{-3} -58 q^{-5} -51 q^{-7} +50 q^{-9} +62 q^{-11} -30 q^{-13} -73 q^{-15} +69 q^{-19} +36 q^{-21} -51 q^{-23} -69 q^{-25} +25 q^{-27} +82 q^{-29} +10 q^{-31} -76 q^{-33} -33 q^{-35} +56 q^{-37} +39 q^{-39} -31 q^{-41} -31 q^{-43} +12 q^{-45} +17 q^{-47} -3 q^{-49} -8 q^{-51} +2 q^{-53} +2 q^{-57} - q^{-61} - q^{-63} +2 q^{-67} - q^{-69} }[/math] |
| 4 | [math]\displaystyle{ q^{96}-2 q^{94}-3 q^{92}+2 q^{90}+5 q^{88}+13 q^{86}-3 q^{84}-23 q^{82}-24 q^{80}-9 q^{78}+55 q^{76}+60 q^{74}+7 q^{72}-71 q^{70}-131 q^{68}-17 q^{66}+112 q^{64}+179 q^{62}+81 q^{60}-177 q^{58}-240 q^{56}-119 q^{54}+180 q^{52}+358 q^{50}+134 q^{48}-188 q^{46}-425 q^{44}-215 q^{42}+262 q^{40}+454 q^{38}+265 q^{36}-288 q^{34}-548 q^{32}-222 q^{30}+304 q^{28}+596 q^{26}+194 q^{24}-407 q^{22}-555 q^{20}-131 q^{18}+493 q^{16}+510 q^{14}-40 q^{12}-514 q^{10}-390 q^8+218 q^6+505 q^4+167 q^2-353-402 q^{-2} +79 q^{-4} +433 q^{-6} +199 q^{-8} -302 q^{-10} -397 q^{-12} +18 q^{-14} +442 q^{-16} +306 q^{-18} -215 q^{-20} -473 q^{-22} -228 q^{-24} +326 q^{-26} +519 q^{-28} +156 q^{-30} -357 q^{-32} -592 q^{-34} -154 q^{-36} +466 q^{-38} +620 q^{-40} +147 q^{-42} -592 q^{-44} -648 q^{-46} - q^{-48} +631 q^{-50} +591 q^{-52} -165 q^{-54} -627 q^{-56} -363 q^{-58} +240 q^{-60} +520 q^{-62} +138 q^{-64} -266 q^{-66} -296 q^{-68} -22 q^{-70} +225 q^{-72} +117 q^{-74} -49 q^{-76} -106 q^{-78} -45 q^{-80} +62 q^{-82} +30 q^{-84} - q^{-86} -18 q^{-88} -18 q^{-90} +19 q^{-92} + q^{-94} -2 q^{-98} -6 q^{-100} +6 q^{-102} - q^{-104} + q^{-106} -2 q^{-110} + q^{-112} }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ -q^{12}+q^{10}+2 q^4-q^2+2- q^{-4} + q^{-6} -2 q^{-8} +2 q^{-10} + q^{-16} - q^{-18} }[/math] |
| 1,1 | [math]\displaystyle{ q^{36}-4 q^{34}+14 q^{32}-38 q^{30}+74 q^{28}-132 q^{26}+196 q^{24}-256 q^{22}+308 q^{20}-318 q^{18}+286 q^{16}-204 q^{14}+95 q^{12}+44 q^{10}-196 q^8+322 q^6-425 q^4+492 q^2-524+502 q^{-2} -427 q^{-4} +330 q^{-6} -200 q^{-8} +74 q^{-10} +48 q^{-12} -134 q^{-14} +180 q^{-16} -198 q^{-18} +178 q^{-20} -152 q^{-22} +130 q^{-24} -104 q^{-26} +83 q^{-28} -70 q^{-30} +70 q^{-32} -64 q^{-34} +48 q^{-36} -42 q^{-38} +38 q^{-40} -28 q^{-42} +18 q^{-44} -12 q^{-46} +8 q^{-48} -4 q^{-50} + q^{-52} }[/math] |
