10 124

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+7 z^6+14 z^4+8 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 1, 8 }
Jones polynomial	[math]\displaystyle{ -q^{10}+q^6+q^4 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -7 z^4 a^{-10} +21 z^2 a^{-8} -14 z^2 a^{-10} +z^2 a^{-12} +7 a^{-8} -8 a^{-10} +2 a^{-12} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-9} +z^7 a^{-11} -8 z^6 a^{-8} -8 z^6 a^{-10} -7 z^5 a^{-9} -7 z^5 a^{-11} +21 z^4 a^{-8} +21 z^4 a^{-10} +14 z^3 a^{-9} +14 z^3 a^{-11} -21 z^2 a^{-8} -22 z^2 a^{-10} -z^2 a^{-12} -8 z a^{-9} -8 z a^{-11} +7 a^{-8} +8 a^{-10} +2 a^{-12} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-14} + q^{-16} +2 q^{-18} +2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-28} -2 q^{-30} -2 q^{-32} - q^{-34} + q^{-40} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} +2 q^{-80} +2 q^{-82} + q^{-84} +2 q^{-86} +2 q^{-88} +2 q^{-90} +2 q^{-92} +2 q^{-94} + q^{-96} +2 q^{-98} + q^{-100} + q^{-104} + q^{-106} - q^{-112} - q^{-114} - q^{-118} -2 q^{-120} -2 q^{-122} - q^{-124} - q^{-126} -2 q^{-128} -2 q^{-130} -2 q^{-132} - q^{-134} - q^{-136} -2 q^{-138} -2 q^{-140} - q^{-142} - q^{-146} - q^{-148} + q^{-160} + q^{-162} + q^{-168} + q^{-170} + q^{-180} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_124.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-7} + q^{-9} + q^{-11} + q^{-13} - q^{-19} - q^{-21} }[/math]
2	[math]\displaystyle{ q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} + q^{-24} + q^{-26} - q^{-36} - q^{-38} - q^{-40} - q^{-42} - q^{-44} + q^{-60} }[/math]
3	[math]\displaystyle{ q^{-21} + q^{-23} + q^{-25} + q^{-27} + q^{-29} + q^{-31} + q^{-33} + q^{-35} + q^{-37} + q^{-39} - q^{-53} - q^{-55} - q^{-57} - q^{-59} - q^{-61} - q^{-63} - q^{-65} - q^{-67} + q^{-97} + q^{-99} + q^{-101} + q^{-103} - q^{-109} - q^{-111} }[/math]
5	[math]\displaystyle{ q^{-35} + q^{-37} + q^{-39} + q^{-41} + q^{-43} + q^{-45} + q^{-47} + q^{-49} + q^{-51} + q^{-53} + q^{-55} + q^{-57} + q^{-59} + q^{-61} + q^{-63} + q^{-65} - q^{-87} - q^{-89} - q^{-91} - q^{-93} - q^{-95} - q^{-97} - q^{-99} - q^{-101} - q^{-103} - q^{-105} - q^{-107} - q^{-109} - q^{-111} - q^{-113} + q^{-171} + q^{-173} + q^{-175} + q^{-177} + q^{-179} + q^{-181} + q^{-183} + q^{-185} + q^{-187} + q^{-189} - q^{-203} - q^{-205} - q^{-207} - q^{-209} - q^{-211} - q^{-213} - q^{-215} - q^{-217} + q^{-247} + q^{-249} + q^{-251} + q^{-253} - q^{-259} - q^{-261} }[/math]
