10 1

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -4 t+9-4 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 1-4 z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 17, 0 }
Jones polynomial	[math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} - q^{-7} + q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^8-z^2 a^6-a^6-z^2 a^4-z^2 a^2-z^2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^7 z^9+a^5 z^9+a^8 z^8+2 a^6 z^8+a^4 z^8-7 a^7 z^7-6 a^5 z^7+a^3 z^7-7 a^8 z^6-12 a^6 z^6-4 a^4 z^6+a^2 z^6+16 a^7 z^5+12 a^5 z^5-3 a^3 z^5+a z^5+15 a^8 z^4+21 a^6 z^4+3 a^4 z^4-2 a^2 z^4+z^4-14 a^7 z^3-11 a^5 z^3+a^3 z^3-a z^3+z^3 a^{-1} -10 a^8 z^2-11 a^6 z^2+z^2 a^{-2} +4 a^7 z+4 a^5 z+a^8+a^6- a^{-2} }[/math]
The A2 invariant	[math]\displaystyle{ q^{26}+q^{24}-q^{18}-q^{16}+ q^{-2} + q^{-6} + q^{-8} }[/math]
The G2 invariant	Data:10 1/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_1.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{17}-q^{11}+ q^{-1} + q^{-5} }[/math]
2	[math]\displaystyle{ q^{50}-q^{46}-q^{40}+q^{36}+q^{16}+q^{14}-q^4-q^2+ q^{-2} + q^{-8} + q^{-14} }[/math]
3	[math]\displaystyle{ q^{99}-q^{95}-q^{93}+q^{89}-q^{85}+q^{81}+q^{79}-q^{75}+q^{49}+q^{47}-q^{43}-q^{37}-q^{35}-q^{29}+q^{25}+q^{23}+q^{15}+q^{13}+q^{11}-q^9+q^5+q^3-q- q^{-1} + q^{-3} + q^{-5} - q^{-7} - q^{-9} + q^{-19} + q^{-27} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{26}+q^{24}-q^{18}-q^{16}+ q^{-2} + q^{-6} + q^{-8} }[/math]
2,0	[math]\displaystyle{ q^{68}+q^{66}+q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+q^{48}+q^{46}+q^{24}+2 q^{22}+q^{20}+q^{18}+q^{16}-q^{10}-2 q^8-2 q^6-q^4+ q^{-4} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} }[/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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -4 t+9-4 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-4 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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^7 z^9+a^5 z^9+a^8 z^8+2 a^6 z^8+a^4 z^8-7 a^7 z^7-6 a^5 z^7+a^3 z^7-7 a^8 z^6-12 a^6 z^6-4 a^4 z^6+a^2 z^6+16 a^7 z^5+12 a^5 z^5-3 a^3 z^5+a z^5+15 a^8 z^4+21 a^6 z^4+3 a^4 z^4-2 a^2 z^4+z^4-14 a^7 z^3-11 a^5 z^3+a^3 z^3-a z^3+z^3 a^{-1} -10 a^8 z^2-11 a^6 z^2+z^2 a^{-2} +4 a^7 z+4 a^5 z+a^8+a^6- a^{-2} }[/math]