10 147

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+7 t-9+7 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 27, 2 }
Jones polynomial	[math]\displaystyle{ q^5-3 q^4+4 q^3-4 q^2+5 q-4+3 q^{-1} -2 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^{-2} -z^4+a^2 z^2-z^2 a^{-2} +z^2 a^{-4} -2 z^2+a^2+ a^{-2} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+4 z^7 a^{-1} +2 z^7 a^{-3} +a^2 z^6-z^6 a^{-2} +z^6 a^{-4} -z^6-8 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} -4 a^2 z^4-2 z^4 a^{-2} -6 z^4+8 a z^3+13 z^3 a^{-1} +8 z^3 a^{-3} +3 z^3 a^{-5} +4 a^2 z^2+z^2 a^{-2} +z^2 a^{-6} +6 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^2- a^{-2} -1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{10}+q^4-q^2+2 q^{-6} + q^{-10} - q^{-12} - q^{-14} + q^{-16} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-q^{44}+3 q^{42}-4 q^{40}+3 q^{38}-q^{36}-3 q^{34}+10 q^{32}-11 q^{30}+12 q^{28}-6 q^{26}-5 q^{24}+12 q^{22}-14 q^{20}+11 q^{18}-5 q^{16}-4 q^{14}+12 q^{12}-9 q^{10}+3 q^8+5 q^6-13 q^4+14 q^2-7-5 q^{-2} +10 q^{-4} -14 q^{-6} +18 q^{-8} -11 q^{-10} +4 q^{-12} +4 q^{-14} -13 q^{-16} +15 q^{-18} -13 q^{-20} +7 q^{-22} -7 q^{-26} +11 q^{-28} -6 q^{-30} +3 q^{-32} +6 q^{-34} -14 q^{-36} +11 q^{-38} - q^{-40} -7 q^{-42} +13 q^{-44} -15 q^{-46} +11 q^{-48} + q^{-50} -6 q^{-52} +7 q^{-54} -11 q^{-56} +7 q^{-58} -3 q^{-62} +2 q^{-64} -2 q^{-66} + q^{-68} + q^{-70} +2 q^{-72} - q^{-74} - q^{-78} - q^{-84} + q^{-86} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_147.

A1 Invariants.

