10 131

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 1-2 z^4 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 31, -2 }
Jones polynomial	[math]\displaystyle{ 2 q^{-1} -3 q^{-2} +5 q^{-3} -5 q^{-4} +5 q^{-5} -5 q^{-6} +3 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^6-2 z^2 a^6-2 a^6-z^4 a^4-z^2 a^4+2 z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+9 z^3 a^9-3 z a^9+z^8 a^8-z^6 a^8-4 z^4 a^8+2 z^2 a^8+a^8+4 z^7 a^7-12 z^5 a^7+10 z^3 a^7-5 z a^7+z^8 a^6-2 z^4 a^6-3 z^2 a^6+2 a^6+2 z^7 a^5-3 z^5 a^5+2 z^3 a^5-z a^5+2 z^6 a^4-2 z^4 a^4+2 z^2 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^2-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{28}+q^{22}-2 q^{20}-q^{18}-q^{16}-q^{14}+q^{12}+2 q^8+q^6+2 q^2 }[/math]
The G2 invariant	[math]\displaystyle{ q^{142}-q^{140}+3 q^{138}-5 q^{136}+3 q^{134}-2 q^{132}-4 q^{130}+10 q^{128}-14 q^{126}+14 q^{124}-7 q^{122}-4 q^{120}+14 q^{118}-19 q^{116}+18 q^{114}-8 q^{112}-2 q^{110}+13 q^{108}-15 q^{106}+12 q^{104}+q^{102}-8 q^{100}+14 q^{98}-11 q^{96}+2 q^{94}+7 q^{92}-15 q^{90}+19 q^{88}-18 q^{86}+9 q^{84}+2 q^{82}-17 q^{80}+21 q^{78}-25 q^{76}+14 q^{74}-4 q^{72}-11 q^{70}+16 q^{68}-18 q^{66}+11 q^{64}+q^{62}-12 q^{60}+14 q^{58}-9 q^{56}-q^{54}+11 q^{52}-15 q^{50}+14 q^{48}-4 q^{46}-3 q^{44}+10 q^{42}-14 q^{40}+14 q^{38}-7 q^{36}+2 q^{34}+4 q^{32}-7 q^{30}+8 q^{28}-5 q^{26}+6 q^{24}-q^{22}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8 }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_131.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{19}-q^{17}+q^{15}-2 q^{13}+2 q^5-q^3+2 q }[/math]
2	[math]\displaystyle{ q^{54}-q^{52}-2 q^{50}+3 q^{48}+q^{46}-5 q^{44}+2 q^{42}+4 q^{40}-5 q^{38}+5 q^{34}-2 q^{32}-3 q^{30}+4 q^{28}+q^{26}-4 q^{24}+3 q^{20}-2 q^{18}-5 q^{16}+5 q^{14}+q^{12}-5 q^{10}+4 q^8+2 q^6-2 q^4+2 q^2+1 }[/math]
3	[math]\displaystyle{ q^{105}-q^{103}-2 q^{101}+4 q^{97}+3 q^{95}-5 q^{93}-7 q^{91}+3 q^{89}+11 q^{87}+2 q^{85}-13 q^{83}-9 q^{81}+11 q^{79}+14 q^{77}-4 q^{75}-17 q^{73}-q^{71}+15 q^{69}+8 q^{67}-14 q^{65}-12 q^{63}+11 q^{61}+14 q^{59}-8 q^{57}-16 q^{55}+6 q^{53}+16 q^{51}-3 q^{49}-16 q^{47}+q^{45}+13 q^{43}+7 q^{41}-11 q^{39}-11 q^{37}+3 q^{35}+16 q^{33}+3 q^{31}-17 q^{29}-10 q^{27}+15 q^{25}+14 q^{23}-11 q^{21}-14 q^{19}+5 q^{17}+10 q^{15}-q^{13}-6 q^{11}+q^9+4 q^7-q^5+q^3+2 q^{-1} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{28}+q^{22}-2 q^{20}-q^{18}-q^{16}-q^{14}+q^{12}+2 q^8+q^6+2 q^2 }[/math]
1,1	[math]\displaystyle{ q^{76}-2 q^{74}+6 q^{72}-14 q^{70}+21 q^{68}-32 q^{66}+44 q^{64}-48 q^{62}+47 q^{60}-40 q^{58}+28 q^{56}-6 q^{54}-22 q^{52}+38 q^{50}-60 q^{48}+76 q^{46}-84 q^{44}+88 q^{42}-74 q^{40}+66 q^{38}-40 q^{36}+18 q^{34}+4 q^{32}-26 q^{30}+36 q^{28}-46 q^{26}+36 q^{24}-38 q^{22}+28 q^{20}-24 q^{18}+14 q^{16}-8 q^{14}+13 q^{12}+4 q^8+2 q^4+2 q^2 }[/math]
2,0	[math]\displaystyle{ q^{72}-q^{68}-q^{66}+q^{64}+2 q^{62}-2 q^{60}-2 q^{58}+q^{56}+q^{54}-2 q^{52}-3 q^{50}+2 q^{48}+4 q^{46}+2 q^{40}+2 q^{38}-q^{30}-2 q^{28}-q^{26}-4 q^{24}-6 q^{22}+q^{20}+2 q^{18}-q^{16}+5 q^{12}+5 q^{10}-q^6+3 q^4+q^2 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{60}-q^{58}+q^{56}-4 q^{52}+q^{50}-q^{48}-2 q^{46}+5 q^{44}+3 q^{42}+6 q^{38}+q^{36}-4 q^{34}-2 q^{32}-3 q^{30}-3 q^{28}-4 q^{26}-q^{24}+2 q^{22}-3 q^{20}+5 q^{16}-2 q^{14}+2 q^{12}+6 q^{10}+3 q^4 }[/math]
