10 2

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+3 t^3-3 t^2+3 t-3+3 t^{-1} -3 t^{-2} +3 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-5 z^6-5 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 23, -6 }
Jones polynomial	[math]\displaystyle{ q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +3 q^{-5} -3 q^{-6} +3 q^{-7} -3 q^{-8} +2 q^{-9} -2 q^{-10} + q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^8+5 z^4 a^8+6 z^2 a^8+a^8-z^8 a^6-7 z^6 a^6-16 z^4 a^6-14 z^2 a^6-4 a^6+z^6 a^4+6 z^4 a^4+10 z^2 a^4+4 a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^2 a^{14}+2 z^3 a^{13}-z a^{13}+2 z^4 a^{12}-z^2 a^{12}+2 z^5 a^{11}-2 z^3 a^{11}-z a^{11}+2 z^6 a^{10}-4 z^4 a^{10}+2 z^7 a^9-6 z^5 a^9+2 z^3 a^9+z a^9+2 z^8 a^8-9 z^6 a^8+11 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-4 z^7 a^7+2 z^5 a^7+3 z^3 a^7-z a^7+3 z^8 a^6-18 z^6 a^6+33 z^4 a^6-21 z^2 a^6+4 a^6+z^9 a^5-6 z^7 a^5+10 z^5 a^5-3 z^3 a^5-2 z a^5+z^8 a^4-7 z^6 a^4+16 z^4 a^4-14 z^2 a^4+4 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ q^{32}-q^{26}-q^{24}-q^{22}-q^{20}+q^{18}+q^{14}+q^{10}+q^8+q^6+q^4 }[/math]
The G2 invariant	[math]\displaystyle{ q^{182}-q^{180}+q^{178}-q^{176}-q^{174}-q^{170}+2 q^{168}-2 q^{166}+q^{164}+q^{158}-q^{156}+q^{154}-q^{152}+q^{150}-q^{146}+2 q^{144}+q^{140}+q^{134}+q^{128}-q^{120}+q^{118}-q^{116}-q^{114}-q^{110}-q^{106}-q^{104}-q^{102}-q^{98}-2 q^{92}+2 q^{90}-2 q^{88}+q^{86}-q^{84}-q^{82}+q^{80}-q^{78}+2 q^{76}-q^{74}-q^{70}-q^{64}+q^{56}+q^{54}-q^{52}+3 q^{50}-2 q^{48}+2 q^{46}+q^{44}-q^{42}+4 q^{40}-2 q^{38}+3 q^{36}+q^{34}+q^{32}+q^{30}-q^{28}+2 q^{26}+q^{22} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_2.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{23}-q^{21}-q^{17}+q^9+q^5+q }[/math]
2	[math]\displaystyle{ q^{62}-q^{60}-q^{58}+q^{56}-q^{54}+q^{50}+q^{48}-q^{36}-q^{30}-q^{28}+q^{18}+2 q^{12}+q^{10}-q^8+q^6+q^4-q^2+ q^{-2} }[/math]
3	[math]\displaystyle{ q^{117}-q^{115}-q^{113}+q^{109}-q^{105}+2 q^{103}+q^{101}-q^{99}-3 q^{97}+2 q^{95}+2 q^{93}-q^{91}-3 q^{89}+3 q^{85}-3 q^{81}-q^{79}+2 q^{77}+q^{75}+q^{73}-q^{71}+q^{67}+3 q^{65}-2 q^{61}-q^{59}+q^{57}-q^{55}-2 q^{53}+q^{49}-q^{47}-q^{45}+q^{41}-q^{37}-q^{35}+q^{27}+q^{21}+2 q^{19}+q^{17}-q^{15}-q^{13}+2 q^{11}+2 q^9-2 q^5+2 q+ q^{-1} - q^{-3} - q^{-5} + q^{-9} }[/math]