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

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{10}+q^4-q^2+2 q^{-6} + q^{-10} - q^{-12} - q^{-14} + q^{-16} }[/math]
1,1	[math]\displaystyle{ q^{28}-2 q^{26}+6 q^{24}-12 q^{22}+19 q^{20}-28 q^{18}+36 q^{16}-38 q^{14}+33 q^{12}-24 q^{10}+12 q^8+14 q^6-31 q^4+48 q^2-60+62 q^{-2} -69 q^{-4} +56 q^{-6} -42 q^{-8} +32 q^{-10} -3 q^{-12} -6 q^{-14} +28 q^{-16} -38 q^{-18} +38 q^{-20} -40 q^{-22} +24 q^{-24} -18 q^{-26} +11 q^{-28} -2 q^{-30} -2 q^{-32} +2 q^{-34} +2 q^{-36} -2 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{28}-q^{24}-q^{22}+q^{20}+2 q^{18}-q^{16}-q^{14}+q^{12}+2 q^{10}+q^8-2 q^6+q^2-1-2 q^{-2} + q^{-6} + q^{-8} +2 q^{-10} +2 q^{-12} +2 q^{-14} +2 q^{-16} +2 q^{-18} - q^{-20} -5 q^{-22} - q^{-24} -2 q^{-28} - q^{-30} +2 q^{-32} +3 q^{-34} - q^{-38} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{20}-q^{18}+q^{16}+q^{14}-2 q^{12}+2 q^{10}-q^6+3 q^4-1+2 q^{-2} -2 q^{-6} + q^{-8} + q^{-12} - q^{-14} +2 q^{-18} -2 q^{-20} + q^{-22} +3 q^{-24} -2 q^{-26} +2 q^{-30} -2 q^{-32} - q^{-34} + q^{-36} }[/math]
1,0,0	[math]\displaystyle{ q^{13}+q^9+q^5-q^3- q^{-1} + q^{-7} +2 q^{-9} + q^{-13} - q^{-15} - q^{-19} + q^{-21} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{26}+2 q^{20}+q^{18}-2 q^{16}+q^{12}-2 q^{10}-2 q^8+3 q^6+4 q^4-q^2+2+6 q^{-2} -4 q^{-6} - q^{-10} -5 q^{-12} - q^{-14} +2 q^{-16} + q^{-18} - q^{-20} +4 q^{-22} +3 q^{-24} -2 q^{-26} +3 q^{-30} -2 q^{-34} + q^{-36} + q^{-38} - q^{-40} -2 q^{-42} + q^{-46} }[/math]
1,0,0,0	[math]\displaystyle{ q^{16}+q^{12}+q^{10}+q^6-q^4-1- q^{-2} + q^{-8} + q^{-10} +2 q^{-12} + q^{-16} - q^{-18} - q^{-24} + q^{-26} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{20}-q^{18}+3 q^{16}-3 q^{14}+4 q^{12}-4 q^{10}+4 q^8-3 q^6+q^4-3+4 q^{-2} -6 q^{-4} +8 q^{-6} -7 q^{-8} +8 q^{-10} -5 q^{-12} +5 q^{-14} -2 q^{-16} +2 q^{-20} -3 q^{-22} +3 q^{-24} -4 q^{-26} +4 q^{-28} -4 q^{-30} +2 q^{-32} - q^{-34} + q^{-36} }[/math]
1,0	[math]\displaystyle{ q^{34}-q^{30}-q^{28}+2 q^{26}+2 q^{24}-2 q^{22}-3 q^{20}+q^{18}+4 q^{16}+q^{14}-4 q^{12}-2 q^{10}+4 q^8+4 q^6-q^4-4 q^2+3 q^{-2} + q^{-4} -2 q^{-6} -2 q^{-8} +2 q^{-10} +2 q^{-12} - q^{-14} -3 q^{-16} +2 q^{-18} +4 q^{-20} -4 q^{-24} - q^{-26} +3 q^{-28} +2 q^{-30} -3 q^{-32} -3 q^{-34} +2 q^{-36} +4 q^{-38} -3 q^{-42} - q^{-44} +2 q^{-46} +3 q^{-48} - q^{-50} -2 q^{-52} - q^{-54} + q^{-58} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{26}-q^{24}+2 q^{22}-2 q^{20}+4 q^{18}-3 q^{16}+3 q^{14}-3 q^{12}+4 q^{10}-2 q^8+q^6+3-3 q^{-2} +4 q^{-4} -5 q^{-6} +5 q^{-8} -6 q^{-10} +6 q^{-12} -6 q^{-14} +5 q^{-16} -4 q^{-18} +3 q^{-20} - q^{-22} + q^{-24} + q^{-26} - q^{-28} +3 q^{-30} -2 q^{-32} +4 q^{-34} -3 q^{-36} +2 q^{-38} -3 q^{-40} +3 q^{-42} -2 q^{-44} - q^{-48} + q^{-50} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{46}-q^{44}+3 q^{42}-4 q^{40}+3 q^{38}-q^{36}-3 q^{34}+10 q^{32}-11 q^{30}+12 q^{28}-6 q^{26}-5 q^{24}+12 q^{22}-14 q^{20}+11 q^{18}-5 q^{16}-4 q^{14}+12 q^{12}-9 q^{10}+3 q^8+5 q^6-13 q^4+14 q^2-7-5 q^{-2} +10 q^{-4} -14 q^{-6} +18 q^{-8} -11 q^{-10} +4 q^{-12} +4 q^{-14} -13 q^{-16} +15 q^{-18} -13 q^{-20} +7 q^{-22} -7 q^{-26} +11 q^{-28} -6 q^{-30} +3 q^{-32} +6 q^{-34} -14 q^{-36} +11 q^{-38} - q^{-40} -7 q^{-42} +13 q^{-44} -15 q^{-46} +11 q^{-48} + q^{-50} -6 q^{-52} +7 q^{-54} -11 q^{-56} +7 q^{-58} -3 q^{-62} +2 q^{-64} -2 q^{-66} + q^{-68} + q^{-70} +2 q^{-72} - q^{-74} - q^{-78} - q^{-84} + q^{-86} }[/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["10 147"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^2+7 t-9+7 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+4 z^7 a^{-1} +2 z^7 a^{-3} +a^2 z^6-z^6 a^{-2} +z^6 a^{-4} -z^6-8 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} -4 a^2 z^4-2 z^4 a^{-2} -6 z^4+8 a z^3+13 z^3 a^{-1} +8 z^3 a^{-3} +3 z^3 a^{-5} +4 a^2 z^2+z^2 a^{-2} +z^2 a^{-6} +6 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^2- a^{-2} -1 }[/math]