8 15

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^2-8 t+11-8 t^{-1} +3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 33, -4 }
Jones polynomial	[math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -5 q^{-5} +6 q^{-6} -6 q^{-7} +4 q^{-8} -3 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^{10}-3 z^2 a^8-4 a^8+2 z^4 a^6+5 z^2 a^6+3 a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-5 z^3 a^{11}+2 z a^{11}+3 z^6 a^{10}-3 z^4 a^{10}-a^{10}+z^7 a^9+6 z^5 a^9-14 z^3 a^9+8 z a^9+6 z^6 a^8-10 z^4 a^8+8 z^2 a^8-4 a^8+z^7 a^7+5 z^5 a^7-11 z^3 a^7+6 z a^7+3 z^6 a^6-5 z^4 a^6+5 z^2 a^6-3 a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]
The A2 invariant	[math]\displaystyle{ q^{32}+q^{30}-2 q^{28}-q^{26}-2 q^{24}-2 q^{22}+q^{20}+3 q^{16}+q^{14}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+3 q^{154}-q^{152}-6 q^{150}+14 q^{148}-18 q^{146}+20 q^{144}-12 q^{142}-q^{140}+17 q^{138}-27 q^{136}+34 q^{134}-24 q^{132}+7 q^{130}+10 q^{128}-21 q^{126}+25 q^{124}-16 q^{122}+q^{120}+14 q^{118}-20 q^{116}+12 q^{114}-24 q^{110}+34 q^{108}-36 q^{106}+18 q^{104}-2 q^{102}-24 q^{100}+40 q^{98}-47 q^{96}+34 q^{94}-18 q^{92}-7 q^{90}+25 q^{88}-33 q^{86}+26 q^{84}-9 q^{82}-2 q^{80}+16 q^{78}-17 q^{76}+9 q^{74}+10 q^{72}-21 q^{70}+29 q^{68}-21 q^{66}+6 q^{64}+17 q^{62}-27 q^{60}+33 q^{58}-23 q^{56}+12 q^{54}+2 q^{52}-14 q^{50}+17 q^{48}-14 q^{46}+10 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_15.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{21}-2 q^{19}+q^{17}-2 q^{15}+q^{11}+3 q^7-q^5+q^3 }[/math]
2	[math]\displaystyle{ q^{58}-2 q^{56}-2 q^{54}+6 q^{52}-2 q^{50}-6 q^{48}+9 q^{46}+q^{44}-9 q^{42}+6 q^{40}+3 q^{38}-6 q^{36}-q^{34}+2 q^{32}-7 q^{28}+2 q^{26}+7 q^{24}-8 q^{22}+10 q^{18}-5 q^{16}-2 q^{14}+6 q^{12}-q^8+q^6 }[/math]
3	[math]\displaystyle{ q^{111}-2 q^{109}-2 q^{107}+3 q^{105}+6 q^{103}-2 q^{101}-13 q^{99}+2 q^{97}+18 q^{95}+4 q^{93}-25 q^{91}-13 q^{89}+27 q^{87}+22 q^{85}-27 q^{83}-29 q^{81}+20 q^{79}+35 q^{77}-12 q^{75}-32 q^{73}+7 q^{71}+27 q^{69}+4 q^{67}-21 q^{65}-9 q^{63}+10 q^{61}+17 q^{59}-5 q^{57}-21 q^{55}-7 q^{53}+27 q^{51}+13 q^{49}-30 q^{47}-23 q^{45}+27 q^{43}+27 q^{41}-22 q^{39}-32 q^{37}+12 q^{35}+32 q^{33}-6 q^{31}-23 q^{29}-2 q^{27}+19 q^{25}+7 q^{23}-8 q^{21}-4 q^{19}+4 q^{17}+3 q^{15}-q^{11}+q^9 }[/math]
4	[math]\displaystyle{ q^{180}-2 q^{178}-2 q^{176}+3 q^{174}+3 q^{172}+6 q^{170}-9 q^{168}-13 q^{166}+2 q^{164}+11 q^{162}+30 q^{160}-11 q^{158}-41 q^{156}-22 q^{154}+14 q^{152}+80 q^{150}+22 q^{148}-62 q^{146}-83 q^{144}-27 q^{142}+123 q^{140}+97 q^{138}-31 q^{136}-135 q^{134}-108 q^{132}+102 q^{130}+151 q^{128}+41 q^{126}-124 q^{124}-159 q^{122}+38 q^{120}+138 q^{118}+88 q^{116}-63 q^{114}-137 q^{112}-19 q^{110}+80 q^{108}+92 q^{106}-2 q^{104}-81 q^{102}-57 q^{100}+17 q^{98}+75 q^{96}+49 q^{94}-23 q^{92}-93 q^{90}-46 q^{88}+62 q^{86}+102 q^{84}+34 q^{82}-121 q^{80}-107 q^{78}+32 q^{76}+141 q^{74}+101 q^{72}-112 q^{70}-148 q^{68}-28 q^{66}+124 q^{64}+148 q^{62}-49 q^{60}-133 q^{58}-86 q^{56}+53 q^{54}+136 q^{52}+19 q^{50}-62 q^{48}-83 q^{46}-12 q^{44}+70 q^{42}+38 q^{40}-38 q^{36}-24 q^{34}+16 q^{32}+14 q^{30}+11 q^{28}-5 q^{26}-7 q^{24}+2 q^{22}+q^{20}+3 q^{18}-q^{14}+q^{12} }[/math]