4	[math]\displaystyle{ q^{188}-q^{186}-q^{184}+2 q^{178}-q^{176}+q^{174}-q^{170}-2 q^{166}+2 q^{164}+2 q^{162}-3 q^{158}-3 q^{156}+3 q^{154}+4 q^{152}-4 q^{148}-4 q^{146}+4 q^{144}+4 q^{142}-3 q^{138}-3 q^{136}+3 q^{134}+3 q^{132}+q^{130}-3 q^{128}-4 q^{126}+2 q^{122}+2 q^{120}-2 q^{116}-3 q^{114}-q^{112}+3 q^{110}+3 q^{108}+q^{106}-2 q^{104}-2 q^{102}+3 q^{98}+2 q^{96}-q^{94}-q^{92}+3 q^{88}+q^{86}-3 q^{84}-2 q^{82}+4 q^{78}+2 q^{76}-4 q^{74}-4 q^{72}-q^{70}+4 q^{68}+3 q^{66}-3 q^{64}-4 q^{62}-2 q^{60}+2 q^{58}+3 q^{56}-q^{54}-2 q^{52}-2 q^{50}+2 q^{46}+q^{44}-q^{40}-q^{38}+q^{36}+q^{30}+q^{28}+2 q^{26}-q^{22}+4 q^{16}+2 q^{14}-q^{12}-2 q^{10}-3 q^8+2 q^6+3 q^4+2 q^2-4 q^{-2} - q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} - q^{-12} - q^{-14} - q^{-16} + q^{-20} }[/math]
5	[math]\displaystyle{ q^{275}-q^{273}-q^{271}+q^{265}+q^{263}+q^{261}-q^{259}-q^{257}-2 q^{255}+q^{251}+3 q^{249}+2 q^{247}-2 q^{245}-4 q^{243}-3 q^{241}+2 q^{239}+5 q^{237}+4 q^{235}-2 q^{233}-5 q^{231}-5 q^{229}+6 q^{225}+5 q^{223}-2 q^{221}-2 q^{219}-2 q^{217}-q^{215}+q^{213}-2 q^{209}+q^{207}+4 q^{205}+3 q^{203}-q^{201}-5 q^{199}-5 q^{197}-q^{195}+6 q^{193}+8 q^{191}+3 q^{189}-5 q^{187}-7 q^{185}-5 q^{183}+q^{181}+6 q^{179}+6 q^{177}+2 q^{175}-q^{173}-6 q^{171}-6 q^{169}-3 q^{167}+q^{165}+6 q^{163}+7 q^{161}-5 q^{157}-6 q^{155}-3 q^{153}+3 q^{151}+6 q^{149}+2 q^{147}-3 q^{145}-3 q^{143}+2 q^{139}+2 q^{137}-2 q^{135}-4 q^{133}-q^{131}+2 q^{129}+5 q^{127}+4 q^{125}+q^{123}-3 q^{121}-4 q^{119}-q^{117}+3 q^{115}+5 q^{113}+4 q^{111}-q^{109}-5 q^{107}-7 q^{105}-q^{103}+5 q^{101}+7 q^{99}+3 q^{97}-4 q^{95}-10 q^{93}-6 q^{91}+3 q^{89}+9 q^{87}+7 q^{85}-q^{83}-9 q^{81}-10 q^{79}-3 q^{77}+7 q^{75}+9 q^{73}+4 q^{71}-3 q^{69}-8 q^{67}-6 q^{65}+5 q^{61}+5 q^{59}+q^{57}-2 q^{55}-4 q^{53}-3 q^{51}+2 q^{47}+3 q^{45}+q^{43}+q^{35}+q^{33}-q^{29}+2 q^{25}+2 q^{23}+3 q^{21}+q^{19}-2 q^{17}-4 q^{15}-2 q^{13}+q^{11}+4 q^9+5 q^7+2 q^5-2 q^3-5 q-4 q^{-1} +3 q^{-5} +5 q^{-7} +3 q^{-9} - q^{-11} -4 q^{-13} -3 q^{-15} - q^{-17} + q^{-19} +3 q^{-21} +2 q^{-23} - q^{-27} - q^{-29} - q^{-31} + q^{-35} }[/math]
