5 1

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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 t^2+ t^{-2} -t- t^{-1} +1}
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 z^4+3 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 { 5, -4 }
Jones 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 - q^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2} }
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^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 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 a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 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^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+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^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30}}
Further Quantum Invariants
Further quantum knot invariants for 5_1.

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^{15}+q^7+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^{40}-q^{32}-q^{30}-q^{28}+q^{14}+q^{12}+q^{10}+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^{75}+q^{67}+q^{65}+q^{63}-q^{49}-q^{47}-q^{45}-q^{43}-q^{41}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9}
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 q^{120}-q^{112}-q^{110}-q^{108}+q^{94}+q^{92}+q^{90}+q^{88}+q^{86}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}}
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 -q^{175}+q^{167}+q^{165}+q^{163}-q^{149}-q^{147}-q^{145}-q^{143}-q^{141}+q^{121}+q^{119}+q^{117}+q^{115}+q^{113}+q^{111}+q^{109}-q^{83}-q^{81}-q^{79}-q^{77}-q^{75}-q^{73}-q^{71}-q^{69}-q^{67}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}}
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 q^{240}-q^{232}-q^{230}-q^{228}+q^{214}+q^{212}+q^{210}+q^{208}+q^{206}-q^{186}-q^{184}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}+q^{148}+q^{146}+q^{144}+q^{142}+q^{140}+q^{138}+q^{136}+q^{134}+q^{132}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}}
8 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^{400}-q^{392}-q^{390}-q^{388}+q^{374}+q^{372}+q^{370}+q^{368}+q^{366}-q^{346}-q^{344}-q^{342}-q^{340}-q^{338}-q^{336}-q^{334}+q^{308}+q^{306}+q^{304}+q^{302}+q^{300}+q^{298}+q^{296}+q^{294}+q^{292}-q^{260}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}-q^{244}-q^{242}-q^{240}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}+q^{186}+q^{184}+q^{182}+q^{180}+q^{178}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}}

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^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6}
1,1 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^{60}-2 q^{36}-2 q^{34}-4 q^{32}-4 q^{30}-3 q^{28}+2 q^{24}+4 q^{22}+5 q^{20}+4 q^{18}+4 q^{16}+2 q^{14}+q^{12}}
2,0 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^{54}+q^{52}+2 q^{50}+q^{48}-2 q^{44}-3 q^{42}-3 q^{40}-3 q^{38}-2 q^{36}-q^{34}+q^{28}+q^{26}+2 q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12}}
3,0 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^{96}-q^{94}-2 q^{92}-2 q^{90}-q^{88}+q^{86}+3 q^{84}+4 q^{82}+5 q^{80}+4 q^{78}+4 q^{76}+2 q^{74}+q^{72}-q^{70}-2 q^{68}-3 q^{66}-4 q^{64}-5 q^{62}-5 q^{60}-5 q^{58}-4 q^{56}-3 q^{54}-2 q^{52}-q^{50}+q^{42}+q^{40}+2 q^{38}+2 q^{36}+3 q^{34}+3 q^{32}+4 q^{30}+3 q^{28}+3 q^{26}+2 q^{24}+2 q^{22}+q^{20}+q^{18}}

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^{50}-q^{36}-2 q^{34}-3 q^{32}-3 q^{30}-2 q^{28}-q^{26}+2 q^{24}+3 q^{22}+4 q^{20}+3 q^{18}+3 q^{16}+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^{29}-q^{27}-2 q^{25}-q^{23}+q^{19}+2 q^{17}+2 q^{15}+2 q^{13}+q^{11}+q^9}
1,0,1 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}+q^{58}+q^{56}+3 q^{54}+3 q^{52}+2 q^{50}-q^{48}-5 q^{46}-9 q^{44}-12 q^{42}-12 q^{40}-9 q^{38}-3 q^{36}+2 q^{34}+7 q^{32}+10 q^{30}+11 q^{28}+10 q^{26}+7 q^{24}+5 q^{22}+2 q^{20}+q^{18}}

A4 Invariants.

