10 13

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-13 t+23-13 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4-5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 53, 0 }
Jones polynomial	[math]\displaystyle{ q^4-2 q^3+5 q^2-7 q+8-9 q^{-1} +8 q^{-2} -6 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-2 z^2 a^4-a^4+z^4 a^2+a^2+z^4-z^2-1-2 z^2 a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+2 a^4 z^8+4 a^2 z^8+2 z^8+2 a^5 z^7+a z^7+3 z^7 a^{-1} +a^6 z^6-5 a^4 z^6-9 a^2 z^6+3 z^6 a^{-2} -7 a^5 z^5-4 a^3 z^5-2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -4 a^6 z^4+a^4 z^4+6 a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+6 a^5 z^3+a^3 z^3+3 z^3 a^{-1} -2 z^3 a^{-3} +4 a^6 z^2+2 a^4 z^2-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +4 z^2-a^5 z+a^3 z-2 z a^{-1} -a^6-a^4-a^2+ a^{-4} -1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}-q^{16}+q^{14}-2 q^{10}+2 q^8-2+ q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
The G2 invariant	Data:10 13/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_13.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-q^{11}+2 q^9-2 q^7+2 q^5-q^3-q+ q^{-1} -2 q^{-3} +3 q^{-5} - q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{38}-q^{36}-q^{34}+4 q^{32}-2 q^{30}-6 q^{28}+8 q^{26}+2 q^{24}-11 q^{22}+7 q^{20}+6 q^{18}-13 q^{16}+3 q^{14}+9 q^{12}-8 q^{10}-2 q^8+7 q^6+2 q^4-6 q^2+11 q^{-2} -7 q^{-4} -7 q^{-6} +13 q^{-8} -5 q^{-10} -8 q^{-12} +9 q^{-14} -2 q^{-16} -4 q^{-18} +4 q^{-20} - q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{75}-q^{73}-q^{71}+q^{69}+3 q^{67}-2 q^{65}-7 q^{63}+q^{61}+13 q^{59}+3 q^{57}-17 q^{55}-12 q^{53}+20 q^{51}+23 q^{49}-17 q^{47}-34 q^{45}+6 q^{43}+41 q^{41}+6 q^{39}-42 q^{37}-19 q^{35}+39 q^{33}+31 q^{31}-31 q^{29}-37 q^{27}+23 q^{25}+40 q^{23}-13 q^{21}-42 q^{19}+7 q^{17}+37 q^{15}+5 q^{13}-32 q^{11}-16 q^9+22 q^7+26 q^5-10 q^3-36 q-5 q^{-1} +40 q^{-3} +22 q^{-5} -40 q^{-7} -29 q^{-9} +32 q^{-11} +36 q^{-13} -24 q^{-15} -30 q^{-17} +14 q^{-19} +26 q^{-21} -9 q^{-23} -18 q^{-25} +7 q^{-27} +9 q^{-29} -4 q^{-31} -7 q^{-33} +4 q^{-35} +4 q^{-37} -3 q^{-39} -3 q^{-41} +2 q^{-43} + q^{-45} - q^{-49} + q^{-51} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{20}+q^{18}-q^{16}+q^{14}-2 q^{10}+2 q^8-2+ q^{-2} -2 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 13"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-13 t+23-13 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^4-5 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]=

{ 53, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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