3 1

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The trefoil knot has only three crossings!

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+ t^{-1} -1}
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 3, -2 }
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^{-4} + q^{-3} + q^{-1} }
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^4+a^2 z^2+2 a^2}
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^5 z+a^4 z^2-a^4+a^3 z+a^2 z^2-2 a^2}
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^{14}-q^{12}+q^8+2 q^6+q^4+q^2}
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^{72}-q^{64}-q^{62}-q^{56}-2 q^{54}-q^{52}+q^{50}-q^{46}-2 q^{44}+2 q^{40}+q^{38}-q^{36}+2 q^{32}+2 q^{30}+q^{28}+2 q^{22}+2 q^{20}+q^{14}+q^{12}+q^{10}}
Further Quantum Invariants
Further quantum knot invariants for 3_1.

The braid index of 3_1 is only 2, so it's easy to calculate lots of quantum invariants. 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^9+q^5+q^3+q}
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^{24}-q^{20}-q^{18}-q^{16}+q^{10}+q^8+q^6+q^4+q^2}
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^{45}+q^{41}+q^{39}+q^{37}-q^{31}-q^{29}-q^{27}-q^{25}-q^{23}+q^{15}+q^{13}+q^{11}+q^9+q^7+q^5+q^3}
4
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^{105}+q^{101}+q^{99}+q^{97}-q^{91}-q^{89}-q^{87}-q^{85}-q^{83}+q^{75}+q^{73}+q^{71}+q^{69}+q^{67}+q^{65}+q^{63}-q^{53}-q^{51}-q^{49}-q^{47}-q^{45}-q^{43}-q^{41}-q^{39}-q^{37}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^7+q^5}
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^{144}-q^{140}-q^{138}-q^{136}+q^{130}+q^{128}+q^{126}+q^{124}+q^{122}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}-q^{104}-q^{102}+q^{92}+q^{90}+q^{88}+q^{86}+q^{84}+q^{82}+q^{80}+q^{78}+q^{76}-q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^8+q^6}
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^{240}-q^{236}-q^{234}-q^{232}+q^{226}+q^{224}+q^{222}+q^{220}+q^{218}-q^{210}-q^{208}-q^{206}-q^{204}-q^{202}-q^{200}-q^{198}+q^{188}+q^{186}+q^{184}+q^{182}+q^{180}+q^{178}+q^{176}+q^{174}+q^{172}-q^{160}-q^{158}-q^{156}-q^{154}-q^{152}-q^{150}-q^{148}-q^{146}-q^{144}-q^{142}-q^{140}+q^{126}+q^{124}+q^{122}+q^{120}+q^{118}+q^{116}+q^{114}+q^{112}+q^{110}+q^{108}+q^{106}+q^{104}+q^{102}-q^{86}-q^{84}-q^{82}-q^{80}-q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^8}

A2 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^{14}-q^{12}+q^8+2 q^6+q^4+q^2}
0,2 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^{34}+q^{32}+q^{30}-q^{28}-2 q^{26}-3 q^{24}-3 q^{22}-q^{20}+2 q^{16}+2 q^{14}+3 q^{12}+2 q^{10}+2 q^8+q^6+q^4}
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^{14}-q^{12}+q^8+2 q^6+q^4+q^2}
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^{36}-2 q^{24}-2 q^{22}-3 q^{20}-2 q^{18}+2 q^{14}+3 q^{12}+4 q^{10}+4 q^8+2 q^6+q^4}
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^{34}+q^{32}+q^{30}-q^{28}-2 q^{26}-3 q^{24}-3 q^{22}-q^{20}+2 q^{16}+2 q^{14}+3 q^{12}+2 q^{10}+2 q^8+q^6+q^4}
3,0

A3 Invariants.

Weight Invariant
0,0,1 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^{19}-q^{17}-q^{15}+q^{11}+2 q^9+2 q^7+q^5+q^3}
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^{30}-q^{24}-2 q^{22}-2 q^{20}-2 q^{18}+q^{14}+3 q^{12}+3 q^{10}+3 q^8+q^6+q^4}
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^{19}-q^{17}-q^{15}+q^{11}+2 q^9+2 q^7+q^5+q^3}
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^{48}+q^{38}+q^{36}+q^{34}-q^{32}-3 q^{30}-5 q^{28}-6 q^{26}-6 q^{24}-3 q^{22}+q^{20}+4 q^{18}+7 q^{16}+8 q^{14}+7 q^{12}+5 q^{10}+2 q^8+q^6}

A4 Invariants.

