10 142

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+9 z^4+8 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature	{ 15, 6 }
Jones polynomial	[math]\displaystyle{ -2 q^{10}+2 q^9-2 q^8+3 q^7-2 q^6+2 q^5-q^4+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +7 z^2 a^{-8} -5 z^2 a^{-10} + a^{-6} +4 a^{-8} -5 a^{-10} + a^{-12} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -6 z^6 a^{-10} -4 z^5 a^{-7} -9 z^5 a^{-9} -5 z^5 a^{-11} -5 z^4 a^{-6} +9 z^4 a^{-8} +15 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-7} +12 z^3 a^{-9} +9 z^3 a^{-11} +6 z^2 a^{-6} -10 z^2 a^{-8} -17 z^2 a^{-10} -z^2 a^{-12} -6 z a^{-9} -4 z a^{-11} +2 z a^{-13} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-10} + q^{-14} + q^{-18} + q^{-20} +2 q^{-22} +3 q^{-24} - q^{-28} -3 q^{-30} -2 q^{-32} - q^{-34} + q^{-38} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-50} + q^{-54} - q^{-56} + q^{-58} +3 q^{-64} -2 q^{-66} +4 q^{-68} - q^{-70} - q^{-72} +3 q^{-74} -2 q^{-76} +2 q^{-78} - q^{-82} +3 q^{-84} +3 q^{-90} -4 q^{-92} +4 q^{-94} -2 q^{-98} +5 q^{-100} -2 q^{-102} +4 q^{-104} + q^{-106} +3 q^{-108} +2 q^{-110} -2 q^{-112} + q^{-114} +3 q^{-118} - q^{-120} -3 q^{-122} - q^{-124} + q^{-126} - q^{-130} -9 q^{-132} + q^{-136} -4 q^{-138} -7 q^{-142} + q^{-144} +3 q^{-146} -3 q^{-148} -2 q^{-150} -2 q^{-152} + q^{-154} +3 q^{-156} - q^{-158} - q^{-160} +2 q^{-162} +2 q^{-166} +2 q^{-168} - q^{-170} + q^{-172} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_142.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-5} + q^{-9} + q^{-13} + q^{-15} -2 q^{-21} }[/math]
2	[math]\displaystyle{ q^{-10} +2 q^{-16} + q^{-18} - q^{-20} + q^{-22} +2 q^{-24} - q^{-28} - q^{-34} + q^{-36} +2 q^{-38} - q^{-40} - q^{-46} -3 q^{-48} - q^{-50} - q^{-54} + q^{-56} + q^{-58} + q^{-60} }[/math]
3	[math]\displaystyle{ q^{-15} + q^{-21} +2 q^{-23} + q^{-25} - q^{-27} - q^{-29} +2 q^{-31} +3 q^{-33} + q^{-35} -2 q^{-37} -2 q^{-39} + q^{-41} +3 q^{-43} + q^{-45} -2 q^{-47} -3 q^{-49} + q^{-51} +4 q^{-53} + q^{-55} -4 q^{-57} -3 q^{-59} +2 q^{-61} +2 q^{-63} -3 q^{-65} -2 q^{-67} +3 q^{-69} + q^{-71} -2 q^{-73} - q^{-75} + q^{-77} - q^{-81} -3 q^{-83} -2 q^{-85} +3 q^{-89} -2 q^{-91} -4 q^{-93} +7 q^{-97} +4 q^{-99} -2 q^{-101} -2 q^{-103} + q^{-105} +4 q^{-107} -2 q^{-111} -2 q^{-113} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-10} + q^{-14} + q^{-18} + q^{-20} +2 q^{-22} +3 q^{-24} - q^{-28} -3 q^{-30} -2 q^{-32} - q^{-34} + q^{-38} }[/math]
1,1	[math]\displaystyle{ q^{-20} +2 q^{-24} -2 q^{-26} +6 q^{-28} -2 q^{-30} +12 q^{-32} -2 q^{-34} +7 q^{-36} +6 q^{-42} -10 q^{-44} +10 q^{-46} -10 q^{-48} +8 q^{-50} -11 q^{-52} +2 q^{-54} -8 q^{-56} -4 q^{-58} -3 q^{-60} -6 q^{-62} +6 q^{-64} -4 q^{-66} +10 q^{-68} +2 q^{-72} +2 q^{-74} -2 q^{-76} -2 q^{-78} -4 q^{-80} -2 q^{-82} +4 q^{-84} +2 q^{-88} }[/math]
2,0	[math]\displaystyle{ q^{-20} + q^{-26} +2 q^{-28} + q^{-30} + q^{-32} +2 q^{-34} +3 q^{-36} +2 q^{-38} + q^{-40} + q^{-42} - q^{-46} + q^{-52} +2 q^{-54} +3 q^{-56} +2 q^{-58} + q^{-60} -4 q^{-62} -6 q^{-64} -10 q^{-66} -9 q^{-68} -5 q^{-70} - q^{-72} +3 q^{-74} +4 q^{-76} +6 q^{-78} +5 q^{-80} +4 q^{-82} -2 q^{-88} -2 q^{-90} - q^{-92} + q^{-96} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-20} + q^{-24} + q^{-26} + q^{-30} +2 q^{-32} +3 q^{-34} +5 q^{-36} +4 q^{-38} +4 q^{-40} +2 q^{-42} -3 q^{-46} -3 q^{-48} -5 q^{-50} -4 q^{-52} -4 q^{-54} -3 q^{-56} - q^{-58} - q^{-60} + q^{-62} + q^{-64} + q^{-66} + q^{-68} +2 q^{-70} }[/math]
