10 134

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-4 t^2+4 t-3+4 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+8 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 23, 6 }
Jones polynomial	[math]\displaystyle{ q^{11}-3 q^{10}+3 q^9-4 q^8+4 q^7-3 q^6+3 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} +4 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-6} +3 z^2 a^{-8} -4 z^2 a^{-10} +3 a^{-6} -3 a^{-10} + a^{-12} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +3 z^7 a^{-9} +2 z^7 a^{-11} +z^6 a^{-6} -3 z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} -3 z^5 a^{-7} -11 z^5 a^{-9} -8 z^5 a^{-11} -5 z^4 a^{-6} +z^4 a^{-8} +5 z^4 a^{-10} -z^4 a^{-12} +11 z^3 a^{-9} +14 z^3 a^{-11} +3 z^3 a^{-13} +7 z^2 a^{-6} -7 z^2 a^{-10} +z^2 a^{-12} +z^2 a^{-14} +2 z a^{-7} -4 z a^{-9} -8 z a^{-11} -2 z a^{-13} -3 a^{-6} +3 a^{-10} + a^{-12} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-10} +2 q^{-14} + q^{-16} +2 q^{-18} + q^{-20} + q^{-24} -2 q^{-26} - q^{-28} -2 q^{-30} - q^{-32} + q^{-38} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-50} +2 q^{-54} - q^{-56} +2 q^{-58} +5 q^{-64} -5 q^{-66} +8 q^{-68} -4 q^{-70} +6 q^{-74} -7 q^{-76} +10 q^{-78} -5 q^{-80} +2 q^{-82} +5 q^{-84} -6 q^{-86} +6 q^{-88} -5 q^{-92} +9 q^{-94} -6 q^{-96} + q^{-98} +5 q^{-100} -9 q^{-102} +12 q^{-104} -8 q^{-106} +3 q^{-108} + q^{-110} -7 q^{-112} +8 q^{-114} -10 q^{-116} +5 q^{-118} -4 q^{-120} -3 q^{-122} +4 q^{-124} -8 q^{-126} +2 q^{-128} - q^{-130} -6 q^{-132} +5 q^{-134} -6 q^{-136} -2 q^{-138} +6 q^{-140} -9 q^{-142} +9 q^{-144} -4 q^{-146} -2 q^{-148} +7 q^{-150} -7 q^{-152} +6 q^{-154} - q^{-156} +3 q^{-160} - q^{-162} + q^{-164} +2 q^{-168} -2 q^{-174} - q^{-180} + q^{-182} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_134.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-5} +2 q^{-9} + q^{-13} - q^{-17} -2 q^{-21} + q^{-23} }[/math]
2	[math]\displaystyle{ q^{-10} +3 q^{-16} + q^{-18} -2 q^{-20} +3 q^{-22} +3 q^{-24} -2 q^{-26} - q^{-28} +2 q^{-30} -2 q^{-32} -3 q^{-34} + q^{-36} -3 q^{-40} +2 q^{-44} - q^{-46} -2 q^{-48} +3 q^{-50} + q^{-52} -3 q^{-54} +2 q^{-56} + q^{-58} - q^{-60} }[/math]
3	[math]\displaystyle{ q^{-15} + q^{-21} +3 q^{-23} + q^{-25} -2 q^{-27} - q^{-29} +5 q^{-31} +5 q^{-33} - q^{-35} -6 q^{-37} +6 q^{-41} +5 q^{-43} -4 q^{-45} -9 q^{-47} -2 q^{-49} +7 q^{-51} +4 q^{-53} -8 q^{-55} -9 q^{-57} +3 q^{-59} +8 q^{-61} -3 q^{-63} -9 q^{-65} + q^{-67} +10 q^{-69} - q^{-71} -6 q^{-73} +7 q^{-77} + q^{-79} -4 q^{-81} -4 q^{-83} +3 q^{-85} +6 q^{-87} +2 q^{-89} -8 q^{-91} -6 q^{-93} +8 q^{-95} +9 q^{-97} -5 q^{-99} -10 q^{-101} +2 q^{-103} +7 q^{-105} +3 q^{-107} -6 q^{-109} -3 q^{-111} + q^{-113} +2 q^{-115} + q^{-117} - q^{-119} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-10} +2 q^{-14} + q^{-16} +2 q^{-18} + q^{-20} + q^{-24} -2 q^{-26} - q^{-28} -2 q^{-30} - q^{-32} + q^{-38} }[/math]