6	[math]\displaystyle{ q^{378}-q^{376}-q^{374}+q^{368}+3 q^{364}-q^{362}-2 q^{360}-2 q^{358}-q^{356}+q^{354}+q^{352}+6 q^{350}-2 q^{346}-5 q^{344}-4 q^{342}+3 q^{338}+9 q^{336}+4 q^{334}-3 q^{332}-10 q^{330}-8 q^{328}-q^{326}+4 q^{324}+13 q^{322}+10 q^{320}-11 q^{316}-14 q^{314}-6 q^{312}-q^{310}+12 q^{308}+17 q^{306}+10 q^{304}-3 q^{302}-14 q^{300}-16 q^{298}-13 q^{296}+4 q^{294}+18 q^{292}+20 q^{290}+11 q^{288}-7 q^{286}-20 q^{284}-24 q^{282}-9 q^{280}+11 q^{278}+23 q^{276}+21 q^{274}+5 q^{272}-13 q^{270}-27 q^{268}-20 q^{266}-3 q^{264}+16 q^{262}+25 q^{260}+19 q^{258}+4 q^{256}-16 q^{254}-21 q^{252}-15 q^{250}-2 q^{248}+11 q^{246}+18 q^{244}+16 q^{242}+3 q^{240}-7 q^{238}-13 q^{236}-14 q^{234}-10 q^{232}+q^{230}+11 q^{228}+15 q^{226}+12 q^{224}+2 q^{222}-8 q^{220}-15 q^{218}-11 q^{216}-q^{214}+9 q^{212}+11 q^{210}+4 q^{208}-2 q^{206}-7 q^{204}-5 q^{202}+6 q^{198}+4 q^{196}-5 q^{194}-9 q^{192}-7 q^{190}+q^{188}+8 q^{186}+14 q^{184}+6 q^{182}-8 q^{180}-14 q^{178}-11 q^{176}+10 q^{172}+17 q^{170}+10 q^{168}-6 q^{166}-13 q^{164}-12 q^{162}-3 q^{160}+7 q^{158}+14 q^{156}+10 q^{154}-2 q^{152}-9 q^{150}-9 q^{148}-6 q^{146}+9 q^{142}+11 q^{140}+6 q^{138}-4 q^{134}-8 q^{132}-9 q^{130}-2 q^{128}+4 q^{126}+9 q^{124}+10 q^{122}+7 q^{120}-3 q^{118}-14 q^{116}-14 q^{114}-10 q^{112}+2 q^{110}+13 q^{108}+18 q^{106}+10 q^{104}-5 q^{102}-15 q^{100}-20 q^{98}-11 q^{96}+4 q^{94}+18 q^{92}+19 q^{90}+9 q^{88}-4 q^{86}-18 q^{84}-19 q^{82}-10 q^{80}+5 q^{78}+14 q^{76}+15 q^{74}+9 q^{72}-4 q^{70}-11 q^{68}-12 q^{66}-6 q^{64}+q^{62}+7 q^{60}+9 q^{58}+4 q^{56}+q^{54}-4 q^{52}-5 q^{50}-3 q^{48}+3 q^{44}+2 q^{42}+3 q^{40}+q^{38}-q^{36}+q^{32}+3 q^{30}+q^{28}+2 q^{26}-q^{24}-4 q^{22}-3 q^{20}-q^{18}+3 q^{16}+3 q^{14}+7 q^{12}+4 q^{10}-2 q^8-5 q^6-7 q^4-4 q^2-2+6 q^{-2} +8 q^{-4} +6 q^{-6} +2 q^{-8} -3 q^{-10} -6 q^{-12} -9 q^{-14} -2 q^{-16} +2 q^{-18} +5 q^{-20} +6 q^{-22} +4 q^{-24} + q^{-26} -5 q^{-28} -4 q^{-30} -3 q^{-32} - q^{-34} + q^{-36} +3 q^{-38} +3 q^{-40} - q^{-46} - q^{-48} - q^{-50} + q^{-54} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{32}-q^{26}-q^{24}-q^{22}-q^{20}+q^{18}+q^{14}+q^{10}+q^8+q^6+q^4 }[/math]
