8 21

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 8_21.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-q^{13}-q^9+q^5+2 q }[/math]
2	[math]\displaystyle{ q^{42}-q^{40}-2 q^{38}+2 q^{36}+q^{34}-2 q^{32}+q^{30}+2 q^{28}-2 q^{26}+q^{22}-q^{20}-q^{18}+2 q^{14}-2 q^{12}-q^{10}+3 q^8-q^6-q^4+3 q^2+1 }[/math]
3	[math]\displaystyle{ q^{81}-q^{79}-2 q^{77}+3 q^{73}+3 q^{71}-3 q^{69}-4 q^{67}+q^{65}+5 q^{63}+q^{61}-6 q^{59}-4 q^{57}+5 q^{55}+4 q^{53}-3 q^{51}-4 q^{49}+4 q^{47}+5 q^{45}-2 q^{43}-3 q^{41}+2 q^{37}-q^{35}-2 q^{33}-3 q^{31}+2 q^{29}+4 q^{27}-6 q^{23}-q^{21}+7 q^{19}+3 q^{17}-6 q^{15}-3 q^{13}+4 q^{11}+4 q^9-2 q^7-4 q^5+2 q^3+2 q+2 q^{-1} }[/math]
4	[math]\displaystyle{ q^{132}-q^{130}-2 q^{128}+q^{124}+5 q^{122}+q^{120}-3 q^{118}-5 q^{116}-6 q^{114}+6 q^{112}+8 q^{110}+4 q^{108}-4 q^{106}-14 q^{104}-4 q^{102}+7 q^{100}+15 q^{98}+8 q^{96}-13 q^{94}-14 q^{92}-3 q^{90}+15 q^{88}+15 q^{86}-6 q^{84}-16 q^{82}-11 q^{80}+9 q^{78}+14 q^{76}-q^{74}-10 q^{72}-8 q^{70}+6 q^{68}+11 q^{66}+2 q^{64}-5 q^{62}-5 q^{60}+2 q^{58}+5 q^{56}+4 q^{54}-2 q^{52}-5 q^{50}-6 q^{48}+11 q^{44}+4 q^{42}-6 q^{40}-15 q^{38}-7 q^{36}+15 q^{34}+14 q^{32}+q^{30}-18 q^{28}-15 q^{26}+11 q^{24}+16 q^{22}+10 q^{20}-9 q^{18}-15 q^{16}+7 q^{12}+10 q^{10}-7 q^6-3 q^4-q^2+3+3 q^{-2} + q^{-4} }[/math]
5	[math]\displaystyle{ q^{195}-q^{193}-2 q^{191}+q^{187}+3 q^{185}+3 q^{183}+q^{181}-5 q^{179}-7 q^{177}-4 q^{175}+q^{173}+8 q^{171}+12 q^{169}+7 q^{167}-8 q^{165}-16 q^{163}-15 q^{161}-3 q^{159}+16 q^{157}+26 q^{155}+18 q^{153}-8 q^{151}-29 q^{149}-32 q^{147}-11 q^{145}+25 q^{143}+42 q^{141}+26 q^{139}-14 q^{137}-45 q^{135}-41 q^{133}+41 q^{129}+49 q^{127}+17 q^{125}-33 q^{123}-49 q^{121}-23 q^{119}+22 q^{117}+46 q^{115}+27 q^{113}-14 q^{111}-40 q^{109}-26 q^{107}+9 q^{105}+30 q^{103}+21 q^{101}-7 q^{99}-24 q^{97}-18 q^{95}+5 q^{93}+18 q^{91}+11 q^{89}-q^{87}-11 q^{85}-12 q^{83}+9 q^{79}+10 q^{77}+9 q^{75}-2 q^{73}-15 q^{71}-15 q^{69}-q^{67}+17 q^{65}+27 q^{63}+12 q^{61}-19 q^{59}-37 q^{57}-25 q^{55}+15 q^{53}+45 q^{51}+36 q^{49}-9 q^{47}-50 q^{45}-48 q^{43}-7 q^{41}+46 q^{39}+54 q^{37}+16 q^{35}-32 q^{33}-52 q^{31}-27 q^{29}+16 q^{27}+43 q^{25}+33 q^{23}-q^{21}-24 q^{19}-25 q^{17}-9 q^{15}+10 q^{13}+21 q^{11}+10 q^9-2 q^7-6 q^5-10 q^3-4 q+2 q^{-1} +4 q^{-3} +2 q^{-5} +2 q^{-7} }[/math]
6	[math]\displaystyle{ q^{270}-q^{268}-2 q^{266}+q^{262}+3 q^{260}+q^{258}+3 q^{256}-q^{254}-7 q^{252}-6 q^{250}-4 q^{248}+2 q^{246}+5 q^{244}+15 q^{242}+11 q^{240}+q^{238}-10 q^{236}-19 q^{234}-19 q^{232}-16 q^{230}+14 q^{228}+31 q^{226}+34 q^{224}+21 q^{222}-5 q^{220}-35 q^{218}-61 q^{216}-40 q^{214}-q^{212}+46 q^{210}+71 q^{208}+68 q^{206}+18 q^{204}-60 q^{202}-94 q^{200}-86 q^{198}-21 q^{196}+59 q^{194}+123 q^{192}+109 q^{190}+19 q^{188}-76 q^{186}-137 q^{184}-114 q^{182}-24 q^{180}+99 q^{178}+150 q^{176}+105 q^{174}-113 q^{170}-148 q^{168}-94 q^{166}+35 q^{164}+126 q^{162}+130 q^{160}+56 q^{158}-58 q^{156}-121 q^{154}-104 q^{152}-6 q^{150}+78 q^{148}+101 q^{146}+57 q^{144}-24 q^{142}-77 q^{140}-75 q^{138}-11 q^{136}+42 q^{134}+58 q^{132}+33 