10 133
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ -t^2+5 t-7+5 t^{-1} - t^{-2} }[/math] |
| Conway polynomial | [math]\displaystyle{ -z^4+z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 19, -2 } |
| Jones polynomial | [math]\displaystyle{ q^{-1} - q^{-2} +3 q^{-3} -3 q^{-4} +3 q^{-5} -3 q^{-6} +2 q^{-7} -2 q^{-8} + q^{-9} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-3 z^2 a^6-3 a^6+2 z^2 a^4+2 a^4+z^2 a^2+a^2 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-9 z^5 a^9+10 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+z^2 a^8+a^8+3 z^7 a^7-13 z^5 a^7+16 z^3 a^7-7 z a^7+z^8 a^6-4 z^6 a^6+6 z^4 a^6-6 z^2 a^6+3 a^6+z^7 a^5-4 z^5 a^5+7 z^3 a^5-4 z a^5+2 z^4 a^4-3 z^2 a^4+2 a^4+z^3 a^3+z^2 a^2-a^2 }[/math] |
| The A2 invariant | [math]\displaystyle{ q^{28}-2 q^{20}-q^{18}-q^{16}+q^{12}+q^{10}+2 q^8+q^6+q^2 }[/math] |
| The G2 invariant | [math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+q^{134}-q^{132}-3 q^{130}+6 q^{128}-6 q^{126}+4 q^{124}-q^{122}-2 q^{120}+5 q^{118}-4 q^{116}+2 q^{114}+3 q^{112}-3 q^{110}+4 q^{108}+q^{106}-3 q^{104}+8 q^{102}-6 q^{100}+3 q^{98}+q^{96}-4 q^{94}+5 q^{92}-6 q^{90}+3 q^{88}-4 q^{86}-q^{82}-5 q^{80}+q^{78}-4 q^{76}-3 q^{70}+q^{66}-4 q^{64}+6 q^{62}-5 q^{60}+2 q^{58}+4 q^{56}-5 q^{54}+7 q^{52}-2 q^{50}+q^{48}+3 q^{46}-2 q^{44}+q^{42}+2 q^{40}+2 q^{36}+q^{34}-q^{32}+2 q^{30}-q^{28}+q^{26}+q^{24}-q^{22}+2 q^{20}+q^{14}+q^{10} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 10_133.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | [math]\displaystyle{ q^{19}-q^{17}-q^{13}+2 q^5+q }[/math] |
| 2 | [math]\displaystyle{ q^{54}-q^{52}-2 q^{50}+2 q^{48}+q^{46}-2 q^{44}+2 q^{40}-q^{36}+q^{34}+q^{32}-2 q^{30}+q^{26}-2 q^{24}-q^{22}+q^{20}-2 q^{16}+q^{14}+2 q^{12}+q^6+q^4+q^2 }[/math] |
| 3 | [math]\displaystyle{ q^{105}-q^{103}-2 q^{101}+3 q^{97}+3 q^{95}-3 q^{93}-4 q^{91}+4 q^{87}+3 q^{85}-2 q^{83}-4 q^{81}-q^{79}+3 q^{77}+4 q^{75}-2 q^{73}-6 q^{71}-2 q^{69}+6 q^{67}+4 q^{65}-5 q^{63}-4 q^{61}+5 q^{59}+5 q^{57}-3 q^{55}-3 q^{53}+3 q^{51}+3 q^{49}-4 q^{47}-3 q^{45}+q^{43}+3 q^{41}-2 q^{37}-5 q^{35}+2 q^{33}+6 q^{31}+q^{29}-8 q^{27}-6 q^{25}+7 q^{23}+7 q^{21}-3 q^{19}-6 q^{17}+q^{15}+5 q^{13}+2 q^{11}-2 q^9+q^5+2 q^3 }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{28}-2 q^{20}-q^{18}-q^{16}+q^{12}+q^{10}+2 q^8+q^6+q^2 }[/math] |
| 1,1 | [math]\displaystyle{ q^{76}-2 q^{74}+4 q^{72}-8 q^{70}+11 q^{68}-14 q^{66}+14 q^{64}-12 q^{62}+8 q^{60}-8 q^{56}+14 q^{54}-19 q^{52}+22 q^{50}-20 q^{48}+20 q^{46}-16 q^{44}+12 q^{42}-6 q^{40}+6 q^{36}-10 q^{34}+12 q^{32}-14 q^{30}+6 q^{28}-8 q^{26}-2 q^{24}-2 q^{20}+2 q^{18}+4 q^{16}+2 q^{14}+6 q^{12}+4 q^8+q^4 }[/math] |
| 2,0 | [math]\displaystyle{ q^{72}-q^{68}-q^{66}+q^{62}-q^{60}-q^{58}+q^{52}+q^{50}+2 q^{48}+3 q^{46}+q^{44}-2 q^{40}-q^{38}-2 q^{36}-2 q^{34}-2 q^{32}-q^{30}-q^{28}-q^{26}-2 q^{22}+2 q^{20}+2 q^{18}+2 q^{16}+q^{14}+2 q^{12}+3 q^{10}+q^8+q^4 }[/math] |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | [math]\displaystyle{ q^{60}-q^{58}-2 q^{52}-q^{48}+q^{46}+2 q^{44}+3 q^{42}+3 q^{40}+3 q^{38}-3 q^{34}-4 q^{32}-6 q^{30}-4 q^{28}-3 q^{26}+2 q^{22}+2 q^{20}+3 q^{18}+3 q^{16}+q^{14}+2 q^{12}+2 q^{10}+q^8+q^4 }[/math] |
