10 156

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 35, -2 }
Jones polynomial	[math]\displaystyle{ -q^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^2 z^6-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^2-2 z^2-a^4+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^4 z^8+a^2 z^8+a^5 z^7+4 a^3 z^7+3 a z^7-a^4 z^6+2 a^2 z^6+3 z^6-a^5 z^5-9 a^3 z^5-7 a z^5+z^5 a^{-1} +3 a^6 z^4+2 a^4 z^4-9 a^2 z^4-8 z^4+a^7 z^3+4 a^5 z^3+8 a^3 z^3+3 a z^3-2 z^3 a^{-1} -2 a^6 z^2+a^4 z^2+7 a^2 z^2+4 z^2-a^7 z-2 a^5 z-2 a^3 z-a z-a^4-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ q^{-4} - q^{-6} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-q^{98}+q^{96}-2 q^{90}+2 q^{88}+2 q^{86}-6 q^{84}+11 q^{82}-16 q^{80}+10 q^{78}-3 q^{76}-13 q^{74}+29 q^{72}-35 q^{70}+27 q^{68}-8 q^{66}-16 q^{64}+35 q^{62}-36 q^{60}+23 q^{58}-20 q^{54}+28 q^{52}-21 q^{50}+2 q^{48}+21 q^{46}-34 q^{44}+32 q^{42}-18 q^{40}-5 q^{38}+27 q^{36}-42 q^{34}+42 q^{32}-31 q^{30}+11 q^{28}+15 q^{26}-35 q^{24}+44 q^{22}-33 q^{20}+15 q^{18}+10 q^{16}-28 q^{14}+31 q^{12}-16 q^{10}-3 q^8+25 q^6-32 q^4+24 q^2-1-22 q^{-2} +36 q^{-4} -35 q^{-6} +22 q^{-8} -3 q^{-10} -16 q^{-12} +25 q^{-14} -22 q^{-16} +16 q^{-18} -5 q^{-20} -3 q^{-22} +5 q^{-24} -7 q^{-26} +4 q^{-28} -2 q^{-30} + q^{-32} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_156.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+2 q^{11}-2 q^9+q^7+2 q- q^{-1} +2 q^{-3} - q^{-5} }[/math]
2	[math]\displaystyle{ -2 q^{34}+2 q^{32}+5 q^{30}-7 q^{28}-2 q^{26}+10 q^{24}-6 q^{22}-6 q^{20}+8 q^{18}-5 q^{14}+q^{12}+5 q^{10}-3 q^8-5 q^6+8 q^4+2 q^2-8+6 q^{-2} +6 q^{-4} -8 q^{-6} - q^{-8} +6 q^{-10} -2 q^{-12} -2 q^{-14} + q^{-16} }[/math]
3	[math]\displaystyle{ q^{71}-q^{67}-5 q^{65}-q^{63}+11 q^{61}+9 q^{59}-10 q^{57}-24 q^{55}+6 q^{53}+35 q^{51}+4 q^{49}-40 q^{47}-22 q^{45}+39 q^{43}+34 q^{41}-28 q^{39}-36 q^{37}+14 q^{35}+35 q^{33}-q^{31}-28 q^{29}-11 q^{27}+20 q^{25}+17 q^{23}-13 q^{21}-27 q^{19}+7 q^{17}+33 q^{15}-39 q^{11}-5 q^9+42 q^7+18 q^5-37 q^3-29 q+30 q^{-1} +35 q^{-3} -12 q^{-5} -37 q^{-7} -3 q^{-9} +30 q^{-11} +15 q^{-13} -16 q^{-15} -18 q^{-17} +3 q^{-19} +14 q^{-21} + q^{-23} -6 q^{-25} -3 q^{-27} +2 q^{-29} +2 q^{-31} - q^{-33} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ q^{-4} - q^{-6} }[/math]
1,1	[math]\displaystyle{ q^{52}-2 q^{50}-2 q^{48}+2 q^{46}+2 q^{44}-4 q^{42}+18 q^{40}-36 q^{38}+62 q^{36}-90 q^{34}+104 q^{32}-114 q^{30}+99 q^{28}-68 q^{26}+26 q^{24}+32 q^{22}-84 q^{20}+132 q^{18}-170 q^{16}+186 q^{14}-194 q^{12}+176 q^{10}-138 q^8+94 q^6-32 q^4-16 q^2+70-96 q^{-2} +107 q^{-4} -104 q^{-6} +84 q^{-8} -62 q^{-10} +40 q^{-12} -22 q^{-14} +10 q^{-16} -4 q^{-18} + q^{-20} }[/math]