6	[math]\displaystyle{ q^{-42} + q^{-44} + q^{-46} + q^{-48} + q^{-50} + q^{-52} + q^{-54} + q^{-56} + q^{-58} + q^{-60} + q^{-62} + q^{-64} + q^{-66} + q^{-68} + q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} - q^{-104} - q^{-106} - q^{-108} - q^{-110} - q^{-112} - q^{-114} - q^{-116} - q^{-118} - q^{-120} - q^{-122} - q^{-124} - q^{-126} - q^{-128} - q^{-130} - q^{-132} - q^{-134} - q^{-136} + q^{-208} + q^{-210} + q^{-212} + q^{-214} + q^{-216} + q^{-218} + q^{-220} + q^{-222} + q^{-224} + q^{-226} + q^{-228} + q^{-230} + q^{-232} - q^{-250} - q^{-252} - q^{-254} - q^{-256} - q^{-258} - q^{-260} - q^{-262} - q^{-264} - q^{-266} - q^{-268} - q^{-270} + q^{-314} + q^{-316} + q^{-318} + q^{-320} + q^{-322} + q^{-324} + q^{-326} - q^{-336} - q^{-338} - q^{-340} - q^{-342} - q^{-344} + q^{-360} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-14} + q^{-16} +2 q^{-18} +2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-28} -2 q^{-30} -2 q^{-32} - q^{-34} + q^{-40} }[/math]
1,1	[math]\displaystyle{ q^{-28} +2 q^{-30} +4 q^{-32} +6 q^{-34} +7 q^{-36} +8 q^{-38} +6 q^{-40} +4 q^{-42} + q^{-44} -2 q^{-46} -6 q^{-48} -8 q^{-50} -9 q^{-52} -8 q^{-54} -6 q^{-56} -4 q^{-58} -2 q^{-60} +2 q^{-64} +2 q^{-66} +4 q^{-68} +4 q^{-70} +4 q^{-72} +2 q^{-74} + q^{-76} -2 q^{-78} -2 q^{-80} -2 q^{-82} - q^{-84} +2 q^{-90} }[/math]
2,0	[math]\displaystyle{ q^{-28} + q^{-30} +2 q^{-32} +2 q^{-34} +3 q^{-36} +3 q^{-38} +4 q^{-40} +3 q^{-42} +3 q^{-44} +2 q^{-46} +2 q^{-48} - q^{-52} -3 q^{-54} -4 q^{-56} -5 q^{-58} -5 q^{-60} -5 q^{-62} -4 q^{-64} -3 q^{-66} -2 q^{-68} - q^{-70} +2 q^{-74} +2 q^{-76} +4 q^{-78} +4 q^{-80} +4 q^{-82} +2 q^{-84} + q^{-86} -2 q^{-88} -2 q^{-90} -2 q^{-92} - q^{-94} + q^{-100} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-28} + q^{-30} +3 q^{-32} +4 q^{-34} +6 q^{-36} +6 q^{-38} +7 q^{-40} +3 q^{-42} + q^{-44} -4 q^{-46} -7 q^{-48} -9 q^{-50} -9 q^{-52} -6 q^{-54} -3 q^{-56} + q^{-58} +3 q^{-60} +4 q^{-62} +2 q^{-64} +2 q^{-66} }[/math]
1,0,0	[math]\displaystyle{ q^{-21} + q^{-23} +2 q^{-25} +3 q^{-27} +3 q^{-29} +3 q^{-31} +2 q^{-33} -2 q^{-37} -3 q^{-39} -4 q^{-41} -3 q^{-43} -2 q^{-45} + q^{-49} + q^{-51} + q^{-53} }[/math]
1,0,1	[math]\displaystyle{ q^{-42} +2 q^{-44} +5 q^{-46} +9 q^{-48} +14 q^{-50} +19 q^{-52} +23 q^{-54} +22 q^{-56} +19 q^{-58} +10 q^{-60} - q^{-62} -14 q^{-64} -26 q^{-66} -34 q^{-68} -38 q^{-70} -35 q^{-72} -26 q^{-74} -13 q^{-76} - q^{-78} +11 q^{-80} +18 q^{-82} +21 q^{-84} +19 q^{-86} +14 q^{-88} +8 q^{-90} +4 q^{-92} -3 q^{-96} -3 q^{-98} -4 q^{-100} -3 q^{-102} -3 q^{-104} - q^{-106} - q^{-108} +2 q^{-120} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{-42} + q^{-44} +3 q^{-46} +5 q^{-48} +8 q^{-50} +10 q^{-52} +14 q^{-54} +13 q^{-56} +13 q^{-58} +8 q^{-60} +2 q^{-62} -7 q^{-64} -14 q^{-66} -21 q^{-68} -23 q^{-70} -21 q^{-72} -16 q^{-74} -8 q^{-76} - q^{-78} +7 q^{-80} +9 q^{-82} +11 q^{-84} +9 q^{-86} +7 q^{-88} +3 q^{-90} +2 q^{-92} - q^{-96} - q^{-98} - q^{-100} - q^{-102} - q^{-104} }[/math]