Weight Invariant
0,0,0,1 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^{24}-q^{22}-q^{20}-q^{18}+q^{14}+2 q^{12}+2 q^{10}+2 q^8+q^6+q^4}
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^{40}+q^{38}+q^{36}-q^{32}-3 q^{30}-4 q^{28}-4 q^{26}-3 q^{24}-q^{22}+q^{20}+4 q^{18}+4 q^{16}+5 q^{14}+4 q^{12}+3 q^{10}+q^8+q^6}
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^{24}-q^{22}-q^{20}-q^{18}+q^{14}+2 q^{12}+2 q^{10}+2 q^8+q^6+q^4}

A5 Invariants.

Weight Invariant
0,0,0,0,1 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^{29}-q^{27}-q^{25}-q^{23}-q^{21}+q^{17}+2 q^{15}+2 q^{13}+2 q^{11}+2 q^9+q^7+q^5}
1,0,0,0,0

A6 Invariants.

Weight Invariant
0,0,0,0,0,1 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^{34}-q^{32}-q^{30}-q^{28}-q^{26}-q^{24}+q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+2 q^{10}+q^8+q^6}
1,0,0,0,0,0 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^{34}-q^{32}-q^{30}-q^{28}-q^{26}-q^{24}+q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+2 q^{10}+q^8+q^6}

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^{30}-q^{24}+q^{14}+q^{12}+q^{10}+q^8+q^6+q^4}
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^{48}-q^{38}-q^{36}-q^{34}-q^{32}-q^{30}-q^{28}+q^{22}+q^{20}+2 q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^6}

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^{72}-q^{58}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}+q^{34}+2 q^{30}+q^{28}+2 q^{26}+q^{24}+2 q^{22}+q^{20}+2 q^{18}+q^{14}+q^{10}}

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^{96}-q^{78}-q^{74}-q^{70}-q^{68}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{54}+q^{46}+2 q^{42}+2 q^{38}+q^{36}+2 q^{34}+q^{32}+2 q^{30}+q^{28}+2 q^{26}+2 q^{22}+q^{18}+q^{14}}

B5 Invariants.

Weight Invariant
1,0,0,0,0 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^{120}-q^{98}-q^{94}-q^{90}-q^{86}-q^{84}-q^{82}-q^{80}-q^{78}-q^{76}-q^{74}-q^{70}-q^{66}+q^{58}+2 q^{54}+2 q^{50}+2 q^{46}+q^{44}+2 q^{42}+q^{40}+2 q^{38}+q^{36}+2 q^{34}+2 q^{30}+2 q^{26}+q^{22}+q^{18}}

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^{42}-q^{34}-q^{32}-q^{24}+q^{20}+2 q^{18}+q^{16}+q^{14}+q^{12}+2 q^{10}+q^8+q^6}

C4 Invariants.

Weight Invariant
1,0,0,0

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^{72}-q^{64}-q^{62}+2 q^{56}+4 q^{54}+5 q^{52}+4 q^{50}+3 q^{48}-q^{46}-5 q^{44}-9 q^{42}-13 q^{40}-14 q^{38}-13 q^{36}-9 q^{34}-4 q^{32}+2 q^{30}+7 q^{28}+12 q^{26}+12 q^{24}+14 q^{22}+11 q^{20}+9 q^{18}+6 q^{16}+4 q^{14}+q^{12}+q^{10}}
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^{42}-q^{34}-q^{32}-2 q^{30}-2 q^{28}-2 q^{26}-q^{24}+q^{20}+2 q^{18}+3 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^8+q^6}

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^{144}-q^{126}+q^{122}-q^{116}+2 q^{112}+q^{110}-q^{108}+2 q^{104}+q^{102}-q^{98}-q^{96}+q^{94}-2 q^{90}-2 q^{88}-q^{86}-q^{84}-2 q^{82}-3 q^{80}-2 q^{78}-2 q^{76}-2 q^{74}-2 q^{72}-2 q^{70}-q^{68}-q^{64}-q^{62}+q^{60}+q^{58}+q^{56}+2 q^{54}+q^{52}+2 q^{50}+3 q^{48}+2 q^{46}+2 q^{44}+3 q^{42}+2 q^{40}+2 q^{38}+3 q^{36}+2 q^{34}+q^{32}+2 q^{30}+q^{28}+q^{26}+q^{24}+q^{18}}
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^{72}-q^{64}-q^{62}-q^{56}-2 q^{54}-q^{52}+q^{50}-q^{46}-2 q^{44}+2 q^{40}+q^{38}-q^{36}+2 q^{32}+2 q^{30}+q^{28}+2 q^{22}+2 q^{20}+q^{14}+q^{12}+q^{10}}

.

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["3 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+ t^{-1} -1}
In[5]:= Conway[K][z]
Out[5]=
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]=
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 3, -2 }
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^{-4} + q^{-3} + q^{-1} }
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^4+a^2 z^2+2 a^2}
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^5 z+a^4 z^2-a^4+a^3 z+a^2 z^2-2 a^2}
Retrieved from "https://katlas.org/index.php?title=3_1&oldid=955"