1,0,0	[math]\displaystyle{ q^{-15} + q^{-19} + q^{-23} +2 q^{-27} +3 q^{-29} +3 q^{-31} +3 q^{-33} - q^{-37} -4 q^{-39} -3 q^{-41} -3 q^{-43} - q^{-45} + q^{-49} + q^{-51} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{-30} + q^{-34} + q^{-36} + q^{-38} +2 q^{-42} +2 q^{-44} +4 q^{-46} +5 q^{-48} +5 q^{-50} +7 q^{-52} +7 q^{-54} +5 q^{-56} +4 q^{-58} +4 q^{-60} +2 q^{-62} -2 q^{-64} -4 q^{-66} -6 q^{-68} -10 q^{-70} -14 q^{-72} -12 q^{-74} -12 q^{-76} -10 q^{-78} -2 q^{-80} +3 q^{-82} +6 q^{-84} +8 q^{-86} +10 q^{-88} +6 q^{-90} +2 q^{-92} -2 q^{-98} -2 q^{-100} }[/math]
1,0,0,0	[math]\displaystyle{ q^{-20} + q^{-24} + q^{-28} + q^{-32} +2 q^{-34} +3 q^{-36} +4 q^{-38} +3 q^{-40} +3 q^{-42} - q^{-46} -4 q^{-48} -4 q^{-50} -4 q^{-52} -3 q^{-54} - q^{-56} + q^{-60} + q^{-62} + q^{-64} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{-20} + q^{-24} - q^{-26} +2 q^{-28} - q^{-30} +2 q^{-32} + q^{-34} + q^{-36} +2 q^{-38} +2 q^{-42} -2 q^{-44} +3 q^{-46} -3 q^{-48} + q^{-50} -2 q^{-52} - q^{-56} - q^{-58} + q^{-60} - q^{-62} + q^{-64} - q^{-66} + q^{-68} -2 q^{-70} }[/math]
1,0	[math]\displaystyle{ q^{-30} + q^{-38} + q^{-40} - q^{-44} + q^{-46} +2 q^{-48} + q^{-50} - q^{-52} +2 q^{-54} +3 q^{-56} +3 q^{-58} + q^{-60} + q^{-62} + q^{-64} +2 q^{-66} - q^{-70} - q^{-72} - q^{-76} -2 q^{-78} -2 q^{-80} - q^{-82} -2 q^{-86} -3 q^{-88} -2 q^{-90} - q^{-94} -2 q^{-96} - q^{-98} + q^{-100} + q^{-102} + q^{-108} + q^{-110} +2 q^{-112} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{-30} + q^{-34} +2 q^{-38} - q^{-40} +2 q^{-42} +3 q^{-46} +3 q^{-48} +3 q^{-50} +5 q^{-52} +5 q^{-54} +6 q^{-56} +3 q^{-58} +4 q^{-60} -2 q^{-62} -6 q^{-66} -4 q^{-68} -8 q^{-70} -5 q^{-72} -6 q^{-74} -3 q^{-76} -2 q^{-78} - q^{-80} + q^{-82} +2 q^{-86} +2 q^{-90} + q^{-94} +2 q^{-98} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-50} + q^{-54} - q^{-56} + q^{-58} +3 q^{-64} -2 q^{-66} +4 q^{-68} - q^{-70} - q^{-72} +3 q^{-74} -2 q^{-76} +2 q^{-78} - q^{-82} +3 q^{-84} +3 q^{-90} -4 q^{-92} +4 q^{-94} -2 q^{-98} +5 q^{-100} -2 q^{-102} +4 q^{-104} + q^{-106} +3 q^{-108} +2 q^{-110} -2 q^{-112} + q^{-114} +3 q^{-118} - q^{-120} -3 q^{-122} - q^{-124} + q^{-126} - q^{-130} -9 q^{-132} + q^{-136} -4 q^{-138} -7 q^{-142} + q^{-144} +3 q^{-146} -3 q^{-148} -2 q^{-150} -2 q^{-152} + q^{-154} +3 q^{-156} - q^{-158} - q^{-160} +2 q^{-162} +2 q^{-166} +2 q^{-168} - q^{-170} + q^{-172} }[/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["10 142"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+9 z^4+8 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{ \left\{2,t^2+t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 15, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -6 z^6 a^{-10} -4 z^5 a^{-7} -9 z^5 a^{-9} -5 z^5 a^{-11} -5 z^4 a^{-6} +9 z^4 a^{-8} +15 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-7} +12 z^3 a^{-9} +9 z^3 a^{-11} +6 z^2 a^{-6} -10 z^2 a^{-8} -17 z^2 a^{-10} -z^2 a^{-12} -6 z a^{-9} -4 z a^{-11} +2 z a^{-13} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12} }[/math]