Weight Invariant
0,1,0,0 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^{64}+q^{62}+q^{60}+q^{58}+q^{56}-q^{50}-2 q^{48}-4 q^{46}-5 q^{44}-6 q^{42}-6 q^{40}-4 q^{38}-q^{36}+2 q^{34}+4 q^{32}+7 q^{30}+6 q^{28}+6 q^{26}+4 q^{24}+3 q^{22}+q^{20}+q^{18}}
1,0,0,0 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^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12}}

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^{50}-q^{36}-q^{32}-q^{30}+q^{26}+q^{22}+2 q^{20}+q^{18}+q^{16}+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^{80}-q^{58}-q^{56}-q^{54}-q^{52}-2 q^{50}-q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2 q^{34}+q^{32}+2 q^{30}+q^{28}+2 q^{26}+q^{24}+q^{22}+q^{18}}

B3 Invariants.

Weight Invariant
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^{120}-q^{86}-q^{82}-q^{80}-2 q^{78}-q^{76}-2 q^{74}-q^{72}-2 q^{70}-q^{68}-q^{66}-q^{64}+q^{58}+q^{56}+2 q^{54}+q^{52}+3 q^{50}+q^{48}+3 q^{46}+q^{44}+2 q^{42}+q^{40}+2 q^{38}+q^{34}+q^{30}}

B4 Invariants.

Weight Invariant
1,0,0,0 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^{160}-q^{114}-q^{110}-2 q^{106}-q^{104}-2 q^{102}-q^{100}-2 q^{98}-q^{96}-2 q^{94}-q^{92}-2 q^{90}-q^{88}-q^{86}+q^{78}+2 q^{74}+q^{72}+3 q^{70}+q^{68}+3 q^{66}+q^{64}+3 q^{62}+q^{60}+3 q^{58}+q^{56}+2 q^{54}+2 q^{50}+q^{46}+q^{42}}

C3 Invariants.

Weight Invariant
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^{70}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+2 q^{30}+2 q^{28}+2 q^{26}+q^{24}+2 q^{22}+q^{20}+q^{18}}

C4 Invariants.

Weight Invariant
1,0,0,0 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^{90}-q^{64}-q^{62}-2 q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}+q^{44}+2 q^{42}+2 q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+2 q^{28}+q^{26}+q^{24}}

D4 Invariants.

Weight Invariant
0,1,0,0 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^{120}-q^{100}-q^{98}-3 q^{96}-3 q^{94}-q^{92}-q^{90}+2 q^{88}+6 q^{86}+9 q^{84}+11 q^{82}+14 q^{80}+11 q^{78}+9 q^{76}+3 q^{74}-4 q^{72}-12 q^{70}-18 q^{68}-24 q^{66}-27 q^{64}-27 q^{62}-24 q^{60}-17 q^{58}-11 q^{56}+7 q^{52}+14 q^{50}+19 q^{48}+22 q^{46}+19 q^{44}+19 q^{42}+14 q^{40}+10 q^{38}+6 q^{36}+4 q^{34}+q^{32}+q^{30}}
1,0,0,0 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^{70}-q^{50}-q^{48}-3 q^{46}-3 q^{44}-3 q^{42}-3 q^{40}-2 q^{38}+q^{34}+3 q^{32}+4 q^{30}+4 q^{28}+4 q^{26}+3 q^{24}+2 q^{22}+q^{20}+q^{18}}

G2 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^{240}-q^{198}-q^{192}+q^{178}+q^{176}+q^{172}+2 q^{170}+q^{168}+q^{166}+q^{164}+q^{162}+2 q^{160}+q^{158}+q^{154}+q^{152}-q^{148}-q^{146}-q^{144}-q^{142}-2 q^{140}-3 q^{138}-2 q^{136}-2 q^{134}-3 q^{132}-4 q^{130}-4 q^{128}-3 q^{126}-3 q^{124}-4 q^{122}-4 q^{120}-3 q^{118}-2 q^{116}-2 q^{114}-3 q^{112}-2 q^{110}+q^{102}+q^{100}+2 q^{98}+3 q^{96}+2 q^{94}+2 q^{92}+4 q^{90}+3 q^{88}+3 q^{86}+4 q^{84}+3 q^{82}+3 q^{80}+4 q^{78}+2 q^{76}+2 q^{74}+3 q^{72}+2 q^{70}+q^{68}+2 q^{66}+q^{64}+q^{62}+q^{60}+q^{54}}
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^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30}}

.

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["5 1"];
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 t^2+ t^{-2} -t- t^{-1} +1}
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 z^4+3 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]= { 5, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= 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^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2} }
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^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 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 a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 a^4}
Retrieved from "https://katlas.org/index.php?title=5_1&oldid=1229"