10 136

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 10_136.

A1 Invariants.

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

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{20}-q^{18}+q^{14}+q^{10}+q^6-q^4-q^2- q^{-2} + q^{-6} +3 q^{-8} + q^{-10} +3 q^{-12} + q^{-14} - q^{-18} - q^{-20} - q^{-24} }[/math]
1,0,0	[math]\displaystyle{ q^{13}+q^9-q^3-q- q^{-1} + q^{-5} +2 q^{-7} +3 q^{-9} + q^{-11} + q^{-13} - q^{-15} - q^{-17} - q^{-19} }[/math]

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{46}-q^{44}+2 q^{42}-2 q^{40}+q^{38}-2 q^{34}+6 q^{32}-4 q^{30}+3 q^{28}-2 q^{24}+3 q^{22}-2 q^{20}-q^{18}+3 q^{16}-3 q^{14}+3 q^{10}-6 q^8+6 q^6-7 q^4+3-5 q^{-2} +4 q^{-4} -3 q^{-6} +3 q^{-8} + q^{-12} - q^{-14} +2 q^{-18} + q^{-20} + q^{-24} +3 q^{-26} +4 q^{-30} -6 q^{-32} +6 q^{-34} -2 q^{-36} +4 q^{-40} -7 q^{-42} +6 q^{-44} -2 q^{-48} + q^{-50} -2 q^{-52} -2 q^{-54} +3 q^{-56} -3 q^{-58} + q^{-60} + q^{-62} -3 q^{-64} +3 q^{-66} -3 q^{-68} + q^{-70} + q^{-72} -2 q^{-74} + q^{-76} }[/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["10 136"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-z^4 }[/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]=

{ 15, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^4+2 q^3-2 q^2+3 q-2+2 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^2+2 z^2 a^{-2} -3 z^2+a^2+3 a^{-2} - a^{-4} -2 }[/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+3 z^7 a^{-1} +z^7 a^{-3} +a^2 z^6-4 z^6 a^{-2} -3 z^6-9 a z^5-14 z^5 a^{-1} -5 z^5 a^{-3} -4 a^2 z^4+2 z^4 a^{-2} -2 z^4+9 a z^3+16 z^3 a^{-1} +7 z^3 a^{-3} +3 a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-2 a z-4 z a^{-1} -2 z a^{-3} -a^2-3 a^{-2} - a^{-4} -2 }[/math]