| 2,0 | [math]\displaystyle{ q^{34}-q^{32}-3 q^{30}+3 q^{26}+q^{24}-4 q^{22}-q^{20}+7 q^{18}+3 q^{16}-5 q^{14}+4 q^{10}+3 q^8-6 q^6-2 q^4+4 q^2-3- q^{-2} +2 q^{-4} -2 q^{-8} +5 q^{-10} +3 q^{-12} -4 q^{-14} - q^{-16} +7 q^{-18} +3 q^{-20} -10 q^{-22} + q^{-24} +6 q^{-26} -2 q^{-28} -4 q^{-30} +3 q^{-34} - q^{-38} + q^{-40} - q^{-42} - q^{-44} + q^{-46} }[/math] |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | [math]\displaystyle{ q^{26}-2 q^{24}+2 q^{22}+q^{20}-6 q^{18}+6 q^{16}-2 q^{14}-6 q^{12}+9 q^{10}-2 q^8-5 q^6+10 q^4-q^2-6+8 q^{-2} + q^{-4} -4 q^{-6} + q^{-8} +3 q^{-10} +2 q^{-12} -8 q^{-14} +3 q^{-16} +6 q^{-18} -11 q^{-20} +3 q^{-22} +9 q^{-24} -9 q^{-26} +8 q^{-30} -5 q^{-32} -3 q^{-34} +5 q^{-36} - q^{-38} -2 q^{-40} + q^{-42} }[/math] |
| 1,0,0 | [math]\displaystyle{ -q^{15}+q^{13}-q^{11}+2 q^9-q^7+2 q^5-q^3+2 q+ q^{-3} - q^{-7} + q^{-9} -2 q^{-11} +2 q^{-13} - q^{-15} +2 q^{-17} - q^{-19} + q^{-21} - q^{-23} }[/math] |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^{32}-q^{30}+3 q^{26}-2 q^{24}-2 q^{22}+4 q^{20}-2 q^{18}-7 q^{16}+3 q^{14}+3 q^{12}-6 q^{10}-3 q^8+12 q^6+5 q^4-9 q^2+6+15 q^{-2} -7 q^{-4} -7 q^{-6} +11 q^{-8} - q^{-10} -9 q^{-12} +2 q^{-14} +5 q^{-16} -6 q^{-18} -5 q^{-20} +8 q^{-22} + q^{-24} -9 q^{-26} +3 q^{-28} +9 q^{-30} -5 q^{-32} -4 q^{-34} +5 q^{-36} +3 q^{-38} -3 q^{-40} -3 q^{-42} +2 q^{-44} +2 q^{-46} -2 q^{-48} - q^{-50} + q^{-52} }[/math] |
| 1,0,0,0 | [math]\displaystyle{ -q^{18}+q^{16}-q^{14}+q^{12}+q^{10}-q^8+2 q^6-q^4+2 q^2+ q^{-2} + q^{-4} - q^{-10} + q^{-12} -2 q^{-14} +2 q^{-16} - q^{-18} + q^{-20} + q^{-22} - q^{-24} + q^{-26} - q^{-28} }[/math] |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ -q^{26}+2 q^{24}-6 q^{22}+9 q^{20}-12 q^{18}+16 q^{16}-16 q^{14}+18 q^{12}-15 q^{10}+12 q^8-5 q^6-2 q^4+11 q^2-20+26 q^{-2} -31 q^{-4} +32 q^{-6} -31 q^{-8} +27 q^{-10} -20 q^{-12} +14 q^{-14} -5 q^{-16} -2 q^{-18} +9 q^{-20} -13 q^{-22} +15 q^{-24} -15 q^{-26} +14 q^{-28} -12 q^{-30} +9 q^{-32} -7 q^{-34} +5 q^{-36} -3 q^{-38} +2 q^{-40} - q^{-42} }[/math] |
| 1,0 | [math]\displaystyle{ q^{44}-2 q^{40}-2 q^{38}+4 q^{36}+5 q^{34}-5 q^{32}-9 q^{30}+q^{28}+13 q^{26}+4 q^{24}-14 q^{22}-11 q^{20}+11 q^{18}+17 q^{16}-2 q^{14}-18 q^{12}-5 q^{10}+15 q^8+13 q^6-8 q^4-14 q^2+2+12 q^{-2} +2 q^{-4} -10 q^{-6} -3 q^{-8} +10 q^{-10} +5 q^{-12} -10 q^{-14} -7 q^{-16} +10 q^{-18} +12 q^{-20} -6 q^{-22} -15 q^{-24} + q^{-26} +15 q^{-28} +4 q^{-30} -14 q^{-32} -10 q^{-34} +8 q^{-36} +14 q^{-38} - q^{-40} -12 q^{-42} -6 q^{-44} +7 q^{-46} +10 q^{-48} -7 q^{-52} -5 q^{-54} +2 q^{-56} +5 q^{-58} + q^{-60} -2 q^{-62} -2 q^{-64} + q^{-68} }[/math] |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^{34}-2 q^{32}+4 q^{30}-6 q^{28}+9 q^{26}-11 q^{24}+11 q^{22}-14 q^{20}+15 q^{18}-15 q^{16}+11 q^{14}-10 q^{12}+9 q^{10}-3 q^8-q^6+7 q^4-8 q^2+18-18 q^{-2} +23 