10 120

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Polynomial invariants

Alexander polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8 t^2-26 t+37-26 t^{-1} +8 t^{-2} }
Conway polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8 z^4+6 z^2+1}
2nd Alexander ideal (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature { 105, -4 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{12}-4 z^2 a^{10}-3 a^{10}+3 z^4 a^8+3 z^2 a^8+4 z^4 a^6+7 z^2 a^6+3 a^6+z^4 a^4}
Kauffman polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{14}-2 z^4 a^{14}+z^2 a^{14}+4 z^7 a^{13}-10 z^5 a^{13}+8 z^3 a^{13}-2 z a^{13}+6 z^8 a^{12}-13 z^6 a^{12}+6 z^4 a^{12}+z^2 a^{12}+a^{12}+3 z^9 a^{11}+7 z^7 a^{11}-33 z^5 a^{11}+29 z^3 a^{11}-8 z a^{11}+16 z^8 a^{10}-33 z^6 a^{10}+17 z^4 a^{10}-7 z^2 a^{10}+3 a^{10}+3 z^9 a^9+16 z^7 a^9-44 z^5 a^9+26 z^3 a^9-4 z a^9+10 z^8 a^8-9 z^6 a^8-3 z^4 a^8+13 z^7 a^7-17 z^5 a^7+5 z^3 a^7+2 z a^7+10 z^6 a^6-11 z^4 a^6+7 z^2 a^6-3 a^6+4 z^5 a^5+z^4 a^4}
The A2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{38}+q^{36}-3 q^{34}+q^{32}-5 q^{28}+2 q^{26}-2 q^{24}+q^{22}+2 q^{20}-q^{18}+5 q^{16}-2 q^{14}+2 q^{12}+3 q^{10}-3 q^8+q^6}
The G2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{190}-3 q^{188}+8 q^{186}-16 q^{184}+21 q^{182}-23 q^{180}+9 q^{178}+25 q^{176}-74 q^{174}+131 q^{172}-159 q^{170}+124 q^{168}-16 q^{166}-151 q^{164}+326 q^{162}-410 q^{160}+358 q^{158}-152 q^{156}-149 q^{154}+426 q^{152}-569 q^{150}+496 q^{148}-227 q^{146}-123 q^{144}+405 q^{142}-489 q^{140}+345 q^{138}-47 q^{136}-260 q^{134}+434 q^{132}-410 q^{130}+162 q^{128}+180 q^{126}-491 q^{124}+639 q^{122}-549 q^{120}+251 q^{118}+157 q^{116}-531 q^{114}+731 q^{112}-704 q^{110}+430 q^{108}-19 q^{106}-371 q^{104}+597 q^{102}-569 q^{100}+321 q^{98}+41 q^{96}-335 q^{94}+427 q^{92}-308 q^{90}+21 q^{88}+288 q^{86}-464 q^{84}+445 q^{82}-224 q^{80}-79 q^{78}+349 q^{76}-482 q^{74}+440 q^{72}-267 q^{70}+46 q^{68}+153 q^{66}-266 q^{64}+286 q^{62}-215 q^{60}+119 q^{58}-14 q^{56}-56 q^{54}+84 q^{52}-91 q^{50}+68 q^{48}-38 q^{46}+13 q^{44}+7 q^{42}-12 q^{40}+12 q^{38}-10 q^{36}+6 q^{34}-3 q^{32}+q^{30}}
Further Quantum Invariants
Further quantum knot invariants for 10_120.

A1 Invariants.