2,0	[math]\displaystyle{ -q^{48}+q^{42}+3 q^{40}-2 q^{38}-q^{36}+q^{34}+2 q^{32}-2 q^{30}-5 q^{28}+3 q^{26}+q^{24}-3 q^{22}-q^{20}+3 q^{18}-q^{16}-q^{14}+2 q^{12}+q^6+5 q^4-q^2+5 q^{-2} + q^{-4} -4 q^{-6} - q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} - q^{-16} + q^{-18} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{42}-q^{40}-q^{38}+4 q^{36}-3 q^{34}-4 q^{32}+7 q^{30}-4 q^{28}-4 q^{26}+6 q^{24}-3 q^{22}-2 q^{20}+2 q^{18}+q^{16}-q^{12}+3 q^{10}+3 q^8-5 q^6+3 q^4+6 q^2-5+3 q^{-2} +4 q^{-4} -5 q^{-6} +2 q^{-8} -2 q^{-12} + q^{-14} }[/math]
1,0,0	[math]\displaystyle{ -q^{23}+q^{21}-2 q^{19}+q^{17}-q^{15}+q^{13}+q^9+q^7+2 q^3+2 q^{-1} - q^{-3} + q^{-5} - q^{-7} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{42}+q^{40}-3 q^{38}+6 q^{36}-7 q^{34}+8 q^{32}-7 q^{30}+6 q^{28}-4 q^{26}+3 q^{22}-8 q^{20}+10 q^{18}-13 q^{16}+14 q^{14}-13 q^{12}+13 q^{10}-7 q^8+5 q^6+q^4-2 q^2+5-7 q^{-2} +8 q^{-4} -7 q^{-6} +6 q^{-8} -4 q^{-10} +2 q^{-12} - q^{-14} }[/math]
1,0	[math]\displaystyle{ q^{68}-q^{64}-2 q^{62}+5 q^{58}+2 q^{56}-5 q^{54}-6 q^{52}+2 q^{50}+8 q^{48}+q^{46}-8 q^{44}-4 q^{42}+6 q^{40}+5 q^{38}-4 q^{36}-6 q^{34}+2 q^{32}+6 q^{30}-6 q^{26}-q^{24}+5 q^{22}+3 q^{20}-3 q^{18}-4 q^{16}+4 q^{14}+5 q^{12}-2 q^{10}-7 q^8+2 q^6+8 q^4+4 q^2-6-6 q^{-2} +5 q^{-4} +8 q^{-6} - q^{-8} -6 q^{-10} -2 q^{-12} +4 q^{-14} +2 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24} }[/math]

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-q^{98}+q^{96}-2 q^{90}+2 q^{88}+2 q^{86}-6 q^{84}+11 q^{82}-16 q^{80}+10 q^{78}-3 q^{76}-13 q^{74}+29 q^{72}-35 q^{70}+27 q^{68}-8 q^{66}-16 q^{64}+35 q^{62}-36 q^{60}+23 q^{58}-20 q^{54}+28 q^{52}-21 q^{50}+2 q^{48}+21 q^{46}-34 q^{44}+32 q^{42}-18 q^{40}-5 q^{38}+27 q^{36}-42 q^{34}+42 q^{32}-31 q^{30}+11 q^{28}+15 q^{26}-35 q^{24}+44 q^{22}-33 q^{20}+15 q^{18}+10 q^{16}-28 q^{14}+31 q^{12}-16 q^{10}-3 q^8+25 q^6-32 q^4+24 q^2-1-22 q^{-2} +36 q^{-4} -35 q^{-6} +22 q^{-8} -3 q^{-10} -16 q^{-12} +25 q^{-14} -22 q^{-16} +16 q^{-18} -5 q^{-20} -3 q^{-22} +5 q^{-24} -7 q^{-26} +4 q^{-28} -2 q^{-30} + q^{-32} }[/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["10 156"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ a^2 z^6-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^2-2 z^2-a^4+2 a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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