1,1	[math]\displaystyle{ q^{-20} +4 q^{-24} -2 q^{-26} +12 q^{-28} -8 q^{-30} +24 q^{-32} -16 q^{-34} +26 q^{-36} -18 q^{-38} +14 q^{-40} -6 q^{-42} -16 q^{-44} +14 q^{-46} -38 q^{-48} +32 q^{-50} -45 q^{-52} +36 q^{-54} -32 q^{-56} +28 q^{-58} -12 q^{-60} +6 q^{-62} +10 q^{-64} -16 q^{-66} +23 q^{-68} -24 q^{-70} +20 q^{-72} -16 q^{-74} +10 q^{-76} -4 q^{-78} +2 q^{-84} -2 q^{-90} + q^{-92} }[/math]
2,0	[math]\displaystyle{ q^{-20} +2 q^{-26} +3 q^{-28} + q^{-30} +2 q^{-32} +4 q^{-34} +5 q^{-36} +2 q^{-38} +2 q^{-40} +2 q^{-42} -3 q^{-46} -2 q^{-48} -4 q^{-50} -5 q^{-52} -4 q^{-54} -5 q^{-56} -4 q^{-58} -2 q^{-60} + q^{-62} +2 q^{-64} + q^{-66} +2 q^{-68} +3 q^{-70} + q^{-72} + q^{-78} +2 q^{-80} + q^{-82} - q^{-84} - q^{-88} - q^{-90} - q^{-92} + q^{-96} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-20} +2 q^{-24} +3 q^{-26} +2 q^{-28} +4 q^{-30} +4 q^{-32} + q^{-34} +4 q^{-36} - q^{-40} -2 q^{-44} -5 q^{-46} -4 q^{-48} -5 q^{-50} -5 q^{-52} -3 q^{-54} - q^{-56} +4 q^{-58} +2 q^{-60} +4 q^{-62} +4 q^{-64} - q^{-66} - q^{-68} + q^{-70} -2 q^{-72} - q^{-74} + q^{-76} }[/math]
1,0,0	[math]\displaystyle{ q^{-15} +2 q^{-19} + q^{-21} +3 q^{-23} + q^{-25} +2 q^{-27} + q^{-29} -2 q^{-35} - q^{-37} -3 q^{-39} - q^{-41} -2 q^{-43} + q^{-49} + q^{-51} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{-30} +2 q^{-34} +3 q^{-36} +4 q^{-38} +4 q^{-40} +7 q^{-42} +5 q^{-44} +6 q^{-46} +5 q^{-48} +2 q^{-50} +2 q^{-52} +3 q^{-54} -3 q^{-58} -2 q^{-60} -4 q^{-62} -10 q^{-64} -14 q^{-66} -10 q^{-68} -11 q^{-70} -11 q^{-72} -2 q^{-74} +4 q^{-76} +4 q^{-78} +10 q^{-80} +12 q^{-82} +7 q^{-84} +3 q^{-86} +3 q^{-88} -5 q^{-92} -4 q^{-94} - q^{-98} -2 q^{-100} + q^{-102} + q^{-104} }[/math]
1,0,0,0	[math]\displaystyle{ q^{-20} +2 q^{-24} + q^{-26} +3 q^{-28} +2 q^{-30} +2 q^{-32} +2 q^{-34} + q^{-36} + q^{-38} - q^{-40} -2 q^{-44} - q^{-46} -3 q^{-48} -2 q^{-50} -2 q^{-52} -2 q^{-54} + q^{-60} + q^{-62} + q^{-64} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{-20} +2 q^{-24} - q^{-26} +4 q^{-28} -2 q^{-30} +4 q^{-32} - q^{-34} +2 q^{-36} - q^{-40} +2 q^{-42} -4 q^{-44} +5 q^{-46} -6 q^{-48} +5 q^{-50} -5 q^{-52} +3 q^{-54} -3 q^{-56} -2 q^{-62} +2 q^{-64} -3 q^{-66} +3 q^{-68} -3 q^{-70} +2 q^{-72} - q^{-74} + q^{-76} }[/math]