5	[math]\displaystyle{ q^{265}-2 q^{263}-2 q^{261}+3 q^{259}+3 q^{257}+3 q^{255}-q^{253}-9 q^{251}-13 q^{249}+2 q^{247}+20 q^{245}+23 q^{243}+6 q^{241}-27 q^{239}-49 q^{237}-33 q^{235}+37 q^{233}+92 q^{231}+71 q^{229}-21 q^{227}-131 q^{225}-155 q^{223}-29 q^{221}+172 q^{219}+255 q^{217}+121 q^{215}-157 q^{213}-369 q^{211}-275 q^{209}+93 q^{207}+447 q^{205}+449 q^{203}+42 q^{201}-464 q^{199}-612 q^{197}-218 q^{195}+403 q^{193}+725 q^{191}+409 q^{189}-288 q^{187}-747 q^{185}-557 q^{183}+129 q^{181}+696 q^{179}+645 q^{177}+23 q^{175}-589 q^{173}-647 q^{171}-147 q^{169}+441 q^{167}+591 q^{165}+229 q^{163}-299 q^{161}-511 q^{159}-260 q^{157}+162 q^{155}+395 q^{153}+289 q^{151}-49 q^{149}-310 q^{147}-287 q^{145}-48 q^{143}+215 q^{141}+319 q^{139}+152 q^{137}-156 q^{135}-340 q^{133}-254 q^{131}+77 q^{129}+394 q^{127}+373 q^{125}-7 q^{123}-426 q^{121}-499 q^{119}-98 q^{117}+448 q^{115}+620 q^{113}+214 q^{111}-413 q^{109}-712 q^{107}-360 q^{105}+339 q^{103}+747 q^{101}+494 q^{99}-202 q^{97}-709 q^{95}-595 q^{93}+27 q^{91}+607 q^{89}+634 q^{87}+132 q^{85}-435 q^{83}-592 q^{81}-272 q^{79}+239 q^{77}+496 q^{75}+313 q^{73}-70 q^{71}-337 q^{69}-309 q^{67}-55 q^{65}+194 q^{63}+240 q^{61}+101 q^{59}-72 q^{57}-154 q^{55}-101 q^{53}+6 q^{51}+77 q^{49}+74 q^{47}+18 q^{45}-27 q^{43}-36 q^{41}-14 q^{39}+4 q^{37}+17 q^{35}+12 q^{33}-q^{31}-4 q^{29}-q^{27}-q^{25}+q^{23}+3 q^{21}-q^{17}+q^{15} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{32}+q^{30}-2 q^{28}-q^{26}-2 q^{24}-2 q^{22}+q^{20}+3 q^{16}+q^{14}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
1,1	[math]\displaystyle{ q^{84}-4 q^{82}+10 q^{80}-20 q^{78}+34 q^{76}-54 q^{74}+74 q^{72}-92 q^{70}+105 q^{68}-104 q^{66}+90 q^{64}-60 q^{62}+18 q^{60}+36 q^{58}-90 q^{56}+142 q^{54}-176 q^{52}+200 q^{50}-202 q^{48}+182 q^{46}-153 q^{44}+96 q^{42}-52 q^{40}-10 q^{38}+48 q^{36}-86 q^{34}+104 q^{32}-104 q^{30}+99 q^{28}-72 q^{26}+62 q^{24}-34 q^{22}+25 q^{20}-10 q^{18}+6 q^{16}-2 q^{14}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{80}+q^{78}-q^{76}-4 q^{74}-3 q^{72}+2 q^{70}+q^{68}+3 q^{64}+8 q^{62}+4 q^{60}-3 q^{58}+2 q^{54}-4 q^{52}-6 q^{50}-3 q^{48}-3 q^{46}-6 q^{44}-2 q^{42}+q^{40}-3 q^{38}+6 q^{34}+3 q^{32}-3 q^{30}+4 q^{28}+6 q^{26}+q^{24}-2 q^{22}+3 q^{20}+4 q^{18}-q^{16}-q^{14}+q^{12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{68}-2 q^{66}+3 q^{62}-6 q^{60}+q^{58}+6 q^{56}-5 q^{54}+4 q^{52}+10 q^{50}-3 q^{48}-q^{46}-q^{44}-7 q^{42}-9 q^{40}-7 q^{38}+q^{36}-q^{34}-2 q^{32}+10 q^{30}+5 q^{28}-4 q^{26}+9 q^{24}+2 q^{22}-3 q^{20}+4 