10 162

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t^2+9 t-11+9 t^{-1} -3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -3 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 35, -2 }
Jones polynomial	[math]\displaystyle{ 2 q-3+5 q^{-1} -6 q^{-2} +6 q^{-3} -6 q^{-4} +4 q^{-5} -2 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6+a^6-z^4 a^4-z^2 a^4-2 z^4 a^2-5 z^2 a^2-3 a^2+2 z^2+3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8-2 z^2 a^8+2 z^5 a^7-3 z^3 a^7+3 z^6 a^6-6 z^4 a^6+5 z^2 a^6-a^6+3 z^7 a^5-8 z^5 a^5+12 z^3 a^5-5 z a^5+z^8 a^4+z^6 a^4-4 z^4 a^4+5 z^2 a^4+4 z^7 a^3-11 z^5 a^3+15 z^3 a^3-7 z a^3+z^8 a^2-2 z^6 a^2+6 z^4 a^2-9 z^2 a^2+3 a^2+z^7 a-z^5 a-2 z a+3 z^4-7 z^2+3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}+2 q^{16}-q^{14}-q^{10}-2 q^8-2 q^4+2 q^2+1+ q^{-2} +2 q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+2 q^{106}-q^{104}-2 q^{102}+6 q^{100}-8 q^{98}+10 q^{96}-10 q^{94}+6 q^{92}+q^{90}-12 q^{88}+23 q^{86}-26 q^{84}+21 q^{82}-9 q^{80}-10 q^{78}+25 q^{76}-30 q^{74}+27 q^{72}-8 q^{70}-10 q^{68}+24 q^{66}-21 q^{64}+9 q^{62}+14 q^{60}-28 q^{58}+32 q^{56}-19 q^{54}-3 q^{52}+26 q^{50}-41 q^{48}+42 q^{46}-33 q^{44}+8 q^{42}+12 q^{40}-33 q^{38}+37 q^{36}-34 q^{34}+15 q^{32}+2 q^{30}-20 q^{28}+24 q^{26}-20 q^{24}+3 q^{22}+15 q^{20}-26 q^{18}+24 q^{16}-9 q^{14}-12 q^{12}+32 q^{10}-35 q^8+28 q^6-9 q^4-11 q^2+25-26 q^{-2} +22 q^{-4} -8 q^{-6} - q^{-8} +9 q^{-10} -10 q^{-12} +8 q^{-14} - q^{-16} + q^{-18} + q^{-20} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_162.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-q^{13}+2 q^{11}-2 q^9-q^3+2 q- q^{-1} +2 q^{-3} }[/math]
2	[math]\displaystyle{ q^{42}-q^{40}+3 q^{36}-4 q^{34}-3 q^{32}+8 q^{30}-3 q^{28}-8 q^{26}+9 q^{24}+q^{22}-7 q^{20}+3 q^{18}+4 q^{16}-q^{14}-4 q^{12}+5 q^{10}+3 q^8-9 q^6+3 q^4+7 q^2-8- q^{-2} +7 q^{-4} -3 q^{-6} -3 q^{-8} +3 q^{-10} + q^{-12} }[/math]
3	[math]\displaystyle{ q^{81}-q^{79}+q^{75}-3 q^{71}-q^{69}+6 q^{67}+3 q^{65}-11 q^{63}-10 q^{61}+13 q^{59}+23 q^{57}-9 q^{55}-34 q^{53}+q^{51}+38 q^{49}+13 q^{47}-40 q^{45}-19 q^{43}+32 q^{41}+25 q^{39}-19 q^{37}-26 q^{35}+7 q^{33}+21 q^{31}+3 q^{29}-19 q^{27}-13 q^{25}+14 q^{23}+22 q^{21}-13 q^{19}-29 q^{17}+9 q^{15}+37 q^{13}-2 q^{11}-36 q^9-7 q^7+38 q^5+17 q^3-30 q-24 q^{-1} +17 q^{-3} +27 q^{-5} -5 q^{-7} -23 q^{-9} -4 q^{-11} +14 q^{-13} +8 q^{-15} -5 q^{-17} -8 q^{-19} + q^{-21} +2 q^{-23} +2 q^{-25} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}+2 q^{16}-q^{14}-q^{10}-2 q^8-2 q^4+2 q^2+1+ q^{-2} +2 q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{60}-2 q^{58}+4 q^{56}-8 q^{54}+15 q^{52}-20 q^{50}+28 q^{48}-44 q^{46}+62 q^{44}-70 q^{42}+82 q^{40}-90 q^{38}+74 q^{36}-54 q^{34}+8 q^{32}+32 q^{30}-82 q^{28}+132 q^{26}-152 q^{24}+180 q^{22}-170 q^{20}+158 q^{18}-124 q^{16}+82 q^{14}-41 q^{12}-14 q^{10}+52 q^8-84 q^6+96 q^4-102 q^2+90-66 q^{-2} +45 q^{-4} -28 q^{-6} +14 q^{-8} +2 q^{-12} +2 q^{-14} }[/math]
