10 158

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+4 t^2-10 t+15-10 t^{-1} +4 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-2 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 45, 0 }
Jones polynomial	[math]\displaystyle{ 2 q^4-4 q^3+6 q^2-8 q+8-7 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -4 z^4+2 a^2 z^2+z^2 a^{-2} -6 z^2+2 a^2+ a^{-4} -2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8+4 a z^7+5 z^7 a^{-1} +z^7 a^{-3} +5 a^2 z^6+z^6 a^{-2} +6 z^6+3 a^3 z^5-5 a z^5-7 z^5 a^{-1} +z^5 a^{-3} +a^4 z^4-8 a^2 z^4-z^4 a^{-2} +3 z^4 a^{-4} -13 z^4-3 a^3 z^3+3 a z^3+2 z^3 a^{-1} -4 z^3 a^{-3} -a^4 z^2+5 a^2 z^2-2 z^2 a^{-2} -5 z^2 a^{-4} +9 z^2-a z+z a^{-1} +2 z a^{-3} -2 a^2+ a^{-4} -2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-q^{10}+2 q^8+q^6-q^4+2 q^2-2+ q^{-2} -2 q^{-4} - q^{-6} + q^{-8} - q^{-10} +2 q^{-12} + q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+13 q^{52}-22 q^{50}+30 q^{48}-30 q^{46}+13 q^{44}+9 q^{42}-37 q^{40}+62 q^{38}-64 q^{36}+48 q^{34}-7 q^{32}-35 q^{30}+66 q^{28}-67 q^{26}+42 q^{24}+q^{22}-39 q^{20}+53 q^{18}-36 q^{16}+q^{14}+47 q^{12}-75 q^{10}+70 q^8-35 q^6-22 q^4+69 q^2-99+94 q^{-2} -61 q^{-4} +11 q^{-6} +39 q^{-8} -77 q^{-10} +85 q^{-12} -65 q^{-14} +20 q^{-16} +22 q^{-18} -52 q^{-20} +52 q^{-22} -24 q^{-24} -13 q^{-26} +50 q^{-28} -63 q^{-30} +45 q^{-32} -4 q^{-34} -45 q^{-36} +77 q^{-38} -77 q^{-40} +50 q^{-42} -6 q^{-44} -31 q^{-46} +52 q^{-48} -51 q^{-50} +37 q^{-52} -11 q^{-54} -8 q^{-56} +15 q^{-58} -17 q^{-60} +11 q^{-62} -2 q^{-64} + q^{-68} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_158.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-2 q^7+3 q^5-q^3+q-2 q^{-3} +2 q^{-5} -2 q^{-7} +2 q^{-9} }[/math]
2	[math]\displaystyle{ q^{26}-2 q^{24}+7 q^{20}-7 q^{18}-7 q^{16}+16 q^{14}-3 q^{12}-14 q^{10}+13 q^8+3 q^6-12 q^4+3 q^2+7-2 q^{-2} -8 q^{-4} +8 q^{-6} +8 q^{-8} -15 q^{-10} +4 q^{-12} +14 q^{-14} -13 q^{-16} -4 q^{-18} +11 q^{-20} -4 q^{-22} -5 q^{-24} +3 q^{-26} + q^{-28} }[/math]
3	[math]\displaystyle{ q^{51}-2 q^{49}+4 q^{45}+q^{43}-9 q^{41}-8 q^{39}+19 q^{37}+21 q^{35}-24 q^{33}-41 q^{31}+16 q^{29}+68 q^{27}-2 q^{25}-83 q^{23}-23 q^{21}+84 q^{19}+48 q^{17}-73 q^{15}-63 q^{13}+52 q^{11}+67 q^9-26 q^7-59 q^5+q^3+52 q+20 q^{-1} -38 q^{-3} -41 q^{-5} +28 q^{-7} +57 q^{-9} -14 q^{-11} -74 q^{-13} - q^{-15} +85 q^{-17} +18 q^{-19} -83 q^{-21} -43 q^{-23} +73 q^{-25} +59 q^{-27} -50 q^{-29} -65 q^{-31} +23 q^{-33} +60 q^{-35} +3 q^{-37} -44 q^{-39} -14 q^{-41} +22 q^{-43} +16 q^{-45} -6 q^{-47} -12 q^{-49} +2 q^{-53} +2 q^{-55} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-q^{10}+2 q^8+q^6-q^4+2 q^2-2+ q^{-2} -2 q^{-4} - q^{-6} + q^{-8} - q^{-10} +2 q^{-12} + q^{-14} }[/math]
1,1	[math]\displaystyle{ q^{36}-4 q^{34}+10 q^{32}-20 q^{30}+40 q^{28}-68 q^{26}+108 q^{24}-154 q^{22}+201 q^{20}-240 q^{18}+256 q^{16}-236 q^{14}+179 q^{12}-80 q^{10}-42 q^8+178 q^6-313 q^4+412 q^2-480+494 q^{-2} -462 q^{-4} +394 q^{-6} -278 q^{-8} +154 q^{-10} -12 q^{-12} -106 q^{-14} +200 q^{-16} -260 q^{-18} +274 q^{-20} -256 q^{-22} +206 q^{-24} -150 q^{-26} +94 q^{-28} -52 q^{-30} +22 q^{-32} -4 q^{-34} +2 q^{-38} }[/math]