Weight Invariant
1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{25}-3 q^{23}+4 q^{21}-5 q^{19}+3 q^{17}-2 q^{15}-q^{13}+4 q^{11}-3 q^9+6 q^7-3 q^5+q^3}
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{70}-3 q^{68}-q^{66}+13 q^{64}-10 q^{62}-20 q^{60}+35 q^{58}+3 q^{56}-52 q^{54}+36 q^{52}+33 q^{50}-58 q^{48}+10 q^{46}+45 q^{44}-33 q^{42}-20 q^{40}+31 q^{38}+8 q^{36}-39 q^{34}-q^{32}+49 q^{30}-34 q^{28}-33 q^{26}+62 q^{24}-9 q^{22}-41 q^{20}+36 q^{18}+5 q^{16}-20 q^{14}+9 q^{12}+3 q^{10}-3 q^8+q^6}
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{135}-3 q^{133}-q^{131}+8 q^{129}+8 q^{127}-18 q^{125}-34 q^{123}+28 q^{121}+81 q^{119}-5 q^{117}-149 q^{115}-69 q^{113}+203 q^{111}+194 q^{109}-198 q^{107}-344 q^{105}+112 q^{103}+474 q^{101}+35 q^{99}-525 q^{97}-217 q^{95}+496 q^{93}+374 q^{91}-400 q^{89}-477 q^{87}+262 q^{85}+518 q^{83}-120 q^{81}-507 q^{79}-9 q^{77}+462 q^{75}+135 q^{73}-379 q^{71}-260 q^{69}+287 q^{67}+369 q^{65}-143 q^{63}-477 q^{61}-25 q^{59}+518 q^{57}+214 q^{55}-507 q^{53}-381 q^{51}+404 q^{49}+479 q^{47}-253 q^{45}-490 q^{43}+100 q^{41}+415 q^{39}+18 q^{37}-294 q^{35}-57 q^{33}+165 q^{31}+68 q^{29}-87 q^{27}-39 q^{25}+40 q^{23}+18 q^{21}-13 q^{19}-8 q^{17}+6 q^{15}+3 q^{13}-3 q^{11}+q^9}

A2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{38}+q^{36}-3 q^{34}+q^{32}-5 q^{28}+2 q^{26}-2 q^{24}+q^{22}+2 q^{20}-q^{18}+5 q^{16}-2 q^{14}+2 q^{12}+3 q^{10}-3 q^8+q^6}
2,0

A3 Invariants.

Weight Invariant
0,1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{80}-3 q^{78}+2 q^{76}+5 q^{74}-15 q^{72}+11 q^{70}+11 q^{68}-32 q^{66}+26 q^{64}+17 q^{62}-40 q^{60}+29 q^{58}+15 q^{56}-39 q^{54}+9 q^{52}+14 q^{50}-18 q^{48}-9 q^{46}+3 q^{44}+14 q^{42}-21 q^{40}-14 q^{38}+43 q^{36}-22 q^{34}-17 q^{32}+49 q^{30}-15 q^{28}-18 q^{26}+31 q^{24}-8 q^{22}-12 q^{20}+11 q^{18}-3 q^{14}+q^{12}}
1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{51}+q^{49}+q^{47}-3 q^{45}+q^{43}-3 q^{41}-5 q^{37}+2 q^{35}-3 q^{33}+q^{31}+q^{29}+2 q^{27}+2 q^{25}+5 q^{21}-2 q^{19}+3 q^{17}-q^{15}+3 q^{13}-3 q^{11}+q^9}

B2 Invariants.

Weight Invariant
0,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{80}-3 q^{78}+8 q^{76}-15 q^{74}+25 q^{72}-37 q^{70}+45 q^{68}-52 q^{66}+54 q^{64}-47 q^{62}+34 q^{60}-15 q^{58}-11 q^{56}+37 q^{54}-65 q^{52}+84 q^{50}-100 q^{48}+103 q^{46}-97 q^{44}+80 q^{42}-57 q^{40}+32 q^{38}-3 q^{36}-18 q^{34}+39 q^{32}-49 q^{30}+55 q^{28}-50 q^{26}+43 q^{24}-32 q^{22}+24 q^{20}-13 q^{18}+6 q^{16}-3 q^{14}+q^{12}}
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{130}-3 q^{126}-3 q^{124}+5 q^{122}+10 q^{120}-3 q^{118}-20 q^{116}-9 q^{114}+27 q^{112}+27 q^{110}-20 q^{108}-45 q^{106}-q^{104}+55 q^{102}+29 q^{100}-43 q^{98}-47 q^{96}+21 q^{94}+56 q^{92}+3 q^{90}-49 q^{88}-21 q^{86}+34 q^{84}+25 q^{82}-25 q^{80}-31 q^{78}+16 q^{76}+32 q^{74}-12 q^{72}-41 q^{70}+q^{68}+41 q^{66}+8 q^{64}-44 q^{62}-26 q^{60}+40 q^{58}+42 q^{56}-23 q^{54}-53 q^{52}+5 q^{50}+56 q^{48}+26 q^{46}-35 q^{44}-37 q^{42}+13 q^{40}+37 q^{38}+9 q^{36}-20 q^{34}-18 q^{32}+5 q^{30}+12 q^{28}+3 q^{26}-3 q^{24}-3 q^{22}+q^{18}}

