10 35

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-12 t+21-12 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4-4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 49, 0 }
Jones polynomial	[math]\displaystyle{ q^6-2 q^5+4 q^4-6 q^3+7 q^2-8 q+8-6 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^4-2 z^2 a^2-a^2+z^4+1+z^4 a^{-2} -2 z^2 a^{-4} - a^{-4} + a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+2 a z^7+2 z^7 a^{-5} +2 a^2 z^6-11 z^6 a^{-2} -5 z^6 a^{-4} +z^6 a^{-6} -3 z^6+2 a^3 z^5-z^5 a^{-1} -6 z^5 a^{-3} -7 z^5 a^{-5} +a^4 z^4+10 z^4 a^{-2} -4 z^4 a^{-6} +5 z^4-3 a^3 z^3-2 a z^3+5 z^3 a^{-3} +6 z^3 a^{-5} -2 a^4 z^2-3 a^2 z^2-3 z^2 a^{-2} +3 z^2 a^{-4} +4 z^2 a^{-6} -3 z^2+a^3 z+a z+z a^{-1} -z a^{-3} -2 z a^{-5} +a^4+a^2- a^{-4} - a^{-6} +1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{14}+q^{12}-q^{10}+q^8-2 q^4+2 q^2+ q^{-2} - q^{-6} + q^{-8} -2 q^{-10} + q^{-14} - q^{-16} + q^{-18} + q^{-20} }[/math]
The G2 invariant	Data:10 35/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_35.

A1 Invariants.

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

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{14}+q^{12}-q^{10}+q^8-2 q^4+2 q^2+ q^{-2} - q^{-6} + q^{-8} -2 q^{-10} + q^{-14} - 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 35"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-12 t+21-12 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 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]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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