10 20

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{ q-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-a^6-z^4 a^4-z^2 a^4-z^4 a^2-2 z^2 a^2-a^2+z^2+2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+7 z^3 a^9-z a^9+2 z^8 a^8-8 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+3 z^5 a^7-4 z^3 a^7+2 z a^7+3 z^8 a^6-12 z^6 a^6+17 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+9 z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+z^7 a^3-z^5 a^3+2 z^3 a^3-z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{28}+q^{22}-q^{20}-q^{14}-q^{10}-q^4+2 q^2+1+ q^{-2} + q^{-4} }[/math]
The G2 invariant	[math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-2 q^{132}-q^{130}+6 q^{128}-9 q^{126}+10 q^{124}-9 q^{122}+3 q^{120}+4 q^{118}-11 q^{116}+16 q^{114}-13 q^{112}+9 q^{110}+q^{108}-8 q^{106}+12 q^{104}-9 q^{102}+3 q^{100}+3 q^{98}-7 q^{96}+7 q^{94}-2 q^{92}-4 q^{90}+11 q^{88}-12 q^{86}+9 q^{84}-4 q^{82}-5 q^{80}+10 q^{78}-15 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+5 q^{68}-12 q^{66}+13 q^{64}-10 q^{62}+4 q^{60}+q^{58}-7 q^{56}+7 q^{54}-3 q^{52}-2 q^{50}+5 q^{48}-6 q^{46}+q^{44}+2 q^{42}-7 q^{40}+7 q^{38}-5 q^{36}+3 q^{34}-q^{32}-3 q^{30}+4 q^{28}-5 q^{26}+5 q^{24}-5 q^{22}+4 q^{20}-3 q^{18}-q^{16}+4 q^{14}-6 q^{12}+8 q^{10}-4 q^8+3 q^6+q^4-q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_20.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{19}-q^{17}+q^{15}-q^{13}+q^{11}-q^9-q^7+q^5-q^3+2 q+ q^{-3} }[/math]
2	[math]\displaystyle{ q^{54}-q^{52}-q^{50}+3 q^{48}-q^{46}-4 q^{44}+3 q^{42}+2 q^{40}-4 q^{38}+q^{36}+3 q^{34}-3 q^{32}-q^{30}+3 q^{28}-q^{26}-q^{24}+2 q^{22}+2 q^{20}-2 q^{18}-2 q^{16}+4 q^{14}-q^{12}-3 q^{10}+3 q^8-q^6-2 q^4+2 q^2-1+2 q^{-4} + q^{-10} }[/math]
3	[math]\displaystyle{ q^{105}-q^{103}-q^{101}+q^{99}+2 q^{97}-q^{95}-5 q^{93}+7 q^{89}+3 q^{87}-6 q^{85}-7 q^{83}+4 q^{81}+9 q^{79}-q^{77}-8 q^{75}-4 q^{73}+6 q^{71}+7 q^{69}-2 q^{67}-8 q^{65}-2 q^{63}+8 q^{61}+6 q^{59}-7 q^{57}-6 q^{55}+6 q^{53}+7 q^{51}-6 q^{49}-7 q^{47}+2 q^{45}+6 q^{43}-2 q^{41}-6 q^{39}+5 q^{35}+6 q^{33}-4 q^{31}-8 q^{29}+10 q^{25}+3 q^{23}-7 q^{21}-7 q^{19}+6 q^{17}+6 q^{15}+q^{13}-5 q^{11}-2 q^9+3 q^7+3 q^5-4 q- q^{-1} +2 q^{-3} +2 q^{-5} -2 q^{-7} - q^{-9} +2 q^{-13} + q^{-21} }[/math]
4	[math]\displaystyle{ q^{172}-q^{170}-q^{168}+q^{166}+2 q^{162}-3 q^{160}-3 q^{158}+2 q^{156}+2 q^{154}+9 q^{152}-3 q^{150}-11 q^{148}-5 q^{146}-q^{144}+18 q^{142}+9 q^{140}-6 q^{138}-13 q^{136}-18 q^{134}+10 q^{132}+16 q^{130}+10 q^{128}-20 q^{124}-8 q^{122}-2 q^{120}+11 q^{118}+21 q^{116}+2 q^{114}-8 q^{112}-25 q^{110}-14 q^{108}+20 q^{106}+27 q^{104}+13 q^{102}-28 q^{100}-36 q^{98}+4 q^{96}+33 q^{94}+30 q^{92}-15 q^{90}-40 q^{88}-10 