10 137

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^2-6 t+11-6 t^{-1} + t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 25, 0 }
Jones polynomial	[math]\displaystyle{ q^2-2 q+4-4 q^{-1} +4 q^{-2} -4 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-2 z^2 a^4-2 a^4+z^4 a^2+2 z^2 a^2+2 a^2-2 z^2-1+ a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^4 z^8+a^2 z^8+2 a^5 z^7+4 a^3 z^7+2 a z^7+a^6 z^6-a^4 z^6-a^2 z^6+z^6-8 a^5 z^5-15 a^3 z^5-7 a z^5-4 a^6 z^4-7 a^4 z^4-5 a^2 z^4-2 z^4+8 a^5 z^3+15 a^3 z^3+9 a z^3+2 z^3 a^{-1} +4 a^6 z^2+8 a^4 z^2+7 a^2 z^2+z^2 a^{-2} +4 z^2-3 a^5 z-5 a^3 z-3 a z-z a^{-1} -a^6-2 a^4-2 a^2- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}-q^{16}-q^{12}-q^{10}+q^8+q^4+ q^{-2} - q^{-4} + q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-q^{92}+3 q^{90}-4 q^{88}+3 q^{86}-q^{84}-4 q^{82}+10 q^{80}-10 q^{78}+10 q^{76}-4 q^{74}-4 q^{72}+11 q^{70}-12 q^{68}+8 q^{66}-q^{64}-6 q^{62}+8 q^{60}-6 q^{58}-2 q^{56}+9 q^{54}-13 q^{52}+10 q^{50}-5 q^{48}-6 q^{46}+12 q^{44}-15 q^{42}+13 q^{40}-9 q^{38}+3 q^{36}+5 q^{34}-9 q^{32}+11 q^{30}-9 q^{28}+5 q^{26}+3 q^{24}-6 q^{22}+6 q^{20}-2 q^{18}-3 q^{16}+10 q^{14}-11 q^{12}+7 q^{10}+q^8-10 q^6+14 q^4-13 q^2+7+ q^{-2} -7 q^{-4} +7 q^{-6} -5 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} + q^{-16} -2 q^{-20} +3 q^{-22} +2 q^{-28} - q^{-30} + q^{-32} + q^{-38} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_137.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-q^{11}+q^9-q^7+2 q^{-1} - q^{-3} + q^{-5} }[/math]
2	[math]\displaystyle{ q^{38}-q^{36}-2 q^{34}+3 q^{32}+q^{30}-4 q^{28}+q^{26}+3 q^{24}-2 q^{22}-2 q^{20}+3 q^{18}+q^{16}-3 q^{14}+2 q^{12}+2 q^{10}-3 q^8-q^6+3 q^4-3+3 q^{-2} +2 q^{-4} -3 q^{-6} +2 q^{-10} }[/math]
3	[math]\displaystyle{ q^{75}-q^{73}-2 q^{71}+4 q^{67}+3 q^{65}-5 q^{63}-6 q^{61}+2 q^{59}+8 q^{57}+3 q^{55}-7 q^{53}-7 q^{51}+2 q^{49}+10 q^{47}+5 q^{45}-10 q^{43}-10 q^{41}+5 q^{39}+14 q^{37}-2 q^{35}-15 q^{33}-q^{31}+14 q^{29}+4 q^{27}-12 q^{25}-4 q^{23}+11 q^{21}+5 q^{19}-10 q^{17}-5 q^{15}+6 q^{13}+7 q^{11}-3 q^9-8 q^7-4 q^5+10 q^3+11 q-5 q^{-1} -15 q^{-3} +18 q^{-7} +3 q^{-9} -14 q^{-11} -7 q^{-13} +9 q^{-15} +8 q^{-17} -4 q^{-19} -5 q^{-21} + q^{-23} +2 q^{-25} + q^{-27} - q^{-29} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{20}+q^{18}-q^{16}-q^{12}-q^{10}+q^8+q^4+ q^{-2} - q^{-4} + q^{-6} + q^{-8} }[/math]