q^{-4} -23 q^{-6} +26 q^{-8} -25 q^{-10} +23 q^{-12} -21 q^{-14} +16 q^{-16} -14 q^{-18} +8 q^{-20} -4 q^{-22} - q^{-24} +5 q^{-26} -8 q^{-28} +10 q^{-30} -10 q^{-32} +13 q^{-34} -12 q^{-36} +9 q^{-38} -9 q^{-40} +10 q^{-42} -7 q^{-44} +4 q^{-46} -5 q^{-48} +5 q^{-50} -2 q^{-52} + q^{-54} -2 q^{-56} + q^{-58} }[/math] |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{60}-2 q^{58}+6 q^{56}-11 q^{54}+13 q^{52}-12 q^{50}-2 q^{48}+25 q^{46}-46 q^{44}+58 q^{42}-47 q^{40}+11 q^{38}+36 q^{36}-81 q^{34}+98 q^{32}-77 q^{30}+24 q^{28}+37 q^{26}-81 q^{24}+89 q^{22}-54 q^{20}+3 q^{18}+48 q^{16}-68 q^{14}+53 q^{12}-11 q^{10}-39 q^8+74 q^6-77 q^4+59 q^2-12-42 q^{-2} +88 q^{-4} -108 q^{-6} +93 q^{-8} -49 q^{-10} -18 q^{-12} +74 q^{-14} -102 q^{-16} +93 q^{-18} -49 q^{-20} -10 q^{-22} +61 q^{-24} -75 q^{-26} +47 q^{-28} - q^{-30} -45 q^{-32} +67 q^{-34} -50 q^{-36} +12 q^{-38} +31 q^{-40} -57 q^{-42} +63 q^{-44} -46 q^{-46} +16 q^{-48} +13 q^{-50} -36 q^{-52} +41 q^{-54} -36 q^{-56} +27 q^{-58} -12 q^{-60} +2 q^{-62} +9 q^{-64} -21 q^{-66} +24 q^{-68} -23 q^{-70} +16 q^{-72} -7 q^{-74} +7 q^{-78} -12 q^{-80} +13 q^{-82} -10 q^{-84} +7 q^{-86} -2 q^{-88} - q^{-90} +2 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }[/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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 108"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ 2 t^3-8 t^2+14 t-15+14 t^{-1} -8 t^{-2} +2 t^{-3} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ 2 z^6+4 z^4+1 }[/math] |
In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 63, 2 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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[math]\displaystyle{ -q^6+3 q^5-5 q^4+8 q^3-10 q^2+10 q-9+8 q^{-1} -5 q^{-2} +3 q^{-3} - q^{-4} }[/math] |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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[math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +3 z^4-2 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2+1 }[/math] |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+6 z^8 a^{-2} +9 z^8+a^3 z^7-3 a z^7+4 z^7 a^{-1} +8 z^7 a^{-3} -13 a^2 z^6-13 z^6 a^{-2} +7 z^6 a^{-4} -33 z^6-4 a^3 z^5-11 a z^5-29 z^5 a^{-1} -17 z^5 a^{-3} +5 z^5 a^{-5} +17 a^2 z^4+4 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} +33 z^4+5 a^3 z^3+19 a z^3+28 z^3 a^{-1} +10 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+2 z^2 a^{-4} -z^2 a^{-6} -10 z^2-2 a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} +1 }[/math] |