q^{86}+24 q^{84}+29 q^{82}-5 q^{80}-30 q^{78}-11 q^{76}+16 q^{74}+22 q^{72}+q^{70}-19 q^{68}-11 q^{66}+10 q^{64}+14 q^{62}+8 q^{60}-8 q^{58}-20 q^{56}-8 q^{54}+8 q^{52}+29 q^{50}+14 q^{48}-26 q^{46}-34 q^{44}-13 q^{42}+37 q^{40}+45 q^{38}-7 q^{36}-43 q^{34}-41 q^{32}+18 q^{30}+51 q^{28}+17 q^{26}-20 q^{24}-43 q^{22}-6 q^{20}+30 q^{18}+22 q^{16}+3 q^{14}-26 q^{12}-13 q^{10}+10 q^8+13 q^6+11 q^4-11 q^2-10+ q^{-2} +5 q^{-4} +9 q^{-6} -4 q^{-8} -5 q^{-10} -2 q^{-12} +5 q^{-16} - q^{-18} - q^{-20} - q^{-22} - q^{-24} +2 q^{-26} + q^{-36} }[/math]
5	[math]\displaystyle{ q^{255}-q^{253}-q^{251}+q^{249}-q^{241}-2 q^{239}+2 q^{237}+5 q^{235}+2 q^{233}-q^{231}-6 q^{229}-9 q^{227}-5 q^{225}+9 q^{223}+17 q^{221}+11 q^{219}-q^{217}-19 q^{215}-25 q^{213}-13 q^{211}+14 q^{209}+31 q^{207}+26 q^{205}+5 q^{203}-24 q^{201}-36 q^{199}-23 q^{197}+8 q^{195}+29 q^{193}+29 q^{191}+16 q^{189}-6 q^{187}-22 q^{185}-28 q^{183}-21 q^{181}-6 q^{179}+18 q^{177}+43 q^{175}+44 q^{173}+10 q^{171}-45 q^{169}-75 q^{167}-54 q^{165}+19 q^{163}+95 q^{161}+99 q^{159}+19 q^{157}-88 q^{155}-130 q^{153}-66 q^{151}+60 q^{149}+144 q^{147}+108 q^{145}-23 q^{143}-139 q^{141}-133 q^{139}-15 q^{137}+115 q^{135}+146 q^{133}+48 q^{131}-91 q^{129}-141 q^{127}-62 q^{125}+62 q^{123}+123 q^{121}+72 q^{119}-42 q^{117}-103 q^{115}-63 q^{113}+28 q^{111}+79 q^{109}+53 q^{107}-20 q^{105}-65 q^{103}-42 q^{101}+16 q^{99}+48 q^{97}+35 q^{95}-6 q^{93}-42 q^{91}-42 q^{89}-8 q^{87}+38 q^{85}+54 q^{83}+36 q^{81}-20 q^{79}-78 q^{77}-75 q^{75}-3 q^{73}+89 q^{71}+121 q^{69}+48 q^{67}-83 q^{65}-158 q^{63}-99 q^{61}+55 q^{59}+181 q^{57}+146 q^{55}-18 q^{53}-164 q^{51}-174 q^{49}-33 q^{47}+134 q^{45}+174 q^{43}+58 q^{41}-87 q^{39}-144 q^{37}-81 q^{35}+46 q^{33}+110 q^{31}+71 q^{29}-17 q^{27}-72 q^{25}-55 q^{23}+45 q^{19}+41 q^{17}+4 q^{15}-28 q^{13}-25 q^{11}-3 q^9+18 q^7+20 q^5+3 q^3-13 q-13 q^{-1} -4 q^{-3} +10 q^{-5} +13 q^{-7} +3 q^{-9} -6 q^{-11} -7 q^{-13} -6 q^{-15} +2 q^{-17} +9 q^{-19} +3 q^{-21} - q^{-23} -2 q^{-25} -4 q^{-27} -2 q^{-29} +3 q^{-31} + q^{-33} - q^{-39} - q^{-41} + q^{-43} + q^{-55} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{28}+q^{22}-q^{20}-q^{14}-q^{10}-q^4+2 q^2+1+ q^{-2} + q^{-4} }[/math]
1,1	[math]\displaystyle{ q^{76}-2 q^{74}+4 q^{72}-8 q^{70}+15 q^{68}-22 q^{66}+30 q^{64}-38 q^{62}+43 q^{60}-44 q^{58}+38 q^{56}-28 q^{54}+11 q^{52}+8 q^{50}-26 q^{48}+44 q^{46}-59 q^{44}+64 q^{42}-68 q^{40}+64 q^{38}-55 q^{36}+46 q^{34}-28 q^{32}+18 q^{30}-3 q^{28}-4 q^{26}+12 q^{24}-12 q^{22}+13 q^{20}-12 q^{18}+10 q^{16}-14 q^{14}+11 q^{12}-18 q^{10}+14 q^8-18 q^6+16 q^4-12 q^2+12-6 q^{-2} +8 q^{-4} -2 q^{-6} +4 q^{-8} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{72}+q^{64}-3 q^{60}-q^{58}+q^{56}+q^{54}-q^{52}-q^{50}+2 