1,1	[math]\displaystyle{ q^{92}-2 q^{90}+2 q^{88}-2 q^{86}+3 q^{84}-4 q^{82}+2 q^{80}-2 q^{78}+2 q^{76}-2 q^{74}+4 q^{72}+q^{68}-2 q^{66}-2 q^{64}-2 q^{60}+2 q^{58}+2 q^{54}+2 q^{52}+2 q^{50}-2 q^{48}+2 q^{46}-6 q^{44}+6 q^{42}-10 q^{40}+8 q^{38}-11 q^{36}+8 q^{34}-10 q^{32}+4 q^{30}-4 q^{28}+2 q^{26}+4 q^{24}-4 q^{22}+9 q^{20}-6 q^{18}+10 q^{16}-6 q^{14}+6 q^{12}-2 q^{10}+4 q^8+q^4 }[/math]
2,0	[math]\displaystyle{ q^{80}-q^{74}-q^{72}-q^{70}-q^{68}+q^{66}+q^{64}+q^{62}+2 q^{58}+q^{56}+q^{54}+q^{50}-2 q^{46}-2 q^{44}-2 q^{42}-2 q^{40}-3 q^{38}-q^{36}-q^{34}+q^{30}+2 q^{28}+q^{26}+q^{24}+2 q^{22}+2 q^{20}+q^{14}+q^{12}+q^8+q^6+q^4 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{76}-q^{74}-q^{72}+q^{70}-q^{68}-q^{66}+q^{64}+q^{62}+q^{58}+q^{50}+q^{48}-q^{40}-2 q^{38}-q^{36}-2 q^{34}-2 q^{32}-2 q^{30}-2 q^{28}+q^{24}+q^{22}+3 q^{20}+3 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^8 }[/math]
1,0,0	[math]\displaystyle{ q^{41}+q^{37}-q^{35}-2 q^{31}-q^{29}-q^{27}-q^{25}+q^{19}+2 q^{15}+q^{13}+2 q^{11}+q^9+q^7 }[/math]
1,0,1	[math]\displaystyle{ q^{122}-2 q^{120}+q^{118}+q^{116}-2 q^{114}+3 q^{112}-2 q^{110}-q^{108}+q^{106}-2 q^{104}+q^{102}-q^{100}+4 q^{96}+q^{92}-2 q^{88}-2 q^{84}+q^{82}-3 q^{80}+2 q^{78}-q^{76}+3 q^{72}-4 q^{70}+5 q^{68}-5 q^{66}+4 q^{64}-q^{62}+5 q^{60}+3 q^{58}+6 q^{54}-6 q^{52}+6 q^{50}-10 q^{48}-q^{46}-8 q^{44}-7 q^{42}-3 q^{40}-7 q^{38}-3 q^{34}+3 q^{32}+q^{30}+6 q^{28}+2 q^{26}+6 q^{24}+4 q^{22}+5 q^{20}+6 q^{18}+2 q^{16}+4 q^{14}+q^{10} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{94}-q^{90}-2 q^{84}-2 q^{82}+q^{74}+3 q^{72}+2 q^{70}+2 q^{68}+2 q^{66}+2 q^{64}+q^{60}-q^{56}-q^{54}-q^{52}-q^{50}-3 q^{48}-4 q^{46}-5 q^{44}-4 q^{42}-6 q^{40}-4 q^{38}-q^{36}+q^{34}+2 q^{32}+4 q^{30}+5 q^{28}+5 q^{26}+4 q^{24}+4 q^{22}+3 q^{20}+2 q^{18}+q^{16}+q^{14} }[/math]
1,0,0,0	[math]\displaystyle{ q^{50}+q^{46}-q^{40}-q^{38}-q^{36}-2 q^{34}-q^{32}-2 q^{30}-q^{26}+q^{24}+q^{22}+2 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+q^{12}+q^{10} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{76}-q^{74}+q^{72}-q^{70}+q^{68}-q^{66}+q^{64}-q^{62}-q^{58}+q^{50}-q^{48}+2 q^{46}-2 q^{44}+2 q^{42}-3 q^{40}+2 q^{38}-3 q^{36}+2 q^{34}-2 q^{32}+q^{24}-q^{22}+3 q^{20}-q^{18}+2 q^{16}+2 q^{12}+q^8 }[/math]