G2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{190}-3 q^{188}+8 q^{186}-16 q^{184}+21 q^{182}-23 q^{180}+9 q^{178}+25 q^{176}-74 q^{174}+131 q^{172}-159 q^{170}+124 q^{168}-16 q^{166}-151 q^{164}+326 q^{162}-410 q^{160}+358 q^{158}-152 q^{156}-149 q^{154}+426 q^{152}-569 q^{150}+496 q^{148}-227 q^{146}-123 q^{144}+405 q^{142}-489 q^{140}+345 q^{138}-47 q^{136}-260 q^{134}+434 q^{132}-410 q^{130}+162 q^{128}+180 q^{126}-491 q^{124}+639 q^{122}-549 q^{120}+251 q^{118}+157 q^{116}-531 q^{114}+731 q^{112}-704 q^{110}+430 q^{108}-19 q^{106}-371 q^{104}+597 q^{102}-569 q^{100}+321 q^{98}+41 q^{96}-335 q^{94}+427 q^{92}-308 q^{90}+21 q^{88}+288 q^{86}-464 q^{84}+445 q^{82}-224 q^{80}-79 q^{78}+349 q^{76}-482 q^{74}+440 q^{72}-267 q^{70}+46 q^{68}+153 q^{66}-266 q^{64}+286 q^{62}-215 q^{60}+119 q^{58}-14 q^{56}-56 q^{54}+84 q^{52}-91 q^{50}+68 q^{48}-38 q^{46}+13 q^{44}+7 q^{42}-12 q^{40}+12 q^{38}-10 q^{36}+6 q^{34}-3 q^{32}+q^{30}}

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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["10 120"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8 t^2-26 t+37-26 t^{-1} +8 t^{-2} }
In[5]:= Conway[K][z]
Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8 z^4+6 z^2+1}
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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 105, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]=
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{12}-4 z^2 a^{10}-3 a^{10}+3 z^4 a^8+3 z^2 a^8+4 z^4 a^6+7 z^2 a^6+3 a^6+z^4 a^4}
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{14}-2 z^4 a^{14}+z^2 a^{14}+4 z^7 a^{13}-10 z^5 a^{13}+8 z^3 a^{13}-2 z a^{13}+6 z^8 a^{12}-13 z^6 a^{12}+6 z^4 a^{12}+z^2 a^{12}+a^{12}+3 z^9 a^{11}+7 z^7 a^{11}-33 z^5 a^{11}+29 z^3 a^{11}-8 z a^{11}+16 z^8 a^{10}-33 z^6 a^{10}+17 z^4 a^{10}-7 z^2 a^{10}+3 a^{10}+3 z^9 a^9+16 z^7 a^9-44 z^5 a^9+26 z^3 a^9-4 z a^9+10 z^8 a^8-9 z^6 a^8-3 z^4 a^8+13 z^7 a^7-17 z^5 a^7+5 z^3 a^7+2 z a^7+10 z^6 a^6-11 z^4 a^6+7 z^2 a^6-3 a^6+4 z^5 a^5+z^4 a^4}
Retrieved from "https://katlas.org/index.php?title=10_120&oldid=4411"