10 144

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t^2+10 t-13+10 t^{-1} -3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -3 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature	{ 39, -2 }
Jones polynomial	[math]\displaystyle{ 2 q-3+5 q^{-1} -7 q^{-2} +7 q^{-3} -6 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6-z^4 a^4+2 a^4-2 z^4 a^2-5 z^2 a^2-4 a^2+2 z^2+3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8-z^2 a^8+3 z^5 a^7-4 z^3 a^7+4 z^6 a^6-6 z^4 a^6+2 z^2 a^6+3 z^7 a^5-4 z^5 a^5+4 z^3 a^5-2 z a^5+z^8 a^4+2 z^6 a^4-2 z^4 a^4-2 z^2 a^4+2 a^4+4 z^7 a^3-8 z^5 a^3+8 z^3 a^3-2 z a^3+z^8 a^2-2 z^6 a^2+8 z^4 a^2-12 z^2 a^2+4 a^2+z^7 a-z^5 a+3 z^4-7 z^2+3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}-q^{20}-q^{18}+2 q^{16}+2 q^{12}-2 q^8-q^6-3 q^4+2 q^2+1+ q^{-2} +2 q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-q^{104}-4 q^{102}+13 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+3 q^{92}+12 q^{90}-29 q^{88}+39 q^{86}-36 q^{84}+24 q^{82}-24 q^{78}+38 q^{76}-35 q^{74}+17 q^{72}+4 q^{70}-24 q^{68}+27 q^{66}-13 q^{64}-5 q^{62}+34 q^{60}-45 q^{58}+42 q^{56}-16 q^{54}-18 q^{52}+48 q^{50}-63 q^{48}+57 q^{46}-32 q^{44}+5 q^{42}+27 q^{40}-49 q^{38}+47 q^{36}-34 q^{34}+7 q^{32}+13 q^{30}-34 q^{28}+23 q^{26}-4 q^{24}-12 q^{22}+28 q^{20}-38 q^{18}+22 q^{16}+3 q^{14}-29 q^{12}+44 q^{10}-46 q^8+30 q^6-17 q^2+29-29 q^{-2} +25 q^{-4} -7 q^{-6} -4 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_144.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-2 q^{13}+2 q^{11}-q^9+q^7-2 q^3+2 q- q^{-1} +2 q^{-3} }[/math]
2	[math]\displaystyle{ q^{42}-2 q^{40}-q^{38}+6 q^{36}-4 q^{34}-6 q^{32}+11 q^{30}-q^{28}-10 q^{26}+9 q^{24}+2 q^{22}-9 q^{20}+q^{18}+4 q^{16}-q^{14}-6 q^{12}+6 q^{10}+8 q^8-10 q^6+2 q^4+10 q^2-9-3 q^{-2} +7 q^{-4} -3 q^{-6} -3 q^{-8} +3 q^{-10} + q^{-12} }[/math]
3	[math]\displaystyle{ q^{81}-2 q^{79}-q^{77}+3 q^{75}+3 q^{73}-4 q^{71}-9 q^{69}+8 q^{67}+16 q^{65}-7 q^{63}-26 q^{61}+37 q^{57}+8 q^{55}-44 q^{53}-20 q^{51}+45 q^{49}+32 q^{47}-37 q^{45}-39 q^{43}+26 q^{41}+39 q^{39}-13 q^{37}-34 q^{35}-2 q^{33}+27 q^{31}+15 q^{29}-17 q^{27}-27 q^{25}+11 q^{23}+35 q^{21}-q^{19}-43 q^{17}-11 q^{15}+47 q^{13}+17 q^{11}-43 q^9-29 q^7+38 q^5+36 q^3-23 q-36 q^{-1} +12 q^{-3} +34 q^{-5} + q^{-7} -24 q^{-9} -8 q^{-11} +13 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}-q^{20}-q^{18}+2 q^{16}+2 q^{12}-2 q^8-q^6-3 q^4+2 q^2+1+ q^{-2} +2 q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{60}-4 q^{58}+10 q^{56}-20 q^{54}+34 q^{52}-54 q^{50}+80 q^{48}-104 q^{46}+124 q^{44}-138 q^{42}+138 q^{40}-116 q^{38}+73 q^{36}-12 q^{34}-56 q^{32}+132 q^{30}-203 q^{28}+250 q^{26}-280 q^{24}+274 q^{22}-250 q^{20}+200 q^{18}-132 q^{16}+64 q^{14}+20 q^{12}-72 q^{10}+122 q^8-146 q^6+150 q^4-142 q^2+106-82 q^{-2} +51 q^{-4} -28 q^{-6} +14 q^{-8} +2 q^{-12} +2 q^{-14} }[/math]