2,0	[math]\displaystyle{ q^{56}+q^{50}+2 q^{48}-q^{46}-4 q^{44}-q^{42}+4 q^{40}-q^{38}-5 q^{36}+2 q^{32}-q^{30}-5 q^{28}+3 q^{26}+4 q^{24}+q^{22}+4 q^{20}+4 q^{18}+3 q^{12}-2 q^{10}-5 q^8-q^6+2 q^4-4 q^2-6+2 q^{-2} +3 q^{-4} - q^{-6} +2 q^{-10} +4 q^{-12} + q^{-14} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-q^{46}+2 q^{42}-3 q^{40}+6 q^{36}-6 q^{34}-2 q^{32}+6 q^{30}-6 q^{28}+6 q^{24}+2 q^{22}+2 q^{20}+2 q^{18}+2 q^{16}-3 q^{14}-8 q^{12}-2 q^8-9 q^6+5 q^4+4 q^2-3+6 q^{-2} +3 q^{-4} - q^{-6} +3 q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{29}+q^{25}+2 q^{21}-q^{19}+q^{17}-q^{15}-q^{13}-2 q^{11}-2 q^9-q^7-2 q^5+2 q^3+q+3 q^{-1} + q^{-3} +2 q^{-5} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{62}-q^{58}+q^{56}+2 q^{54}-q^{52}-2 q^{50}+3 q^{48}+5 q^{46}-4 q^{44}-5 q^{42}+2 q^{40}-3 q^{38}-10 q^{36}-2 q^{34}+6 q^{32}+2 q^{30}+5 q^{28}+13 q^{26}+11 q^{24}+3 q^{20}+3 q^{18}-12 q^{16}-12 q^{14}-3 q^{12}-7 q^{10}-12 q^8-2 q^6+5 q^4+2 q^2+1+7 q^{-2} +7 q^{-4} +2 q^{-6} +2 q^{-8} +3 q^{-10} }[/math]
1,0,0,0	[math]\displaystyle{ q^{36}+q^{32}+q^{30}+2 q^{26}-q^{24}+q^{22}-q^{18}-q^{16}-2 q^{14}-2 q^{12}-3 q^{10}-q^8-2 q^6+2 q^4+q^2+3+3 q^{-2} + q^{-4} +2 q^{-6} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{48}-q^{46}+2 q^{44}-4 q^{42}+5 q^{40}-6 q^{38}+8 q^{36}-6 q^{34}+6 q^{32}-2 q^{30}+4 q^{26}-8 q^{24}+10 q^{22}-14 q^{20}+12 q^{18}-14 q^{16}+9 q^{14}-8 q^{12}+4 q^{10}-3 q^6+7 q^4-6 q^2+7-6 q^{-2} +7 q^{-4} -3 q^{-6} +3 q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{78}-q^{74}-q^{72}+q^{70}+3 q^{68}-4 q^{64}-3 q^{62}+3 q^{60}+7 q^{58}-q^{56}-8 q^{54}-4 q^{52}+6 q^{50}+5 q^{48}-4 q^{46}-6 q^{44}+2 q^{42}+7 q^{40}+q^{38}-4 q^{36}+7 q^{32}+3 q^{30}-3 q^{28}-4 q^{26}+3 q^{24}+3 q^{22}-3 q^{20}-7 q^{18}-q^{16}+5 q^{14}-8 q^{10}-6 q^8+6 q^6+7 q^4-q^2-6- q^{-2} +6 q^{-4} +5 q^{-6} -2 q^{-8} -2 q^{-10} + q^{-12} +3 q^{-14} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{66}-q^{64}+q^{62}-2 q^{60}+3 q^{58}-4 q^{56}+4 q^{54}-4 q^{52}+7 q^{50}-6 q^{48}+4 q^{46}-4 q^{44}+3 q^{42}-q^{40}-3 q^{38}+3 q^{36}-3 q^{34}+11 q^{32}-6 q^{30}+12 q^{28}-8 q^{26}+12 q^{24}-9 q^{22}+5 q^{20}-12 q^{18}-7 q^{14}-3 q^{12}-3 q^{10}-5 q^8+6 q^6-3 q^4+8 q^2-2+9 q^{-2} -2 q^{-4} +6 q^{-6} -2 q^{-8} +3 q^{-10} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+2 q^{106}-q^{104}-2 q^{102}+6 q^{100}-8 q^{98}+10 q^{96}-10 q^{94}+6 q^{92}+q^{90}-12 q^{88}+23 q^{86}-26 q^{84}+21 q^{82}-9 q^{80}-10 q^{78}+25 q^{76}-30 q^{74}+27 q^{72}-8 q^{70}-10 q^{68}+24 q^{66}-21 q^{64}+9 q^{62}+14 q^{60}-28 q^{58}+32 q^{56}-19 q^{54}-3 q^{52}+26 q^{50}-41 q^{48}+42 q^{46}-33 q^{44}+8 q^{42}+12 q^{40}-33 q^{38}+37 q^{36}-34 q^{34}+15 q^{32}+2 q^{30}-20 q^{28}+24 q^{26}-20 q^{24}+3 q^{22}+15 q^{20}-26 q^{18}+24 q^{16}-9 q^{14}-12 q^{12}+32 q^{10}-35 q^8+28 q^6-9 q^4-11 q^2+25-26 q^{-2} +22 q^{-4} -8 q^{-6} - q^{-8} +9 q^{-10} -10 q^{-12} +8 q^{-14} - q^{-16} + q^{-18} + q^{-20} }[/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["10 162"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -3 t^2+9 t-11+9 t^{-1} -3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -3 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]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^4 a^8-2 z^2 a^8+2 z^5 a^7-3 z^3 a^7+3 z^6 a^6-6 z^4 a^6+5 z^2 a^6-a^6+3 z^7 a^5-8 z^5 a^5+12 z^3 a^5-5 z a^5+z^8 a^4+z^6 a^4-4 z^4 a^4+5 z^2 a^4+4 z^7 a^3-11 z^5 a^3+15 z^3 a^3-7 z a^3+z^8 a^2-2 z^6 a^2+6 z^4 a^2-9 z^2 a^2+3 a^2+z^7 a-z^5 a-2 z a+3 z^4-7 z^2+3 }[/math]