| 1,0,0 | [math]\displaystyle{ q^{37}+q^{33}-2 q^{27}-2 q^{25}-2 q^{23}-q^{21}+q^{17}+2 q^{15}+q^{13}+2 q^{11}+q^9+q^7+q^3 }[/math] |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^{78}+q^{72}-q^{70}-3 q^{68}-2 q^{66}-2 q^{64}-4 q^{62}-q^{60}+4 q^{58}+7 q^{56}+6 q^{54}+9 q^{52}+9 q^{50}+2 q^{48}-3 q^{46}-5 q^{44}-11 q^{42}-13 q^{40}-9 q^{38}-7 q^{36}-4 q^{34}+5 q^{30}+5 q^{28}+4 q^{26}+5 q^{24}+5 q^{22}+2 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}+q^6 }[/math] |
| 1,0,0,0 | [math]\displaystyle{ q^{46}+q^{42}+q^{40}-2 q^{34}-2 q^{32}-3 q^{30}-2 q^{28}-q^{26}+q^{22}+2 q^{20}+2 q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^8+q^4 }[/math] |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ q^{60}-q^{58}+2 q^{56}-2 q^{54}+2 q^{52}-2 q^{50}+q^{48}-q^{46}+q^{42}-3 q^{40}+3 q^{38}-4 q^{36}+3 q^{34}-4 q^{32}+2 q^{30}-2 q^{28}+q^{26}+2 q^{20}-q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^8+q^4 }[/math] |
| 1,0 | [math]\displaystyle{ q^{98}-q^{94}-q^{92}+q^{90}+q^{88}-2 q^{86}-2 q^{84}+q^{82}+2 q^{80}-q^{78}-2 q^{76}+3 q^{72}+2 q^{70}+2 q^{64}+2 q^{62}+q^{60}-q^{58}-q^{56}-q^{52}-4 q^{50}-3 q^{48}-q^{46}-2 q^{42}-3 q^{40}+2 q^{36}+q^{34}-q^{32}+q^{30}+2 q^{28}+3 q^{26}+q^{20}+2 q^{18}+q^{16}+q^{14}+q^6 }[/math] |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^{82}-q^{80}+q^{78}-2 q^{76}+2 q^{74}-3 q^{72}-2 q^{68}+q^{66}+q^{62}+3 q^{60}+2 q^{58}+6 q^{56}+q^{54}+4 q^{52}-3 q^{50}+q^{48}-7 q^{46}-3 q^{44}-8 q^{42}-4 q^{40}-5 q^{38}-q^{36}+q^{32}+4 q^{30}+2 q^{28}+4 q^{26}+q^{24}+4 q^{22}+2 q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^6 }[/math] |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+q^{134}-q^{132}-3 q^{130}+6 q^{128}-6 q^{126}+4 q^{124}-q^{122}-2 q^{120}+5 q^{118}-4 q^{116}+2 q^{114}+3 q^{112}-3 q^{110}+4 q^{108}+q^{106}-3 q^{104}+8 q^{102}-6 q^{100}+3 q^{98}+q^{96}-4 q^{94}+5 q^{92}-6 q^{90}+3 q^{88}-4 q^{86}-q^{82}-5 q^{80}+q^{78}-4 q^{76}-3 q^{70}+q^{66}-4 q^{64}+6 q^{62}-5 q^{60}+2 q^{58}+4 q^{56}-5 q^{54}+7 q^{52}-2 q^{50}+q^{48}+3 q^{46}-2 q^{44}+q^{42}+2 q^{40}+2 q^{36}+q^{34}-q^{32}+2 q^{30}-q^{28}+q^{26}+q^{24}-q^{22}+2 q^{20}+q^{14}+q^{10} }[/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["10 133"];
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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{ -t^2+5 t-7+5 t^{-1} - t^{-2} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ -z^4+z^2+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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{ 19, -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^{-1} - q^{-2} +3 q^{-3} -3 q^{-4} +3 q^{-5} -3 q^{-6} +2 q^{-7} -2 q^{-8} + q^{-9} }[/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^2 a^8+a^8-z^4 a^6-3 z^2 a^6-3 a^6+2 z^2 a^4+2 a^4+z^2 a^2+a^2 }[/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{ z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-9 z^5 a^9+10 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+z^2 a^8+a^8+3 z^7 a^7-13 z^5 a^7+16 z^3 a^7-7 z a^7+z^8 a^6-4 z^6 a^6+6 z^4 a^6-6 z^2 a^6+3 a^6+z^7 a^5-4 z^5 a^5+7 z^3 a^5-4 z a^5+2 z^4 a^4-3 z^2 a^4+2 a^4+z^3 a^3+z^2 a^2-a^2 }[/math] |