q^{130}-12 q^{128}-40 q^{126}-40 q^{124}-6 q^{122}+22 q^{120}+32 q^{118}+19 q^{116}-3 q^{114}-20 q^{112}-26 q^{110}-15 q^{108}+3 q^{106}+25 q^{104}+27 q^{102}+19 q^{100}-6 q^{98}-32 q^{96}-47 q^{94}-28 q^{92}+23 q^{90}+55 q^{88}+68 q^{86}+28 q^{84}-36 q^{82}-95 q^{80}-90 q^{78}-11 q^{76}+72 q^{74}+129 q^{72}+96 q^{70}-3 q^{68}-122 q^{66}-156 q^{64}-80 q^{62}+39 q^{60}+148 q^{58}+156 q^{56}+68 q^{54}-81 q^{52}-163 q^{50}-134 q^{48}-39 q^{46}+86 q^{44}+147 q^{42}+118 q^{40}+7 q^{38}-86 q^{36}-112 q^{34}-83 q^{32}-6 q^{30}+64 q^{28}+91 q^{26}+51 q^{24}+3 q^{22}-35 q^{20}-53 q^{18}-35 q^{16}-4 q^{14}+26 q^{12}+27 q^{10}+19 q^8+8 q^6-8 q^4-15 q^2-11-3 q^{-2} + q^{-4} +3 q^{-6} +3 q^{-8} +3 q^{-10} + q^{-12} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}-2 q^{14}-q^{12}-q^{10}+q^8+2 q^6+q^4+2 q^2 }[/math]
1,1	[math]\displaystyle{ q^{60}-2 q^{58}+4 q^{56}-8 q^{54}+9 q^{52}-10 q^{50}+12 q^{48}-8 q^{46}+4 q^{44}-8 q^{40}+10 q^{38}-15 q^{36}+18 q^{34}-16 q^{32}+20 q^{30}-12 q^{28}+12 q^{26}-6 q^{24}-2 q^{22}+q^{20}-14 q^{18}+8 q^{16}-12 q^{14}+10 q^{12}-4 q^{10}+8 q^8+2 q^6+4 q^4+2 q^2 }[/math]
2,0	[math]\displaystyle{ q^{56}-q^{52}-q^{50}+q^{46}-q^{44}+q^{40}-q^{36}+2 q^{32}+2 q^{30}+q^{28}+q^{26}-q^{24}-3 q^{22}-2 q^{20}-2 q^{18}-3 q^{16}-q^{14}+2 q^{12}+2 q^{10}+2 q^8+2 q^6+4 q^4+q^2 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{48}-q^{46}-3 q^{40}+q^{38}+q^{36}+3 q^{32}+4 q^{30}+q^{28}-2 q^{24}-4 q^{22}-5 q^{20}-5 q^{18}+2 q^{12}+6 q^{10}+3 q^8+q^6+3 q^4 }[/math]
1,0,0	[math]\displaystyle{ q^{29}+q^{25}-2 q^{19}-2 q^{17}-2 q^{15}-q^{13}+q^{11}+2 q^9+3 q^7+q^5+2 q^3 }[/math]
1,0,1	[math]\displaystyle{ q^{78}-2 q^{76}+3 q^{74}-3 q^{72}-q^{70}+5 q^{68}-8 q^{66}+10 q^{64}-6 q^{62}+2 q^{60}+5 q^{58}-11 q^{56}+12 q^{54}-12 q^{52}+2 q^{50}-q^{48}-6 q^{46}+q^{44}+q^{42}+6 q^{40}+3 q^{38}+18 q^{36}-3 q^{34}+13 q^{32}-5 q^{30}-11 q^{28}-2 q^{26}-25 q^{24}-2 q^{22}-11 q^{20}-4 q^{18}+9 q^{16}+3 q^{14}+12 q^{12}+10 q^{10}+5 q^8+5 q^6+2 q^4 }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{66}-q^{64}+q^{62}-2 q^{60}+q^{58}-2 q^{56}-q^{52}+q^{50}+q^{48}+q^{46}+4 q^{44}+2 q^{42}+4 q^{40}-q^{38}+2 q^{36}-4 q^{34}-q^{32}-7 q^{30}-4 q^{28}-6 q^{26}-2 q^{24}-q^{22}+3 q^{18}+3 q^{16}+6 q^{14}+3 q^{12}+4 q^{10}+3 q^6 }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{104}-4 q^{102}+5 q^{100}-4 q^{98}+3 q^{96}+q^{94}-4 q^{92}+5 q^{90}-2 q^{88}+2 q^{86}+2 q^{84}-4 q^{82}+4 q^{80}+2 q^{78}-2 q^{76}+4 q^{74}-4 q^{72}+3 q^{70}-4 q^{66}+3 q^{64}-7 q^{62}+5 q^{60}-4 q^{58}-3 q^{56}+2 q^{54}-6 q^{52}+4 q^{50}-5 q^{48}-q^{46}+2 q^{44}-3 q^{42}+2 q^{40}-3 q^{36}+7 q^{34}-2 q^{32}+3 q^{28}-3 q^{26}+7 q^{24}-2 q^{22}+2 q^{20}+q^{18}-q^{16}+4 q^{14}-q^{12}+2 q^{10}+q^8 }[/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["8 21"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^2+4 t-5+4 t^{-1} - t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-z^4 }[/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]=

{ 15, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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