10 146

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-8 t+13-8 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 33, 0 }
Jones polynomial	[math]\displaystyle{ -q^3+3 q^2-4 q+6-6 q^{-1} +5 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4+z^4 a^2+z^2 a^2+z^4+z^2+1-z^2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+4 a z^7+z^7 a^{-1} +3 a^4 z^6+a^2 z^6-2 z^6+a^5 z^5-8 a^3 z^5-11 a z^5-2 z^5 a^{-1} -8 a^4 z^4-6 a^2 z^4+3 z^4 a^{-2} +5 z^4-2 a^5 z^3+5 a^3 z^3+12 a z^3+6 z^3 a^{-1} +z^3 a^{-3} +3 a^4 z^2+3 a^2 z^2-3 z^2 a^{-2} -3 z^2-a^3 z-3 a z-3 z a^{-1} -z a^{-3} +1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-q^{10}+q^8-q^6+q^2+2 q^{-2} - q^{-4} + q^{-6} + q^{-8} - q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-2 q^{70}-6 q^{68}+16 q^{66}-19 q^{64}+20 q^{62}-13 q^{60}-4 q^{58}+19 q^{56}-29 q^{54}+29 q^{52}-14 q^{50}-4 q^{48}+20 q^{46}-22 q^{44}+16 q^{42}-q^{40}-18 q^{38}+23 q^{36}-21 q^{34}+7 q^{32}+13 q^{30}-31 q^{28}+37 q^{26}-24 q^{24}+10 q^{22}+6 q^{20}-27 q^{18}+34 q^{16}-31 q^{14}+22 q^{12}-3 q^{10}-15 q^8+28 q^6-23 q^4+12 q^2+4-18 q^{-2} +20 q^{-4} -13 q^{-6} - q^{-8} +22 q^{-10} -29 q^{-12} +27 q^{-14} -11 q^{-16} -8 q^{-18} +22 q^{-20} -27 q^{-22} +19 q^{-24} -8 q^{-26} + q^{-28} +8 q^{-30} -12 q^{-32} +8 q^{-34} -3 q^{-36} +2 q^{-38} -2 q^{-42} - q^{-44} + q^{-48} - q^{-50} + q^{-52} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_146.

A1 Invariants.

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

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-q^{10}+q^8-q^6+q^2+2 q^{-2} - q^{-4} + q^{-6} + q^{-8} - q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+10 q^{40}-22 q^{38}+38 q^{36}-54 q^{34}+72 q^{32}-86 q^{30}+85 q^{28}-70 q^{26}+42 q^{24}-4 q^{22}-41 q^{20}+86 q^{18}-122 q^{16}+146 q^{14}-153 q^{12}+146 q^{10}-126 q^8+96 q^6-52 q^4+12 q^2+34-62 q^{-2} +80 q^{-4} -92 q^{-6} +80 q^{-8} -64 q^{-10} +43 q^{-12} -22 q^{-14} +14 q^{-16} -2 q^{-24} -2 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{40}-2 q^{38}+3 q^{34}+3 q^{32}-4 q^{30}-2 q^{28}+2 q^{26}+2 q^{24}-3 q^{22}-4 q^{20}+3 q^{18}+2 q^{16}-q^{14}+4 q^{10}+q^8+q^6+2 q^4-2 q^2-1+ q^{-4} -5 q^{-6} -2 q^{-8} +6 q^{-10} +3 q^{-12} -3 q^{-14} +4 q^{-18} +2 q^{-20} -2 q^{-22} -2 q^{-24} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-2 q^{32}+2 q^{28}-4 q^{26}+4 q^{24}+2 q^{22}-5 q^{20}+4 q^{18}+q^{16}-5 q^{14}+q^{12}+2 q^{10}-2 q^8+q^4+3 q^2- q^{-2} +7 q^{-4} -3 q^{-6} -2 q^{-8} +6 q^{-10} -3 q^{-12} -3 q^{-14} +3 q^{-16} - q^{-18} - q^{-20} + q^{-22} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}+q^{19}+q^{15}-q^{13}+q^{11}-q^9+q^3+q+2 q^{-3} - q^{-5} + q^{-7} + q^{-11} - q^{-13} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-2 q^{70}-6 q^{68}+16 q^{66}-19 q^{64}+20 q^{62}-13 q^{60}-4 q^{58}+19 q^{56}-29 q^{54}+29 q^{52}-14 q^{50}-4 q^{48}+20 q^{46}-22 q^{44}+16 q^{42}-q^{40}-18 q^{38}+23 q^{36}-21 q^{34}+7 q^{32}+13 q^{30}-31 q^{28}+37 q^{26}-24 q^{24}+10 q^{22}+6 q^{20}-27 q^{18}+34 q^{16}-31 q^{14}+22 q^{12}-3 q^{10}-15 q^8+28 q^6-23 q^4+12 q^2+4-18 q^{-2} +20 q^{-4} -13 q^{-6} - q^{-8} +22 q^{-10} -29 q^{-12} +27 q^{-14} -11 q^{-16} -8 q^{-18} +22 q^{-20} -27 q^{-22} +19 q^{-24} -8 q^{-26} + q^{-28} +8 q^{-30} -12 q^{-32} +8 q^{-34} -3 q^{-36} +2 q^{-38} -2 q^{-42} - q^{-44} + q^{-48} - q^{-50} + 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["10 146"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-8 t+13-8 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 33, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^3+3 q^2-4 q+6-6 q^{-1} +5 q^{-2} -4 q^{-3} +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^2 a^4+z^4 a^2+z^2 a^2+z^4+z^2+1-z^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+3 a^3 z^7+4 a z^7+z^7 a^{-1} +3 a^4 z^6+a^2 z^6-2 z^6+a^5 z^5-8 a^3 z^5-11 a z^5-2 z^5 a^{-1} -8 a^4 z^4-6 a^2 z^4+3 z^4 a^{-2} +5 z^4-2 a^5 z^3+5 a^3 z^3+12 a z^3+6 z^3 a^{-1} +z^3 a^{-3} +3 a^4 z^2+3 a^2 z^2-3 z^2 a^{-2} -3 z^2-a^3 z-3 a z-3 z a^{-1} -z a^{-3} +1 }[/math]