2,0	[math]\displaystyle{ q^{32}-q^{30}+4 q^{26}-4 q^{22}-q^{20}+6 q^{18}+2 q^{16}-9 q^{14}+2 q^{12}+6 q^{10}-3 q^8-6 q^6+2 q^4+2 q^2-3+2 q^{-2} +3 q^{-4} - q^{-6} -2 q^{-8} +8 q^{-10} + q^{-12} -5 q^{-14} +5 q^{-16} +6 q^{-18} -3 q^{-20} -7 q^{-22} +2 q^{-26} -4 q^{-28} -4 q^{-30} +2 q^{-32} +3 q^{-34} +2 q^{-36} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}-2 q^{26}+5 q^{22}-6 q^{20}+11 q^{16}-10 q^{14}+q^{12}+12 q^{10}-8 q^8-q^6+6 q^4-3 q^2-4-2 q^{-2} +3 q^{-4} - q^{-6} -7 q^{-8} +10 q^{-10} +4 q^{-12} -11 q^{-14} +9 q^{-16} +2 q^{-18} -10 q^{-20} +6 q^{-22} + q^{-24} -3 q^{-26} +3 q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{15}-q^{13}+3 q^{11}+2 q^7-q^5+q^3-q- q^{-1} -2 q^{-5} -2 q^{-9} +2 q^{-11} - q^{-13} +2 q^{-15} + q^{-17} + q^{-19} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-7 q^{22}+10 q^{20}-12 q^{18}+13 q^{16}-10 q^{14}+9 q^{12}-2 q^{10}-2 q^8+9 q^6-14 q^4+19 q^2-24+22 q^{-2} -21 q^{-4} +15 q^{-6} -11 q^{-8} +4 q^{-10} +2 q^{-12} -7 q^{-14} +11 q^{-16} -12 q^{-18} +12 q^{-20} -10 q^{-22} +9 q^{-24} -5 q^{-26} +3 q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{46}-2 q^{42}-2 q^{40}+2 q^{38}+6 q^{36}+q^{34}-8 q^{32}-6 q^{30}+7 q^{28}+12 q^{26}-2 q^{24}-13 q^{22}-4 q^{20}+13 q^{18}+8 q^{16}-7 q^{14}-9 q^{12}+4 q^{10}+9 q^8-q^6-10 q^4-2 q^2+9+2 q^{-2} -8 q^{-4} -6 q^{-6} +6 q^{-8} +7 q^{-10} -4 q^{-12} -9 q^{-14} +4 q^{-16} +13 q^{-18} +2 q^{-20} -12 q^{-22} -7 q^{-24} +10 q^{-26} +11 q^{-28} -5 q^{-30} -12 q^{-32} -2 q^{-34} +8 q^{-36} +5 q^{-38} -4 q^{-40} -4 q^{-42} + q^{-44} +3 q^{-46} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{38}-2 q^{36}+2 q^{34}-3 q^{32}+6 q^{30}-8 q^{28}+8 q^{26}-8 q^{24}+12 q^{22}-9 q^{20}+8 q^{18}-4 q^{16}+6 q^{14}+3 q^{12}-5 q^{10}+6 q^8-9 q^6+14 q^4-18 q^2+14-20 q^{-2} +16 q^{-4} -15 q^{-6} +11 q^{-8} -12 q^{-10} +8 q^{-12} + q^{-16} +3 q^{-18} -6 q^{-20} +10 q^{-22} -9 q^{-24} +9 q^{-26} -10 q^{-28} +9 q^{-30} -6 q^{-32} +6 q^{-34} -4 q^{-36} +3 q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+13 q^{52}-22 q^{50}+30 q^{48}-30 q^{46}+13 q^{44}+9 q^{42}-37 q^{40}+62 q^{38}-64 q^{36}+48 q^{34}-7 q^{32}-35 q^{30}+66 q^{28}-67 q^{26}+42 q^{24}+q^{22}-39 q^{20}+53 q^{18}-36 q^{16}+q^{14}+47 q^{12}-75 q^{10}+70 q^8-35 q^6-22 q^4+69 q^2-99+94 q^{-2} -61 q^{-4} +11 q^{-6} +39 q^{-8} -77 q^{-10} +85 q^{-12} -65 q^{-14} +20 q^{-16} +22 q^{-18} -52 q^{-20} +52 q^{-22} -24 q^{-24} -13 q^{-26} +50 q^{-28} -63 q^{-30} +45 q^{-32} -4 q^{-34} -45 q^{-36} +77 q^{-38} -77 q^{-40} +50 q^{-42} -6 q^{-44} -31 q^{-46} +52 q^{-48} -51 q^{-50} +37 q^{-52} -11 q^{-54} -8 q^{-56} +15 q^{-58} -17 q^{-60} +11 q^{-62} -2 q^{-64} + q^{-68} }[/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["10 158"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+4 t^2-10 t+15-10 t^{-1} +4 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-2 z^4-3 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]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 2 q^4-4 q^3+6 q^2-8 q+8-7 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -4 z^4+2 a^2 z^2+z^2 a^{-2} -6 z^2+2 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+4 a z^7+5 z^7 a^{-1} +z^7 a^{-3} +5 a^2 z^6+z^6 a^{-2} +6 z^6+3 a^3 z^5-5 a z^5-7 z^5 a^{-1} +z^5 a^{-3} +a^4 z^4-8 a^2 z^4-z^4 a^{-2} +3 z^4 a^{-4} -13 z^4-3 a^3 z^3+3 a z^3+2 z^3 a^{-1} -4 z^3 a^{-3} -a^4 z^2+5 a^2 z^2-2 z^2 a^{-2} -5 z^2 a^{-4} +9 z^2-a z+z a^{-1} +2 z a^{-3} -2 a^2+ a^{-4} -2 }[/math]