q^{18}+q^{16}-q^{14}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ q^{43}+q^{41}+q^{39}-2 q^{37}-q^{35}-4 q^{33}-2 q^{31}-2 q^{29}+q^{27}+q^{25}+2 q^{23}+3 q^{21}+q^{19}+2 q^{17}+2 q^{13}-q^{11}+q^9 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{90}+q^{88}-q^{86}-3 q^{84}-q^{82}-5 q^{78}-4 q^{76}+6 q^{74}+8 q^{72}+3 q^{70}+9 q^{68}+14 q^{66}+5 q^{64}-5 q^{62}-4 q^{60}-9 q^{58}-20 q^{56}-17 q^{54}-7 q^{52}-10 q^{50}-6 q^{48}+9 q^{46}+9 q^{44}+2 q^{42}+7 q^{40}+12 q^{38}+4 q^{36}-q^{34}+4 q^{32}+6 q^{30}-q^{28}+3 q^{24}+q^{22}-q^{20}+q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ q^{54}+q^{52}+q^{50}+q^{48}-2 q^{46}-q^{44}-4 q^{42}-4 q^{40}-2 q^{38}-2 q^{36}+q^{34}+q^{32}+3 q^{30}+2 q^{28}+3 q^{26}+q^{24}+2 q^{22}+q^{20}+2 q^{16}-q^{14}+q^{12} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{68}-2 q^{66}+4 q^{64}-5 q^{62}+6 q^{60}-7 q^{58}+6 q^{56}-5 q^{54}+2 q^{52}-5 q^{48}+7 q^{46}-11 q^{44}+11 q^{42}-13 q^{40}+11 q^{38}-9 q^{36}+7 q^{34}-2 q^{32}+5 q^{28}-4 q^{26}+7 q^{24}-6 q^{22}+7 q^{20}-4 q^{18}+3 q^{16}-q^{14}+q^{12} }[/math]
1,0	[math]\displaystyle{ q^{110}-2 q^{106}-2 q^{104}+2 q^{102}+4 q^{100}-q^{98}-6 q^{96}-3 q^{94}+6 q^{92}+6 q^{90}-2 q^{88}-6 q^{86}+q^{84}+8 q^{82}+5 q^{80}-4 q^{78}-3 q^{76}+3 q^{74}+3 q^{72}-5 q^{70}-7 q^{68}-2 q^{66}+2 q^{64}-3 q^{62}-7 q^{60}-3 q^{58}+3 q^{56}+3 q^{54}-4 q^{52}-3 q^{50}+5 q^{48}+9 q^{46}-5 q^{42}-q^{40}+8 q^{38}+6 q^{36}-2 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+q^{18} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{94}-2 q^{92}+2 q^{90}-3 q^{88}+4 q^{86}-6 q^{84}+4 q^{82}-5 q^{80}+6 q^{78}-3 q^{76}+4 q^{74}+2 q^{72}+5 q^{70}+6 q^{68}-4 q^{66}+5 q^{64}-10 q^{62}+4 q^{60}-17 q^{58}-16 q^{54}+4 q^{52}-8 q^{50}+5 q^{48}-2 q^{46}+6 q^{44}+7 q^{42}+2 q^{40}+6 q^{38}-2 q^{36}+8 q^{34}-3 q^{32}+6 q^{30}-4 q^{28}+5 q^{26}-q^{24}+2 q^{22}-q^{20}+q^{18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+3 q^{154}-q^{152}-6 q^{150}+14 q^{148}-18 q^{146}+20 q^{144}-12 q^{142}-q^{140}+17 q^{138}-27 q^{136}+34 q^{134}-24 q^{132}+7 q^{130}+10 q^{128}-21 q^{126}+25 q^{124}-16 q^{122}+q^{120}+14 q^{118}-20 q^{116}+12 q^{114}-24 q^{110}+34 q^{108}-36 q^{106}+18 q^{104}-2 q^{102}-24 q^{100}+40 q^{98}-47 q^{96}+34 q^{94}-18 q^{92}-7 q^{90}+25 q^{88}-33 q^{86}+26 q^{84}-9 q^{82}-2 q^{80}+16 q^{78}-17 q^{76}+9 q^{74}+10 q^{72}-21 q^{70}+29 q^{68}-21 q^{66}+6 q^{64}+17 q^{62}-27 q^{60}+33 q^{58}-23 q^{56}+12 q^{54}+2 q^{52}-14 q^{50}+17 q^{48}-14 q^{46}+10 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/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["8 15"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 3 t^2-8 t+11-8 t^{-1} +3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 3 z^4+4 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]=

{ 33, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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