10 3

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -6 t+13-6 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 1-6 z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 25, 0 }
Jones polynomial	[math]\displaystyle{ q^4-q^3+2 q^2-3 q+4-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-z^2 a^4-2 z^2 a^2-a^2-2 z^2-z^2 a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-6 a z^7+z^7 a^{-1} +a^6 z^6-4 a^4 z^6-10 a^2 z^6+z^6 a^{-2} -4 z^6-4 a^5 z^5+15 a^3 z^5+15 a z^5-3 z^5 a^{-1} +z^5 a^{-3} -5 a^6 z^4+4 a^4 z^4+18 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} +6 z^4+3 a^5 z^3-18 a^3 z^3-15 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +6 a^6 z^2-2 a^4 z^2-12 a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +6 a^3 z+6 a z-a^6+a^2+ a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}+q^{14}-q^{10}-q^6-q^4+ q^{-2} - q^{-4} + q^{-8} + q^{-12} + q^{-14} }[/math]
The G2 invariant	Data:10 3/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_3.

A1 Invariants.

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

A2 Invariants.

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

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -6 t+13-6 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-6 z^2 }[/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^4-q^3+2 q^2-3 q+4-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - 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-z^2 a^4-2 z^2 a^2-a^2-2 z^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+a^4 z^8+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-6 a z^7+z^7 a^{-1} +a^6 z^6-4 a^4 z^6-10 a^2 z^6+z^6 a^{-2} -4 z^6-4 a^5 z^5+15 a^3 z^5+15 a z^5-3 z^5 a^{-1} +z^5 a^{-3} -5 a^6 z^4+4 a^4 z^4+18 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} +6 z^4+3 a^5 z^3-18 a^3 z^3-15 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +6 a^6 z^2-2 a^4 z^2-12 a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +6 a^3 z+6 a z-a^6+a^2+ a^{-4} }[/math]