10 164

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^2-11 t+17-11 t^{-1} +3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^4+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^3+5 q^2-6 q+8-8 q^{-1} +7 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^4+z^4 a^2-a^2+2 z^4+4 z^2+3-2 z^2 a^{-2} - a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^2 z^8+2 z^8+4 a^3 z^7+7 a z^7+3 z^7 a^{-1} +3 a^4 z^6-a^2 z^6+z^6 a^{-2} -3 z^6+a^5 z^5-10 a^3 z^5-17 a z^5-6 z^5 a^{-1} -7 a^4 z^4-3 a^2 z^4+4 z^4 a^{-2} +8 z^4-2 a^5 z^3+7 a^3 z^3+16 a z^3+10 z^3 a^{-1} +3 z^3 a^{-3} +3 a^4 z^2-6 z^2 a^{-2} -9 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-2 q^{10}+q^8-q^6+2 q^2+3 q^{-2} - q^{-4} + q^{-6} + q^{-8} -2 q^{-10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+6 q^{72}-5 q^{70}-2 q^{68}+15 q^{66}-26 q^{64}+36 q^{62}-32 q^{60}+11 q^{58}+17 q^{56}-50 q^{54}+70 q^{52}-61 q^{50}+29 q^{48}+17 q^{46}-54 q^{44}+69 q^{42}-52 q^{40}+12 q^{38}+30 q^{36}-61 q^{34}+50 q^{32}-15 q^{30}-32 q^{28}+69 q^{26}-75 q^{24}+52 q^{22}-7 q^{20}-40 q^{18}+74 q^{16}-93 q^{14}+78 q^{12}-34 q^{10}-12 q^8+56 q^6-77 q^4+75 q^2-36-6 q^{-2} +40 q^{-4} -58 q^{-6} +45 q^{-8} + q^{-10} -43 q^{-12} +69 q^{-14} -56 q^{-16} +21 q^{-18} +28 q^{-20} -66 q^{-22} +72 q^{-24} -55 q^{-26} +22 q^{-28} +13 q^{-30} -40 q^{-32} +45 q^{-34} -33 q^{-36} +16 q^{-38} + q^{-40} -10 q^{-42} +7 q^{-44} -9 q^{-46} +5 q^{-48} -2 q^{-50} + q^{-52} + q^{-54} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_164.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-2 q^7+2 q^5-q^3+2 q^{-1} - q^{-3} +3 q^{-5} -2 q^{-7} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+7 q^{26}-4 q^{24}-10 q^{22}+11 q^{20}+4 q^{18}-15 q^{16}+7 q^{14}+11 q^{12}-11 q^{10}-q^8+9 q^6-2 q^4-8 q^2+3+10 q^{-2} -11 q^{-4} -4 q^{-6} +17 q^{-8} -6 q^{-10} -9 q^{-12} +12 q^{-14} -7 q^{-18} + q^{-20} + q^{-22} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+q^{59}-4 q^{57}-4 q^{55}+7 q^{53}+15 q^{51}-8 q^{49}-29 q^{47}-5 q^{45}+40 q^{43}+30 q^{41}-40 q^{39}-57 q^{37}+23 q^{35}+77 q^{33}+6 q^{31}-85 q^{29}-33 q^{27}+75 q^{25}+54 q^{23}-58 q^{21}-65 q^{19}+40 q^{17}+64 q^{15}-19 q^{13}-56 q^{11}+q^9+52 q^7+16 q^5-42 q^3-38 q+33 q^{-1} +58 q^{-3} -17 q^{-5} -78 q^{-7} -4 q^{-9} +88 q^{-11} +28 q^{-13} -80 q^{-15} -51 q^{-17} +61 q^{-19} +63 q^{-21} -30 q^{-23} -59 q^{-25} +8 q^{-27} +38 q^{-29} +11 q^{-31} -22 q^{-33} -13 q^{-35} +6 q^{-37} +5 q^{-39} +2 q^{-41} -2 q^{-43} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{14}+q^{12}-2 q^{10}+q^8-q^6+2 q^2+3 q^{-2} - q^{-4} + q^{-6} + q^{-8} -2 q^{-10} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+10 q^{40}-22 q^{38}+46 q^{36}-80 q^{34}+122 q^{32}-172 q^{30}+214 q^{28}-236 q^{26}+224 q^{24}-176 q^{22}+94 q^{20}+18 q^{18}-148 q^{16}+264 q^{14}-353 q^{12}+418 q^{10}-434 q^8+418 q^6-352 q^4+256 q^2-144+16 q^{-2} +92 q^{-4} -182 q^{-6} +242 q^{-8} -246 q^{-10} +227 q^{-12} -176 q^{-14} +128 q^{-16} -80 q^{-18} +35 q^{-20} -14 q^{-22} +2 q^{-24} -2 q^{-28} +2 q^{-30} }[/math]
2,0	[math]\displaystyle{ q^{42}-q^{40}-2 q^{38}+q^{36}+5 q^{34}+2 q^{32}-8 q^{30}-5 q^{28}+4 q^{26}+4 q^{24}-6 q^{22}-3 q^{20}+8 q^{18}+6 q^{16}-3 q^{14}-2 q^{12}+5 q^{10}-3 q^8-2 q^6-2 q^2-1+4 q^{-2} +5 q^{-4} -6 q^{-6} +9 q^{-10} +4 q^{-12} -9 q^{-14} - q^{-16} +7 q^{-18} + q^{-20} -5 q^{-22} -4 q^{-24} +2 q^{-26} + q^{-28} }[/math]