1,0	[math]\displaystyle{ q^{-30} +2 q^{-38} +2 q^{-40} + q^{-42} - q^{-44} +2 q^{-46} +4 q^{-48} +3 q^{-50} -2 q^{-52} +3 q^{-56} +4 q^{-58} -3 q^{-62} - q^{-64} +2 q^{-66} -3 q^{-70} -3 q^{-72} - q^{-74} - q^{-76} -3 q^{-78} -4 q^{-80} -2 q^{-82} + q^{-84} - q^{-86} -3 q^{-88} -2 q^{-90} +3 q^{-92} +3 q^{-94} - q^{-98} +3 q^{-100} +4 q^{-102} + q^{-104} -2 q^{-106} - q^{-108} + q^{-110} +2 q^{-112} - q^{-114} -2 q^{-116} - q^{-118} + q^{-122} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{-30} +2 q^{-34} + q^{-36} +5 q^{-38} + q^{-40} +6 q^{-42} + q^{-44} +6 q^{-46} + q^{-48} +2 q^{-50} + q^{-54} + q^{-56} -2 q^{-58} +2 q^{-60} -5 q^{-62} + q^{-64} -8 q^{-66} - q^{-68} -10 q^{-70} -2 q^{-72} -8 q^{-74} -2 q^{-78} +3 q^{-80} +3 q^{-82} +3 q^{-84} +5 q^{-86} + q^{-88} +4 q^{-90} -2 q^{-92} + q^{-94} -3 q^{-96} +2 q^{-98} -2 q^{-100} - q^{-104} + q^{-106} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-50} +2 q^{-54} - q^{-56} +2 q^{-58} +5 q^{-64} -5 q^{-66} +8 q^{-68} -4 q^{-70} +6 q^{-74} -7 q^{-76} +10 q^{-78} -5 q^{-80} +2 q^{-82} +5 q^{-84} -6 q^{-86} +6 q^{-88} -5 q^{-92} +9 q^{-94} -6 q^{-96} + q^{-98} +5 q^{-100} -9 q^{-102} +12 q^{-104} -8 q^{-106} +3 q^{-108} + q^{-110} -7 q^{-112} +8 q^{-114} -10 q^{-116} +5 q^{-118} -4 q^{-120} -3 q^{-122} +4 q^{-124} -8 q^{-126} +2 q^{-128} - q^{-130} -6 q^{-132} +5 q^{-134} -6 q^{-136} -2 q^{-138} +6 q^{-140} -9 q^{-142} +9 q^{-144} -4 q^{-146} -2 q^{-148} +7 q^{-150} -7 q^{-152} +6 q^{-154} - q^{-156} +3 q^{-160} - q^{-162} + q^{-164} +2 q^{-168} -2 q^{-174} - q^{-180} + q^{-182} }[/math]

.

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["10 134"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+8 z^4+6 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]=

{ 23, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{11}-3 q^{10}+3 q^9-4 q^8+4 q^7-3 q^6+3 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} +4 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-6} +3 z^2 a^{-8} -4 z^2 a^{-10} +3 a^{-6} -3 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} +3 z^7 a^{-9} +2 z^7 a^{-11} +z^6 a^{-6} -3 z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} -3 z^5 a^{-7} -11 z^5 a^{-9} -8 z^5 a^{-11} -5 z^4 a^{-6} +z^4 a^{-8} +5 z^4 a^{-10} -z^4 a^{-12} +11 z^3 a^{-9} +14 z^3 a^{-11} +3 z^3 a^{-13} +7 z^2 a^{-6} -7 z^2 a^{-10} +z^2 a^{-12} +z^2 a^{-14} +2 z a^{-7} -4 z a^{-9} -8 z a^{-11} -2 z a^{-13} -3 a^{-6} +3 a^{-10} + a^{-12} }[/math]