1,0,0	[math]\displaystyle{ q^{37}+q^{33}+q^{29}-2 q^{27}-q^{25}-2 q^{23}-q^{21}-q^{19}+q^{15}+2 q^{11}+q^9+2 q^7+2 q^3 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{78}+2 q^{72}-4 q^{68}-2 q^{66}-q^{64}-5 q^{62}-5 q^{60}+3 q^{58}+8 q^{56}+3 q^{54}+6 q^{52}+12 q^{50}+6 q^{48}-3 q^{46}-q^{44}-4 q^{42}-10 q^{40}-8 q^{38}-4 q^{36}-4 q^{34}-6 q^{32}+q^{30}+3 q^{28}-3 q^{26}-q^{24}+5 q^{22}+3 q^{20}+3 q^{16}+6 q^{14}+4 q^{12}+q^{10}+q^8+3 q^6 }[/math]
1,0,0,0	[math]\displaystyle{ q^{46}+q^{42}+q^{40}+q^{36}-2 q^{34}-q^{32}-2 q^{30}-2 q^{28}-q^{26}-q^{24}+q^{18}+2 q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^4 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{60}-q^{58}+3 q^{56}-4 q^{54}+4 q^{52}-5 q^{50}+5 q^{48}-4 q^{46}+3 q^{44}-q^{42}-2 q^{40}+4 q^{38}-7 q^{36}+8 q^{34}-10 q^{32}+9 q^{30}-9 q^{28}+6 q^{26}-5 q^{24}+2 q^{22}+q^{20}-2 q^{18}+5 q^{16}-4 q^{14}+6 q^{12}-4 q^{10}+4 q^8-2 q^6+3 q^4 }[/math]
1,0	[math]\displaystyle{ q^{98}-q^{94}-q^{92}+2 q^{90}+2 q^{88}-3 q^{86}-4 q^{84}+4 q^{80}+q^{78}-5 q^{76}-3 q^{74}+4 q^{72}+6 q^{70}+q^{68}-4 q^{66}+5 q^{62}+4 q^{60}-2 q^{58}-3 q^{56}+q^{54}+2 q^{52}-3 q^{50}-5 q^{48}-q^{46}+3 q^{44}-q^{42}-5 q^{40}-3 q^{38}+3 q^{36}+3 q^{34}-3 q^{32}-4 q^{30}+q^{28}+6 q^{26}+q^{24}-3 q^{22}-2 q^{20}+4 q^{18}+5 q^{16}+q^{14}-2 q^{12}-q^{10}+q^8+3 q^6 }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{82}-q^{80}+2 q^{78}-3 q^{76}+3 q^{74}-5 q^{72}+2 q^{70}-5 q^{68}+4 q^{66}-3 q^{64}+3 q^{62}+q^{60}+4 q^{58}+5 q^{56}+6 q^{52}-5 q^{50}+5 q^{48}-10 q^{46}+4 q^{44}-11 q^{42}+3 q^{40}-9 q^{38}+3 q^{36}-4 q^{34}+2 q^{32}-q^{30}-q^{28}+2 q^{26}-2 q^{24}+5 q^{22}-3 q^{20}+5 q^{18}-q^{16}+7 q^{14}+3 q^{10}-q^8+3 q^6 }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{142}-q^{140}+3 q^{138}-5 q^{136}+3 q^{134}-2 q^{132}-4 q^{130}+10 q^{128}-14 q^{126}+14 q^{124}-7 q^{122}-4 q^{120}+14 q^{118}-19 q^{116}+18 q^{114}-8 q^{112}-2 q^{110}+13 q^{108}-15 q^{106}+12 q^{104}+q^{102}-8 q^{100}+14 q^{98}-11 q^{96}+2 q^{94}+7 q^{92}-15 q^{90}+19 q^{88}-18 q^{86}+9 q^{84}+2 q^{82}-17 q^{80}+21 q^{78}-25 q^{76}+14 q^{74}-4 q^{72}-11 q^{70}+16 q^{68}-18 q^{66}+11 q^{64}+q^{62}-12 q^{60}+14 q^{58}-9 q^{56}-q^{54}+11 q^{52}-15 q^{50}+14 q^{48}-4 q^{46}-3 q^{44}+10 q^{42}-14 q^{40}+14 q^{38}-7 q^{36}+2 q^{34}+4 q^{32}-7 q^{30}+8 q^{28}-5 q^{26}+6 q^{24}-q^{22}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8 }[/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]:=

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["10 131"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-2 z^4 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 31, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ 2 q^{-1} -3 q^{-2} +5 q^{-3} -5 q^{-4} +5 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4-z^2 a^4+2 z^2 a^2+2 a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+9 z^3 a^9-3 z a^9+z^8 a^8-z^6 a^8-4 z^4 a^8+2 z^2 a^8+a^8+4 z^7 a^7-12 z^5 a^7+10 z^3 a^7-5 z a^7+z^8 a^6-2 z^4 a^6-3 z^2 a^6+2 a^6+2 z^7 a^5-3 z^5 a^5+2 z^3 a^5-z a^5+2 z^6 a^4-2 z^4 a^4+2 z^2 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^2-2 a^2 }[/math]