2,0	[math]\displaystyle{ q^{56}-q^{54}-2 q^{52}+q^{50}+4 q^{48}+q^{46}-6 q^{44}-2 q^{42}+5 q^{40}+q^{38}-3 q^{36}+3 q^{34}+7 q^{32}-q^{30}-7 q^{28}-3 q^{26}-3 q^{24}-7 q^{22}+4 q^{18}+2 q^{16}+7 q^{14}+11 q^{12}+4 q^{10}-5 q^8-q^6+2 q^4-6 q^2-9+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}-2 q^{46}+4 q^{42}-5 q^{40}+q^{38}+7 q^{36}-9 q^{34}-q^{32}+7 q^{30}-6 q^{28}+7 q^{24}+q^{22}+3 q^{16}-2 q^{14}-8 q^{12}+5 q^{10}-q^8-10 q^6+6 q^4+2 q^2-5+6 q^{-2} +3 q^{-4} - q^{-6} +3 q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{29}-q^{27}-q^{23}+2 q^{21}+3 q^{17}+q^{15}-2 q^{11}-3 q^9-2 q^7-3 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^{60}-2 q^{58}+2 q^{56}+3 q^{54}-2 q^{52}-3 q^{50}+4 q^{48}+3 q^{46}-7 q^{44}-4 q^{42}+7 q^{40}+q^{38}-6 q^{36}+4 q^{34}+6 q^{32}-4 q^{30}-3 q^{28}+5 q^{26}+q^{24}-4 q^{22}+7 q^{20}+9 q^{18}-5 q^{16}-3 q^{14}+5 q^{12}-6 q^{10}-14 q^8-4 q^6+2 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^{34}-q^{28}+2 q^{26}+3 q^{22}+2 q^{20}+q^{18}-2 q^{14}-3 q^{12}-4 q^{10}-2 q^8-3 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}-2 q^{46}+4 q^{44}-6 q^{42}+7 q^{40}-9 q^{38}+9 q^{36}-7 q^{34}+5 q^{32}-q^{30}-2 q^{28}+8 q^{26}-11 q^{24}+15 q^{22}-16 q^{20}+16 q^{18}-15 q^{16}+10 q^{14}-8 q^{12}+q^{10}+q^8-6 q^6+8 q^4-8 q^2+9-6 q^{-2} +7 q^{-4} -3 q^{-6} +3 q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+5 q^{68}-6 q^{64}-4 q^{62}+6 q^{60}+8 q^{58}-3 q^{56}-10 q^{54}-3 q^{52}+8 q^{50}+5 q^{48}-5 q^{46}-6 q^{44}+4 q^{42}+7 q^{40}-6 q^{36}+7 q^{32}+2 q^{30}-6 q^{28}-4 q^{26}+5 q^{24}+4 q^{22}-4 q^{20}-7 q^{18}+3 q^{16}+8 q^{14}-10 q^{10}-5 q^8+7 q^6+7 q^4-3 q^2-8- 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}-2 q^{64}+2 q^{62}-3 q^{60}+5 q^{58}-6 q^{56}+6 q^{54}-6 q^{52}+8 q^{50}-7 q^{48}+3 q^{46}-4 q^{44}+2 q^{42}+q^{40}-5 q^{38}+6 q^{36}-5 q^{34}+14 q^{32}-9 q^{30}+14 q^{28}-11 q^{26}+14 q^{24}-10 q^{22}+6 q^{20}-11 q^{18}+q^{16}-4 q^{14}-4 q^{12}-q^{10}-7 q^8+7 q^6-5 q^4+8 q^2-4+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}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-q^{104}-4 q^{102}+13 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+3 q^{92}+12 q^{90}-29 q^{88}+39 q^{86}-36 q^{84}+24 q^{82}-24 q^{78}+38 q^{76}-35 q^{74}+17 q^{72}+4 q^{70}-24 q^{68}+27 q^{66}-13 q^{64}-5 q^{62}+34 q^{60}-45 q^{58}+42 q^{56}-16 q^{54}-18 q^{52}+48 q^{50}-63 q^{48}+57 q^{46}-32 q^{44}+5 q^{42}+27 q^{40}-49 q^{38}+47 q^{36}-34 q^{34}+7 q^{32}+13 q^{30}-34 q^{28}+23 q^{26}-4 q^{24}-12 q^{22}+28 q^{20}-38 q^{18}+22 q^{16}+3 q^{14}-29 q^{12}+44 q^{10}-46 q^8+30 q^6-17 q^2+29-29 q^{-2} +25 q^{-4} -7 q^{-6} -4 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["10 144"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -3 t^2+10 t-13+10 t^{-1} -3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -3 z^4-2 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{ \left\{2,t^2+t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 39, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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