10 153

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-t^2-t+3- t^{-1} - t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+5 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 1, 0 }
Jones polynomial	[math]\displaystyle{ -q^4+q^3-q^2+q+1+ q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6-z^4 a^{-2} +6 z^4-a^4 z^2-a^2 z^2-4 z^2 a^{-2} +10 z^2-a^4-a^2-3 a^{-2} +6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 z^6 a^{-2} -7 z^6+a^5 z^5+a^3 z^5-7 a z^5-13 z^5 a^{-1} -6 z^5 a^{-3} -4 a^4 z^4+10 z^4 a^{-2} +14 z^4-4 a^5 z^3-4 a^3 z^3+12 a z^3+22 z^3 a^{-1} +10 z^3 a^{-3} +3 a^4 z^2-2 a^2 z^2-7 z^2 a^{-2} -12 z^2+3 a^5 z+2 a^3 z-6 a z-10 z a^{-1} -5 z a^{-3} -a^4+a^2+3 a^{-2} +6 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}-q^{12}-q^{10}+2 q^4+2 q^2+3+2 q^{-2} - q^{-8} - q^{-10} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}+q^{76}-q^{74}+q^{70}-2 q^{68}+q^{64}-3 q^{62}+q^{60}-q^{58}-4 q^{56}+5 q^{54}-5 q^{52}+q^{50}+q^{48}-5 q^{46}+5 q^{44}-3 q^{42}-2 q^{40}+2 q^{38}-4 q^{36}+q^{34}+3 q^{32}-5 q^{30}+3 q^{28}-q^{24}+2 q^{22}-2 q^{20}+2 q^{18}+4 q^{14}-2 q^{12}+3 q^{10}+4 q^8-q^6+5 q^4-q^2+2+6 q^{-2} - q^{-4} +2 q^{-6} +4 q^{-8} -2 q^{-10} +6 q^{-12} - q^{-14} -3 q^{-16} +4 q^{-18} -4 q^{-20} +2 q^{-22} -3 q^{-26} + q^{-28} - q^{-30} -3 q^{-32} -2 q^{-36} - q^{-38} -3 q^{-42} - q^{-48} - q^{-52} + q^{-56} + q^{-60} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_153.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^3+q+2 q^{-1} - q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-q^{28}+q^{24}-3 q^{20}-q^{18}+2 q^{16}-2 q^{14}-q^{12}+2 q^{10}+q^6+q^4+2 q^2+1+ q^{-2} +2 q^{-4} -2 q^{-8} + q^{-10} + q^{-12} -2 q^{-14} - q^{-16} - q^{-24} + q^{-28} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{59}+q^{57}-2 q^{53}-q^{51}+q^{49}+4 q^{47}+3 q^{45}-2 q^{43}-5 q^{41}-2 q^{39}+5 q^{37}+4 q^{35}-4 q^{33}-7 q^{31}-q^{29}+5 q^{27}+q^{25}-4 q^{23}-2 q^{21}+q^{19}+q^{17}+q^{15}+q^9+q^7-q^5+q^3+5 q+4 q^{-1} -2 q^{-3} -3 q^{-5} +6 q^{-7} +5 q^{-9} -3 q^{-11} -7 q^{-13} - q^{-15} +5 q^{-17} +2 q^{-19} -3 q^{-21} -5 q^{-23} -2 q^{-25} +3 q^{-27} +3 q^{-29} -3 q^{-33} -2 q^{-35} + q^{-37} + q^{-39} + q^{-41} - q^{-47} + q^{-51} + q^{-53} - q^{-57} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}+q^{30}-q^{24}-3 q^{22}-q^{20}-3 q^{18}-3 q^{16}-q^{14}-2 q^{12}-2 q^{10}+q^8+3 q^6+5 q^4+8 q^2+10+7 q^{-2} +4 q^{-4} - q^{-6} -2 q^{-8} -7 q^{-10} -5 q^{-12} -3 q^{-14} -3 q^{-16} + q^{-20} + q^{-22} + q^{-26} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{17}-q^{15}-q^{13}-q^{11}+2 q^5+3 q^3+4 q+3 q^{-1} +3 q^{-3} - q^{-7} - q^{-9} -2 q^{-11} - q^{-13} - q^{-15} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}+q^{40}+2 