10 129

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 25, 0 }
Jones polynomial	[math]\displaystyle{ -q^3+2 q^2-3 q+5-4 q^{-1} +4 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4-a^4+z^4 a^2+2 z^2 a^2+a^2+z^4+2 z^2+2-z^2 a^{-2} - a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+3 a z^7+z^7 a^{-1} +2 a^4 z^6-2 a^2 z^6-4 z^6+a^5 z^5-6 a^3 z^5-11 a z^5-4 z^5 a^{-1} -6 a^4 z^4+2 z^4 a^{-2} +8 z^4-3 a^5 z^3+4 a^3 z^3+15 a z^3+9 z^3 a^{-1} +z^3 a^{-3} +3 a^4 z^2+2 a^2 z^2-3 z^2 a^{-2} -4 z^2+a^5 z-a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} -a^4-a^2+ a^{-2} +2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}-q^{10}+q^8+q^4+2 q^2+1+2 q^{-2} - q^{-4} - q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-q^{78}+2 q^{76}-3 q^{74}+2 q^{72}-q^{70}-3 q^{68}+6 q^{66}-8 q^{64}+8 q^{62}-7 q^{60}-q^{58}+6 q^{56}-11 q^{54}+12 q^{52}-7 q^{50}+6 q^{46}-9 q^{44}+7 q^{42}-q^{40}-7 q^{38}+11 q^{36}-10 q^{34}+4 q^{32}+6 q^{30}-13 q^{28}+16 q^{26}-12 q^{24}+5 q^{22}+2 q^{20}-10 q^{18}+14 q^{16}-12 q^{14}+9 q^{12}-3 q^8+10 q^6-8 q^4+6 q^2+2-4 q^{-2} +10 q^{-4} -6 q^{-6} + q^{-8} +10 q^{-10} -12 q^{-12} +13 q^{-14} -7 q^{-16} -4 q^{-18} +8 q^{-20} -11 q^{-22} +9 q^{-24} -5 q^{-26} - q^{-28} +3 q^{-30} -5 q^{-32} +2 q^{-34} - q^{-36} - q^{-38} - q^{-44} + q^{-52} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_129.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^9-q^7+q^5+q+2 q^{-1} - q^{-3} + q^{-5} - q^{-7} }[/math]
2	[math]\displaystyle{ q^{32}-q^{30}-q^{28}+3 q^{26}-q^{24}-4 q^{22}+2 q^{20}+2 q^{18}-4 q^{16}+4 q^{12}-2 q^{10}-q^8+3 q^6+q^4-q^2+1+5 q^{-2} -3 q^{-4} -3 q^{-6} +4 q^{-8} - q^{-10} -3 q^{-12} +2 q^{-14} + q^{-16} - q^{-18} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{61}+q^{59}-q^{57}-2 q^{55}+q^{53}+5 q^{51}-6 q^{47}-3 q^{45}+5 q^{43}+8 q^{41}-2 q^{39}-11 q^{37}-4 q^{35}+9 q^{33}+10 q^{31}-9 q^{29}-14 q^{27}+4 q^{25}+15 q^{23}-q^{21}-13 q^{19}-q^{17}+13 q^{15}+3 q^{13}-8 q^{11}-4 q^9+6 q^7+5 q^5-q^3-7 q+12 q^{-3} +4 q^{-5} -10 q^{-7} -11 q^{-9} +10 q^{-11} +12 q^{-13} -6 q^{-15} -14 q^{-17} + q^{-19} +12 q^{-21} +4 q^{-23} -8 q^{-25} -5 q^{-27} +3 q^{-29} +5 q^{-31} -3 q^{-35} - q^{-37} + q^{-41} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}-q^{10}+q^8+q^4+2 q^2+1+2 q^{-2} - q^{-4} - q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+4 q^{40}-8 q^{38}+15 q^{36}-18 q^{34}+24 q^{32}-32 q^{30}+29 q^{28}-24 q^{26}+14 