1,0,0,0	[math]\displaystyle{ q^{-28} + q^{-30} +2 q^{-32} +3 q^{-34} +4 q^{-36} +4 q^{-38} +4 q^{-40} +2 q^{-42} + q^{-44} -2 q^{-46} -4 q^{-48} -5 q^{-50} -5 q^{-52} -4 q^{-54} -2 q^{-56} + q^{-60} +2 q^{-62} + q^{-64} + q^{-66} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{-28} + q^{-30} + q^{-32} +2 q^{-34} +2 q^{-36} +2 q^{-38} + q^{-40} + q^{-42} + q^{-44} - q^{-48} - q^{-50} - q^{-52} -2 q^{-54} - q^{-56} - q^{-58} - q^{-60} }[/math]
1,0	[math]\displaystyle{ q^{-42} + q^{-46} + q^{-48} +2 q^{-50} + q^{-52} +3 q^{-54} +2 q^{-56} +3 q^{-58} +2 q^{-60} +3 q^{-62} +2 q^{-64} +2 q^{-66} - q^{-72} -2 q^{-74} -3 q^{-76} -3 q^{-78} -3 q^{-80} -4 q^{-82} -3 q^{-84} -3 q^{-86} -2 q^{-88} -2 q^{-90} + q^{-96} + q^{-98} +2 q^{-100} + q^{-102} + q^{-104} + q^{-106} + q^{-108} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{-42} + q^{-44} +2 q^{-46} +4 q^{-48} +5 q^{-50} +7 q^{-52} +8 q^{-54} +8 q^{-56} +7 q^{-58} +5 q^{-60} + q^{-62} -3 q^{-64} -7 q^{-66} -10 q^{-68} -11 q^{-70} -11 q^{-72} -9 q^{-74} -6 q^{-76} -2 q^{-78} + q^{-80} +3 q^{-82} +4 q^{-84} +4 q^{-86} +3 q^{-88} +2 q^{-90} + q^{-92} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} +2 q^{-80} +2 q^{-82} + q^{-84} +2 q^{-86} +2 q^{-88} +2 q^{-90} +2 q^{-92} +2 q^{-94} + q^{-96} +2 q^{-98} + q^{-100} + q^{-104} + q^{-106} - q^{-112} - q^{-114} - q^{-118} -2 q^{-120} -2 q^{-122} - q^{-124} - q^{-126} -2 q^{-128} -2 q^{-130} -2 q^{-132} - q^{-134} - q^{-136} -2 q^{-138} -2 q^{-140} - q^{-142} - q^{-146} - q^{-148} + q^{-160} + q^{-162} + q^{-168} + q^{-170} + q^{-180} }[/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["10 124"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+7 z^6+14 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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 1, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{10}+q^6+q^4 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -7 z^4 a^{-10} +21 z^2 a^{-8} -14 z^2 a^{-10} +z^2 a^{-12} +7 a^{-8} -8 a^{-10} +2 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^{-9} +z^7 a^{-11} -8 z^6 a^{-8} -8 z^6 a^{-10} -7 z^5 a^{-9} -7 z^5 a^{-11} +21 z^4 a^{-8} +21 z^4 a^{-10} +14 z^3 a^{-9} +14 z^3 a^{-11} -21 z^2 a^{-8} -22 z^2 a^{-10} -z^2 a^{-12} -8 z a^{-9} -8 z a^{-11} +7 a^{-8} +8 a^{-10} +2 a^{-12} }[/math]