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-4 q^{30}+7 q^{28}-10 q^{26}+12 q^{24}-12 q^{22}+10 q^{20}-8 q^{18}+4 q^{16}+2 q^{14}-8 q^{12}+15 q^{10}-19 q^8+22 q^6-22 q^4+21 q^2-17+11 q^{-2} -3 q^{-4} - q^{-6} +7 q^{-8} -9 q^{-10} +13 q^{-12} -12 q^{-14} +11 q^{-16} -9 q^{-18} +5 q^{-20} -3 q^{-22} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+2 q^{48}+5 q^{46}-q^{44}-8 q^{42}-3 q^{40}+10 q^{38}+9 q^{36}-7 q^{34}-12 q^{32}+q^{30}+13 q^{28}+4 q^{26}-12 q^{24}-8 q^{22}+6 q^{20}+8 q^{18}-4 q^{16}-8 q^{14}+2 q^{12}+9 q^{10}+q^8-8 q^6-2 q^4+9 q^2+7-6 q^{-2} -8 q^{-4} +7 q^{-6} +11 q^{-8} -2 q^{-10} -12 q^{-12} - q^{-14} +12 q^{-16} +8 q^{-18} -9 q^{-20} -11 q^{-22} +2 q^{-24} +9 q^{-26} + q^{-28} -6 q^{-30} -4 q^{-32} + q^{-34} +3 q^{-36} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{46}-2 q^{44}+2 q^{42}-4 q^{40}+6 q^{38}-8 q^{36}+9 q^{34}-9 q^{32}+11 q^{30}-8 q^{28}+7 q^{26}-5 q^{24}+3 q^{22}-q^{20}-8 q^{18}+6 q^{16}-12 q^{14}+12 q^{12}-18 q^{10}+17 q^8-14 q^6+21 q^4-11 q^2+16-6 q^{-2} +11 q^{-4} + q^{-6} -2 q^{-8} +2 q^{-10} -7 q^{-12} +8 q^{-14} -11 q^{-16} +7 q^{-18} -12 q^{-20} +9 q^{-22} -6 q^{-24} +4 q^{-26} -4 q^{-28} +3 q^{-30} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+6 q^{72}-5 q^{70}-2 q^{68}+15 q^{66}-26 q^{64}+36 q^{62}-32 q^{60}+11 q^{58}+17 q^{56}-50 q^{54}+70 q^{52}-61 q^{50}+29 q^{48}+17 q^{46}-54 q^{44}+69 q^{42}-52 q^{40}+12 q^{38}+30 q^{36}-61 q^{34}+50 q^{32}-15 q^{30}-32 q^{28}+69 q^{26}-75 q^{24}+52 q^{22}-7 q^{20}-40 q^{18}+74 q^{16}-93 q^{14}+78 q^{12}-34 q^{10}-12 q^8+56 q^6-77 q^4+75 q^2-36-6 q^{-2} +40 q^{-4} -58 q^{-6} +45 q^{-8} + q^{-10} -43 q^{-12} +69 q^{-14} -56 q^{-16} +21 q^{-18} +28 q^{-20} -66 q^{-22} +72 q^{-24} -55 q^{-26} +22 q^{-28} +13 q^{-30} -40 q^{-32} +45 q^{-34} -33 q^{-36} +16 q^{-38} + q^{-40} -10 q^{-42} +7 q^{-44} -9 q^{-46} +5 q^{-48} -2 q^{-50} + q^{-52} + q^{-54} }[/math]

.

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["10 164"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 3 t^2-11 t+17-11 t^{-1} +3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 3 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]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a^2 z^8+2 z^8+4 a^3 z^7+7 a z^7+3 z^7 a^{-1} +3 a^4 z^6-a^2 z^6+z^6 a^{-2} -3 z^6+a^5 z^5-10 a^3 z^5-17 a z^5-6 z^5 a^{-1} -7 a^4 z^4-3 a^2 z^4+4 z^4 a^{-2} +8 z^4-2 a^5 z^3+7 a^3 z^3+16 a z^3+10 z^3 a^{-1} +3 z^3 a^{-3} +3 a^4 z^2-6 z^2 a^{-2} -9 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]