q^{38}+q^{36}+q^{34}+q^{32}-q^{30}-3 q^{28}-3 q^{26}-5 q^{24}-6 q^{22}-7 q^{20}-6 q^{18}-7 q^{16}-8 q^{14}-2 q^{12}+q^{10}+4 q^8+13 q^6+19 q^4+20 q^2+20+17 q^{-2} +8 q^{-4} -8 q^{-8} -11 q^{-10} -15 q^{-12} -13 q^{-14} -8 q^{-16} -5 q^{-18} -2 q^{-20} +2 q^{-22} +3 q^{-24} +2 q^{-26} +2 q^{-28} + q^{-30} + q^{-32} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{22}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}+2 q^6+3 q^4+5 q^2+4+4 q^{-2} +3 q^{-4} - q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} - q^{-16} - q^{-18} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}-q^{30}-q^{24}+q^{22}-q^{20}+q^{18}-q^{16}+q^{14}+q^8-q^6+3 q^4+2+ q^{-2} +2 q^{-4} + q^{-6} + q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-20} - q^{-22} - q^{-26} }[/math]
1,0	[math]\displaystyle{ q^{56}+q^{48}-q^{44}+q^{40}-q^{38}-2 q^{36}-q^{34}-q^{30}-3 q^{28}-2 q^{26}-2 q^{20}-q^{18}+q^{14}+q^{12}+q^{10}+2 q^8+4 q^6+3 q^4+4 q^2+3+4 q^{-2} +3 q^{-4} +3 q^{-6} - q^{-8} - q^{-14} -3 q^{-16} -3 q^{-18} - q^{-20} - q^{-22} -3 q^{-24} -2 q^{-26} + q^{-36} + q^{-44} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{46}+q^{42}+q^{38}-2 q^{34}-q^{32}-3 q^{30}-q^{28}-3 q^{26}-q^{24}-3 q^{22}-q^{20}-q^{18}-2 q^{16}-2 q^{14}-2 q^{12}+q^{10}+q^8+6 q^6+6 q^4+10 q^2+10+10 q^{-2} +6 q^{-4} +4 q^{-6} -4 q^{-10} -5 q^{-12} -7 q^{-14} -5 q^{-16} -6 q^{-18} -2 q^{-20} -2 q^{-22} + q^{-26} + q^{-28} + q^{-30} + q^{-34} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}+q^{76}-q^{74}+q^{70}-2 q^{68}+q^{64}-3 q^{62}+q^{60}-q^{58}-4 q^{56}+5 q^{54}-5 q^{52}+q^{50}+q^{48}-5 q^{46}+5 q^{44}-3 q^{42}-2 q^{40}+2 q^{38}-4 q^{36}+q^{34}+3 q^{32}-5 q^{30}+3 q^{28}-q^{24}+2 q^{22}-2 q^{20}+2 q^{18}+4 q^{14}-2 q^{12}+3 q^{10}+4 q^8-q^6+5 q^4-q^2+2+6 q^{-2} - q^{-4} +2 q^{-6} +4 q^{-8} -2 q^{-10} +6 q^{-12} - q^{-14} -3 q^{-16} +4 q^{-18} -4 q^{-20} +2 q^{-22} -3 q^{-26} + q^{-28} - q^{-30} -3 q^{-32} -2 q^{-36} - q^{-38} -3 q^{-42} - q^{-48} - q^{-52} + q^{-56} + q^{-60} }[/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["10 153"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+5 z^4+4 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, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^4+q^3-q^2+q+1+ q^{-2} - q^{-3} + q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6-z^4 a^{-2} +6 z^4-a^4 z^2-a^2 z^2-4 z^2 a^{-2} +10 z^2-a^4-a^2-3 a^{-2} +6 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 z^6 a^{-2} -7 z^6+a^5 z^5+a^3 z^5-7 a z^5-13 z^5 a^{-1} -6 z^5 a^{-3} -4 a^4 z^4+10 z^4 a^{-2} +14 z^4-4 a^5 z^3-4 a^3 z^3+12 a z^3+22 z^3 a^{-1} +10 z^3 a^{-3} +3 a^4 z^2-2 a^2 z^2-7 z^2 a^{-2} -12 z^2+3 a^5 z+2 a^3 z-6 a z-10 z a^{-1} -5 z a^{-3} -a^4+a^2+3 a^{-2} +6 }[/math]