q^{48}+q^{46}-2 q^{44}-2 q^{42}+q^{40}+q^{38}-q^{36}+2 q^{32}+2 q^{30}+3 q^{26}+q^{24}-q^{22}+2 q^{20}+q^{18}-3 q^{16}-3 q^{14}+q^{12}-5 q^8-3 q^6+2 q^4+2 q^{-2} +3 q^{-4} + q^{-6} + q^{-8} + q^{-10} + q^{-12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{60}-q^{58}+q^{54}-2 q^{52}+q^{50}+q^{48}-2 q^{46}+2 q^{44}-3 q^{40}+2 q^{38}+2 q^{36}-2 q^{34}+2 q^{32}+3 q^{30}-q^{28}+q^{22}-3 q^{20}-q^{18}+2 q^{16}-4 q^{14}-3 q^{12}+2 q^{10}-2 q^8-3 q^6+4 q^4+2 q^2+1+3 q^{-2} +2 q^{-4} + q^{-8} }[/math]
1,0,0	[math]\displaystyle{ q^{37}+q^{33}+q^{29}-q^{27}-q^{23}-q^{19}-q^{13}-q^9-q^5+2 q^3+q+2 q^{-1} + q^{-3} + q^{-5} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{60}-q^{58}+2 q^{56}-3 q^{54}+4 q^{52}-5 q^{50}+5 q^{48}-4 q^{46}+4 q^{44}-2 q^{42}+q^{40}+2 q^{38}-4 q^{36}+6 q^{34}-8 q^{32}+7 q^{30}-9 q^{28}+8 q^{26}-8 q^{24}+5 q^{22}-3 q^{20}+q^{18}-2 q^{14}+3 q^{12}-4 q^{10}+4 q^8-3 q^6+4 q^4-2 q^2+3- q^{-2} +2 q^{-4} + q^{-8} }[/math]
1,0	[math]\displaystyle{ q^{98}-q^{94}-q^{92}+q^{90}+2 q^{88}-q^{86}-3 q^{84}+4 q^{80}+2 q^{78}-4 q^{76}-4 q^{74}+2 q^{72}+5 q^{70}-5 q^{66}-3 q^{64}+3 q^{62}+4 q^{60}-3 q^{56}+3 q^{52}+q^{50}-q^{48}-q^{46}+3 q^{44}+2 q^{42}-2 q^{40}-3 q^{38}+2 q^{36}+3 q^{34}-q^{32}-4 q^{30}-q^{28}+3 q^{26}+q^{24}-4 q^{22}-4 q^{20}+3 q^{16}+q^{14}-3 q^{12}-3 q^{10}+4 q^6+2 q^4-1+ q^{-2} +2 q^{-4} +2 q^{-6} + q^{-14} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-2 q^{132}-q^{130}+6 q^{128}-9 q^{126}+10 q^{124}-9 q^{122}+3 q^{120}+4 q^{118}-11 q^{116}+16 q^{114}-13 q^{112}+9 q^{110}+q^{108}-8 q^{106}+12 q^{104}-9 q^{102}+3 q^{100}+3 q^{98}-7 q^{96}+7 q^{94}-2 q^{92}-4 q^{90}+11 q^{88}-12 q^{86}+9 q^{84}-4 q^{82}-5 q^{80}+10 q^{78}-15 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+5 q^{68}-12 q^{66}+13 q^{64}-10 q^{62}+4 q^{60}+q^{58}-7 q^{56}+7 q^{54}-3 q^{52}-2 q^{50}+5 q^{48}-6 q^{46}+q^{44}+2 q^{42}-7 q^{40}+7 q^{38}-5 q^{36}+3 q^{34}-q^{32}-3 q^{30}+4 q^{28}-5 q^{26}+5 q^{24}-5 q^{22}+4 q^{20}-3 q^{18}-q^{16}+4 q^{14}-6 q^{12}+8 q^{10}-4 q^8+3 q^6+q^4-q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18} }[/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 20"];

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{ q-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-a^6-z^4 a^4-z^2 a^4-z^4 a^2-2 z^2 a^2-a^2+z^2+2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+7 z^3 a^9-z a^9+2 z^8 a^8-8 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+3 z^5 a^7-4 z^3 a^7+2 z a^7+3 z^8 a^6-12 z^6 a^6+17 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+9 z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+z^7 a^3-z^5 a^3+2 z^3 a^3-z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2 }[/math]