10 58

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^2-16 t+27-16 t^{-1} +3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^4-4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 65, 0 }
Jones polynomial	[math]\displaystyle{ q^4-2 q^3+5 q^2-8 q+10-11 q^{-1} +10 q^{-2} -8 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-3 z^2 a^4-2 a^4+2 z^4 a^2+3 z^2 a^2+3 a^2+z^4-2 z^2-2-2 z^2 a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+3 a^5 z^7+6 a^3 z^7+7 a z^7+4 z^7 a^{-1} +a^6 z^6-5 a^4 z^6-10 a^2 z^6+3 z^6 a^{-2} -z^6-9 a^5 z^5-23 a^3 z^5-22 a z^5-6 z^5 a^{-1} +2 z^5 a^{-3} -3 a^6 z^4-5 a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -5 z^4+7 a^5 z^3+18 a^3 z^3+21 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} +3 a^6 z^2+7 a^4 z^2+10 a^2 z^2-2 z^2 a^{-4} +8 z^2-2 a^5 z-4 a^3 z-6 a z-4 z a^{-1} -a^6-2 a^4-3 a^2+ a^{-4} -2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}-2 q^{16}+q^{14}-2 q^{10}+3 q^8+q^4-2+ q^{-2} -3 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
The G2 invariant	Data:10 58/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_58.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-2 q^{11}+3 q^9-2 q^7+2 q^5-q^3-q+2 q^{-1} -3 q^{-3} +3 q^{-5} - q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{38}-2 q^{36}-2 q^{34}+8 q^{32}-2 q^{30}-12 q^{28}+13 q^{26}+6 q^{24}-20 q^{22}+7 q^{20}+15 q^{18}-17 q^{16}-2 q^{14}+16 q^{12}-7 q^{10}-9 q^8+8 q^6+8 q^4-11 q^2-5+20 q^{-2} -7 q^{-4} -16 q^{-6} +18 q^{-8} -13 q^{-12} +9 q^{-14} + q^{-16} -5 q^{-18} +3 q^{-20} - q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{75}-2 q^{73}-2 q^{71}+3 q^{69}+8 q^{67}-2 q^{65}-19 q^{63}-4 q^{61}+28 q^{59}+21 q^{57}-31 q^{55}-46 q^{53}+25 q^{51}+67 q^{49}-82 q^{45}-34 q^{43}+82 q^{41}+67 q^{39}-66 q^{37}-93 q^{35}+38 q^{33}+108 q^{31}-10 q^{29}-109 q^{27}-12 q^{25}+100 q^{23}+36 q^{21}-87 q^{19}-50 q^{17}+65 q^{15}+64 q^{13}-42 q^{11}-75 q^9+8 q^7+81 q^5+32 q^3-79 q-66 q^{-1} +64 q^{-3} +96 q^{-5} -38 q^{-7} -109 q^{-9} +9 q^{-11} +104 q^{-13} +12 q^{-15} -79 q^{-17} -28 q^{-19} +55 q^{-21} +27 q^{-23} -31 q^{-25} -18 q^{-27} +14 q^{-29} +11 q^{-31} -7 q^{-33} -4 q^{-35} +3 q^{-37} + q^{-39} -2 q^{-41} + q^{-43} - q^{-49} + q^{-51} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{20}+q^{18}-2 q^{16}+q^{14}-2 q^{10}+3 q^8+q^4-2+ q^{-2} -3 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/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 58"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 3 t^2-16 t+27-16 t^{-1} +3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 3 z^4-4 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]=

{ 65, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-2 q^3+5 q^2-8 q+10-11 q^{-1} +10 q^{-2} -8 q^{-3} +6 q^{-4} -3 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^6-3 z^2 a^4-2 a^4+2 z^4 a^2+3 z^2 a^2+3 a^2+z^4-2 z^2-2-2 z^2 a^{-2} + a^{-4} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+3 a^5 z^7+6 a^3 z^7+7 a z^7+4 z^7 a^{-1} +a^6 z^6-5 a^4 z^6-10 a^2 z^6+3 z^6 a^{-2} -z^6-9 a^5 z^5-23 a^3 z^5-22 a z^5-6 z^5 a^{-1} +2 z^5 a^{-3} -3 a^6 z^4-5 a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -5 z^4+7 a^5 z^3+18 a^3 z^3+21 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} +3 a^6 z^2+7 a^4 z^2+10 a^2 z^2-2 z^2 a^{-4} +8 z^2-2 a^5 z-4 a^3 z-6 a z-4 z a^{-1} -a^6-2 a^4-3 a^2+ a^{-4} -2 }[/math]