2,0	[math]\displaystyle{ q^{52}+q^{50}-3 q^{46}-2 q^{44}+q^{42}+2 q^{40}-q^{36}+2 q^{34}+2 q^{32}-3 q^{28}-q^{26}+q^{22}+q^{18}+3 q^{16}-2 q^{10}-q^8+q^4-q^2-1+2 q^{-2} +3 q^{-4} -3 q^{-8} + q^{-10} +2 q^{-12} + q^{-20} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}-q^{38}+q^{36}+q^{34}-2 q^{32}+q^{30}-q^{28}-q^{26}+2 q^{24}+2 q^{18}-q^{14}-q^{10}-q^6+2 q^2-1+ q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8} +2 q^{-10} + q^{-16} }[/math]
1,0,0	[math]\displaystyle{ q^{27}+q^{25}+q^{23}-q^{21}-2 q^{17}-q^{15}-q^{13}+q^{11}+q^9+q^7+q^5- q^{-1} + q^{-3} - q^{-5} + q^{-7} + q^{-9} + q^{-11} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{40}-q^{38}+3 q^{36}-3 q^{34}+4 q^{32}-3 q^{30}+3 q^{28}-3 q^{26}-4 q^{20}+4 q^{18}-6 q^{16}+7 q^{14}-6 q^{12}+7 q^{10}-4 q^8+3 q^6+1-3 q^{-2} +4 q^{-4} -4 q^{-6} +3 q^{-8} -2 q^{-10} +2 q^{-12} + q^{-16} }[/math]
1,0	[math]\displaystyle{ q^{66}-q^{62}-q^{60}+2 q^{58}+2 q^{56}-2 q^{54}-3 q^{52}+q^{50}+3 q^{48}-4 q^{44}-2 q^{42}+4 q^{40}+3 q^{38}-q^{36}-3 q^{34}+q^{32}+3 q^{30}+q^{28}-2 q^{26}-q^{24}+2 q^{22}+q^{20}-2 q^{18}-3 q^{16}+q^{14}+3 q^{12}-q^{10}-4 q^8+3 q^4+2 q^2-2-2 q^{-2} +2 q^{-4} +4 q^{-6} - q^{-8} -3 q^{-10} +2 q^{-14} +2 q^{-16} - q^{-20} + q^{-26} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{94}-q^{92}+3 q^{90}-4 q^{88}+3 q^{86}-q^{84}-4 q^{82}+10 q^{80}-10 q^{78}+10 q^{76}-4 q^{74}-4 q^{72}+11 q^{70}-12 q^{68}+8 q^{66}-q^{64}-6 q^{62}+8 q^{60}-6 q^{58}-2 q^{56}+9 q^{54}-13 q^{52}+10 q^{50}-5 q^{48}-6 q^{46}+12 q^{44}-15 q^{42}+13 q^{40}-9 q^{38}+3 q^{36}+5 q^{34}-9 q^{32}+11 q^{30}-9 q^{28}+5 q^{26}+3 q^{24}-6 q^{22}+6 q^{20}-2 q^{18}-3 q^{16}+10 q^{14}-11 q^{12}+7 q^{10}+q^8-10 q^6+14 q^4-13 q^2+7+ q^{-2} -7 q^{-4} +7 q^{-6} -5 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} + q^{-16} -2 q^{-20} +3 q^{-22} +2 q^{-28} - q^{-30} + q^{-32} + q^{-38} }[/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["10 137"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^2-6 t+11-6 t^{-1} + t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ a^6-2 z^2 a^4-2 a^4+z^4 a^2+2 z^2 a^2+2 a^2-2 z^2-1+ a^{-2} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^4 z^8+a^2 z^8+2 a^5 z^7+4 a^3 z^7+2 a z^7+a^6 z^6-a^4 z^6-a^2 z^6+z^6-8 a^5 z^5-15 a^3 z^5-7 a z^5-4 a^6 z^4-7 a^4 z^4-5 a^2 z^4-2 z^4+8 a^5 z^3+15 a^3 z^3+9 a z^3+2 z^3 a^{-1} +4 a^6 z^2+8 a^4 z^2+7 a^2 z^2+z^2 a^{-2} +4 z^2-3 a^5 z-5 a^3 z-3 a z-z a^{-1} -a^6-2 a^4-2 a^2- a^{-2} -1 }[/math]