1,0	[math]\displaystyle{ q^{122}-q^{118}-q^{116}+q^{112}-q^{108}-q^{106}+q^{102}+q^{100}+q^{92}+q^{82}+q^{80}-q^{76}+q^{72}-q^{68}-q^{66}-q^{62}-q^{60}-q^{58}-2 q^{52}-2 q^{50}+q^{46}-q^{42}-q^{40}+q^{38}+2 q^{36}+q^{34}-q^{32}+q^{30}+2 q^{28}+2 q^{26}+q^{20}+2 q^{18}+q^{10} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{106}-q^{104}-q^{100}+q^{98}-q^{96}-q^{92}+q^{90}+q^{86}+q^{82}+q^{68}+2 q^{64}-q^{62}+2 q^{60}-2 q^{58}+2 q^{56}-2 q^{54}+q^{52}-4 q^{50}-4 q^{46}-2 q^{44}-4 q^{42}-3 q^{40}-2 q^{38}-q^{36}+q^{34}+q^{32}+5 q^{30}+2 q^{28}+5 q^{26}+2 q^{24}+4 q^{22}+q^{20}+2 q^{18}+q^{14} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{182}-q^{180}+q^{178}-q^{176}-q^{174}-q^{170}+2 q^{168}-2 q^{166}+q^{164}+q^{158}-q^{156}+q^{154}-q^{152}+q^{150}-q^{146}+2 q^{144}+q^{140}+q^{134}+q^{128}-q^{120}+q^{118}-q^{116}-q^{114}-q^{110}-q^{106}-q^{104}-q^{102}-q^{98}-2 q^{92}+2 q^{90}-2 q^{88}+q^{86}-q^{84}-q^{82}+q^{80}-q^{78}+2 q^{76}-q^{74}-q^{70}-q^{64}+q^{56}+q^{54}-q^{52}+3 q^{50}-2 q^{48}+2 q^{46}+q^{44}-q^{42}+4 q^{40}-2 q^{38}+3 q^{36}+q^{34}+q^{32}+q^{30}-q^{28}+2 q^{26}+q^{22} }[/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 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-5 z^6-5 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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 23, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +3 q^{-5} -3 q^{-6} +3 q^{-7} -3 q^{-8} +2 q^{-9} -2 q^{-10} + q^{-11} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6 a^8+5 z^4 a^8+6 z^2 a^8+a^8-z^8 a^6-7 z^6 a^6-16 z^4 a^6-14 z^2 a^6-4 a^6+z^6 a^4+6 z^4 a^4+10 z^2 a^4+4 a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^2 a^{14}+2 z^3 a^{13}-z a^{13}+2 z^4 a^{12}-z^2 a^{12}+2 z^5 a^{11}-2 z^3 a^{11}-z a^{11}+2 z^6 a^{10}-4 z^4 a^{10}+2 z^7 a^9-6 z^5 a^9+2 z^3 a^9+z a^9+2 z^8 a^8-9 z^6 a^8+11 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-4 z^7 a^7+2 z^5 a^7+3 z^3 a^7-z a^7+3 z^8 a^6-18 z^6 a^6+33 z^4 a^6-21 z^2 a^6+4 a^6+z^9 a^5-6 z^7 a^5+10 z^5 a^5-3 z^3 a^5-2 z a^5+z^8 a^4-7 z^6 a^4+16 z^4 a^4-14 z^2 a^4+4 a^4 }[/math]