10 138

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-5 t^2+8 t-7+8 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 35, 2 }
Jones polynomial	[math]\displaystyle{ 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} -6 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+3 z^6 a^{-2} +3 z^6 a^{-4} +z^6-7 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4-13 z^4 a^{-2} -5 z^4 a^{-4} -12 z^4+6 a z^3+8 z^3 a^{-1} +5 z^3 a^{-3} +3 z^3 a^{-5} +5 a^2 z^2+10 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2 a^{-6} +12 z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{10}+q^8+q^4-q^2- q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-2 q^{36}-4 q^{34}+14 q^{32}-18 q^{30}+20 q^{28}-11 q^{26}-4 q^{24}+19 q^{22}-27 q^{20}+29 q^{18}-16 q^{16}+17 q^{12}-25 q^{10}+19 q^8-5 q^6-13 q^4+20 q^2-21+7 q^{-2} +8 q^{-4} -25 q^{-6} +33 q^{-8} -29 q^{-10} +14 q^{-12} +5 q^{-14} -25 q^{-16} +35 q^{-18} -34 q^{-20} +24 q^{-22} -4 q^{-24} -12 q^{-26} +28 q^{-28} -27 q^{-30} +17 q^{-32} + q^{-34} -15 q^{-36} +22 q^{-38} -17 q^{-40} +16 q^{-44} -24 q^{-46} +25 q^{-48} -15 q^{-50} -5 q^{-52} +18 q^{-54} -26 q^{-56} +22 q^{-58} -13 q^{-60} + q^{-62} +9 q^{-64} -12 q^{-66} +11 q^{-68} -8 q^{-70} +5 q^{-72} + q^{-74} -3 q^{-76} + q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_138.

A1 Invariants.

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

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{10}+q^8+q^4-q^2- q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{28}+q^{26}-2 q^{22}+3 q^{18}+q^{16}-3 q^{14}-q^{12}+3 q^{10}+2 q^8-3 q^6+3 q^2-2 q^{-2} - q^{-6} -2 q^{-8} + q^{-10} +2 q^{-16} +7 q^{-18} +2 q^{-20} -4 q^{-22} + q^{-26} -3 q^{-28} -4 q^{-30} +2 q^{-34} +2 q^{-36} - q^{-48} + q^{-52} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{20}-q^{18}+2 q^{16}+2 q^{14}-2 q^{12}+4 q^{10}+q^8-4 q^6+4 q^4-2 q^2-5+3 q^{-2} -3 q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} - q^{-14} +4 q^{-18} -4 q^{-20} +5 q^{-24} -5 q^{-26} - q^{-28} +4 q^{-30} -3 q^{-32} - q^{-34} +3 q^{-36} }[/math]
1,0,0	[math]\displaystyle{ q^{13}+q^{11}+2 q^9+q^5-2 q^3-2 q^{-1} + q^{-7} +2 q^{-9} + q^{-13} -2 q^{-15} -2 q^{-19} + q^{-21} + q^{-25} + q^{-27} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{20}-q^{18}+4 q^{16}-4 q^{14}+6 q^{12}-6 q^{10}+7 q^8-6 q^6+4 q^4-2 q^2-3+5 q^{-2} -8 q^{-4} +11 q^{-6} -12 q^{-8} +14 q^{-10} -11 q^{-12} +9 q^{-14} -6 q^{-16} +2 q^{-18} -4 q^{-22} +5 q^{-24} -7 q^{-26} +7 q^{-28} -6 q^{-30} +5 q^{-32} -3 q^{-34} +3 q^{-36} }[/math]
1,0	[math]\displaystyle{ q^{34}-q^{30}-q^{28}+3 q^{26}+3 q^{24}-2 q^{22}-4 q^{20}+q^{18}+6 q^{16}+3 q^{14}-6 q^{12}-5 q^{10}+4 q^8+7 q^6-q^4-8 q^2-2+5 q^{-2} +3 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +4 q^{-12} - q^{-14} -5 q^{-16} + q^{-18} +6 q^{-20} + q^{-22} -5 q^{-24} -3 q^{-26} +5 q^{-28} +5 q^{-30} -3 q^{-32} -6 q^{-34} +2 q^{-36} +7 q^{-38} + q^{-40} -6 q^{-42} -4 q^{-44} +3 q^{-46} +5 q^{-48} - q^{-50} -4 q^{-52} -2 q^{-54} + q^{-56} +3 q^{-58} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-2 q^{36}-4 q^{34}+14 q^{32}-18 q^{30}+20 q^{28}-11 q^{26}-4 q^{24}+19 q^{22}-27 q^{20}+29 q^{18}-16 q^{16}+17 q^{12}-25 q^{10}+19 q^8-5 q^6-13 q^4+20 q^2-21+7 q^{-2} +8 q^{-4} -25 q^{-6} +33 q^{-8} -29 q^{-10} +14 q^{-12} +5 q^{-14} -25 q^{-16} +35 q^{-18} -34 q^{-20} +24 q^{-22} -4 q^{-24} -12 q^{-26} +28 q^{-28} -27 q^{-30} +17 q^{-32} + q^{-34} -15 q^{-36} +22 q^{-38} -17 q^{-40} +16 q^{-44} -24 q^{-46} +25 q^{-48} -15 q^{-50} -5 q^{-52} +18 q^{-54} -26 q^{-56} +22 q^{-58} -13 q^{-60} + q^{-62} +9 q^{-64} -12 q^{-66} +11 q^{-68} -8 q^{-70} +5 q^{-72} + q^{-74} -3 q^{-76} + q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/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["10 138"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+z^4-3 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]=

{ 35, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 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^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} -6 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3 }[/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+5 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+3 z^6 a^{-2} +3 z^6 a^{-4} +z^6-7 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4-13 z^4 a^{-2} -5 z^4 a^{-4} -12 z^4+6 a z^3+8 z^3 a^{-1} +5 z^3 a^{-3} +3 z^3 a^{-5} +5 a^2 z^2+10 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2 a^{-6} +12 z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3 }[/math]