q^{24}-4 q^{22}-19 q^{20}+32 q^{18}-48 q^{16}+54 q^{14}-54 q^{12}+58 q^{10}-38 q^8+36 q^6-13 q^4+2 q^2+14-26 q^{-2} +30 q^{-4} -38 q^{-6} +32 q^{-8} -22 q^{-10} +15 q^{-12} -8 q^{-14} +2 q^{-16} +4 q^{-18} -3 q^{-20} -2 q^{-24} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{38}+2 q^{34}+2 q^{32}-2 q^{30}-3 q^{28}-q^{26}-q^{24}-3 q^{22}-3 q^{20}+q^{18}+2 q^{16}+q^{14}+q^{12}+3 q^{10}+2 q^8+2 q^6+4 q^4+q^2+2+ q^{-2} + q^{-4} -4 q^{-6} -3 q^{-8} + q^{-10} + q^{-12} -2 q^{-14} - q^{-16} +2 q^{-18} + q^{-20} - q^{-24} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-q^{32}+q^{28}-2 q^{26}+q^{24}-4 q^{20}+q^{18}-3 q^{14}+q^{12}+q^{10}+q^6+3 q^4+4 q^2+4+2 q^{-2} +5 q^{-4} -2 q^{-6} -3 q^{-8} + q^{-10} -4 q^{-12} -3 q^{-14} + q^{-16} + q^{-22} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{17}-q^{13}+q^{11}+q^7+q^5+2 q^3+2 q+ q^{-1} +2 q^{-3} - q^{-5} - q^{-9} - q^{-13} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}+q^{38}-2 q^{34}-q^{32}+q^{30}-2 q^{28}-4 q^{26}+2 q^{22}-3 q^{20}-3 q^{18}+2 q^{16}-q^{14}-4 q^{12}+3 q^8+3 q^6+5 q^4+12 q^2+9+5 q^{-2} +5 q^{-4} +3 q^{-6} -6 q^{-8} -7 q^{-10} -3 q^{-12} -4 q^{-14} -5 q^{-16} -2 q^{-18} +2 q^{-20} + q^{-22} + q^{-26} + q^{-28} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{22}-q^{20}-q^{16}+q^{14}+q^{10}+q^8+q^6+2 q^4+2 q^2+2+ q^{-2} +2 q^{-4} - q^{-6} - q^{-10} - q^{-12} - q^{-16} }[/math]

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-q^{78}+2 q^{76}-3 q^{74}+2 q^{72}-q^{70}-3 q^{68}+6 q^{66}-8 q^{64}+8 q^{62}-7 q^{60}-q^{58}+6 q^{56}-11 q^{54}+12 q^{52}-7 q^{50}+6 q^{46}-9 q^{44}+7 q^{42}-q^{40}-7 q^{38}+11 q^{36}-10 q^{34}+4 q^{32}+6 q^{30}-13 q^{28}+16 q^{26}-12 q^{24}+5 q^{22}+2 q^{20}-10 q^{18}+14 q^{16}-12 q^{14}+9 q^{12}-3 q^8+10 q^6-8 q^4+6 q^2+2-4 q^{-2} +10 q^{-4} -6 q^{-6} + q^{-8} +10 q^{-10} -12 q^{-12} +13 q^{-14} -7 q^{-16} -4 q^{-18} +8 q^{-20} -11 q^{-22} +9 q^{-24} -5 q^{-26} - q^{-28} +3 q^{-30} -5 q^{-32} +2 q^{-34} - q^{-36} - q^{-38} - q^{-44} + q^{-52} }[/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["10 129"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+3 a z^7+z^7 a^{-1} +2 a^4 z^6-2 a^2 z^6-4 z^6+a^5 z^5-6 a^3 z^5-11 a z^5-4 z^5 a^{-1} -6 a^4 z^4+2 z^4 a^{-2} +8 z^4-3 a^5 z^3+4 a^3 z^3+15 a z^3+9 z^3 a^{-1} +z^3 a^{-3} +3 a^4 z^2+2 a^2 z^2-3 z^2 a^{-2} -4 z^2+a^5 z-a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} -a^4-a^2+ a^{-2} +2 }[/math]