10 152

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-t^3-t^2+4 t-5+4 t^{-1} - t^{-2} - t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+7 z^6+13 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 11, -6 }
Jones polynomial	[math]\displaystyle{ q^{-4} + q^{-6} + q^{-7} -2 q^{-8} +2 q^{-9} -3 q^{-10} +2 q^{-11} -2 q^{-12} + q^{-13} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ 2 z^2 a^{12}+3 a^{12}-z^6 a^{10}-8 z^4 a^{10}-17 z^2 a^{10}-10 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+22 z^2 a^8+8 a^8 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{16}-2 z^2 a^{16}+2 z^5 a^{15}-5 z^3 a^{15}+2 z a^{15}+z^6 a^{14}-z^4 a^{14}-z^2 a^{14}+2 z^5 a^{13}-3 z^3 a^{13}+z a^{13}+2 z^4 a^{12}-3 z^2 a^{12}+3 a^{12}+z^7 a^{11}-8 z^5 a^{11}+19 z^3 a^{11}-11 z a^{11}+z^8 a^{10}-9 z^6 a^{10}+25 z^4 a^{10}-26 z^2 a^{10}+10 a^{10}+z^7 a^9-8 z^5 a^9+17 z^3 a^9-10 z a^9+z^8 a^8-8 z^6 a^8+21 z^4 a^8-22 z^2 a^8+8 a^8 }[/math]
The A2 invariant	[math]\displaystyle{ 2 q^{40}-q^{34}-3 q^{32}-2 q^{30}-3 q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+q^{16}+q^{14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{210}-q^{208}+2 q^{206}-3 q^{204}+q^{200}-4 q^{198}+5 q^{196}-5 q^{194}+2 q^{192}+2 q^{190}-6 q^{188}+6 q^{186}-2 q^{184}-q^{182}+8 q^{180}-7 q^{178}+5 q^{176}+3 q^{174}-6 q^{172}+11 q^{170}-8 q^{168}+3 q^{166}+4 q^{164}-6 q^{162}+9 q^{160}-6 q^{158}+2 q^{156}+2 q^{154}-4 q^{152}+3 q^{150}-5 q^{148}+q^{146}+q^{144}-5 q^{142}+3 q^{140}-6 q^{138}+2 q^{134}-11 q^{132}+6 q^{130}-8 q^{128}-q^{126}+3 q^{124}-12 q^{122}+7 q^{120}-4 q^{118}-3 q^{116}+3 q^{114}-7 q^{112}+2 q^{110}+3 q^{108}-3 q^{106}+5 q^{104}-q^{102}+q^{100}+5 q^{98}-2 q^{96}+4 q^{94}+2 q^{92}+2 q^{90}+2 q^{88}+2 q^{86}+q^{84}+3 q^{82}+2 q^{80}+2 q^{76}+q^{74}+q^{72}+q^{70} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_152.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{27}-q^{25}-q^{21}-q^{19}-q^{15}+2 q^{13}+q^{11}+q^9+q^7 }[/math]
2	[math]\displaystyle{ q^{74}-q^{72}-2 q^{70}+2 q^{68}+q^{66}-3 q^{64}+4 q^{60}-q^{58}-q^{56}+2 q^{54}+q^{52}-q^{50}+2 q^{46}-3 q^{44}-3 q^{42}+2 q^{40}-2 q^{38}-4 q^{36}+q^{34}+q^{32}-q^{30}-q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14} }[/math]
3	[math]\displaystyle{ q^{141}-q^{139}-2 q^{137}+3 q^{133}+3 q^{131}-4 q^{129}-5 q^{127}+2 q^{125}+9 q^{123}+4 q^{121}-10 q^{119}-10 q^{117}+6 q^{115}+11 q^{113}-2 q^{111}-12 q^{109}+9 q^{105}+2 q^{103}-6 q^{101}-3 q^{99}+3 q^{97}+3 q^{95}-q^{93}-4 q^{91}+q^{89}+6 q^{87}+3 q^{85}-7 q^{83}-3 q^{81}+10 q^{79}+9 q^{77}-8 q^{75}-8 q^{73}+3 q^{71}+10 q^{69}-2 q^{67}-11 q^{65}-5 q^{63}+3 q^{61}+5 q^{59}-2 q^{57}-6 q^{55}-4 q^{53}+q^{51}+2 q^{49}+q^{47}-q^{45}-q^{43}-q^{41}+2 q^{39}+2 q^{37}+2 q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21} }[/math]
5	[math]\displaystyle{ q^{335}-q^{333}-2 q^{331}+q^{327}+3 q^{325}+3 q^{323}+q^{321}-6 q^{319}-8 q^{317}-3 q^{315}+4 q^{313}+12 q^{311}+15 q^{309}+5 q^{307}-18 q^{305}-29 q^{303}-24 q^{301}+q^{299}+38 q^{297}+57 q^{295}+37 q^{293}-24 q^{291}-77 q^{289}-86 q^{287}-29 q^{285}+70 q^{283}+132 q^{281}+99 q^{279}-21 q^{277}-139 q^{275}-166 q^{273}-58 q^{271}+104 q^{269}+193 q^{267}+131 q^{265}-44 q^{263}-178 q^{261}-168 q^{259}-23 q^{257}+132 q^{255}+167 q^{253}+64 q^{251}-79 q^{249}-132 q^{247}-73 q^{245}+34 q^{243}+93 q^{241}+63 q^{239}-8 q^{237}-54 q^{235}-43 q^{233}-q^{231}+29 q^{229}+28 q^{227}+4 q^{225}-19 q^{223}-21 q^{221}-7 q^{219}+13 q^{217}+25 q^{215}+15 q^{213}-16 q^{211}-39 q^{209}-29 q^{207}+14 q^{205}+58 q^{203}+54 q^{201}-12 q^{199}-81 q^{197}-87 q^{195}-q^{193}+101 q^{191}+125 q^{189}+29 q^{187}-104 q^{185}-155 q^{183}-75 q^{181}+88 q^{179}+174 q^{177}+110 q^{175}-42 q^{173}-165 q^{171}-155 q^{169}-19 q^{167}+128 q^{165}+163 q^{163}+77 q^{161}-56 q^{159}-141 q^{157}-115 q^{155}-13 q^{153}+89 q^{151}+113 q^{149}+66 q^{147}-14 q^{145}-75 q^{143}-79 q^{141}-33 q^{139}+25 q^{137}+58 q^{135}+55 q^{133}+23 q^{131}-16 q^{129}-37 q^{127}-37 q^{125}-16 q^{123}+7 q^{121}+23 q^{119}+23 q^{117}+14 q^{115}-4 q^{113}-15 q^{111}-16 q^{109}-14 q^{107}-5 q^{105}+4 q^{103}+7 q^{101}+6 q^{99}+5 q^{97}-2 q^{95}-6 q^{93}-6 q^{91}-6 q^{89}-4 q^{87}+q^{85}+2 q^{83}+2 q^{81}+2 q^{79}+q^{77}-q^{75}-q^{73}-q^{71}-q^{69}-q^{67}+2 q^{65}+2 q^{63}+2 q^{61}+2 q^{59}+2 q^{57}+q^{55}+q^{53}+q^{51}+q^{49}+q^{47}+q^{45}+q^{43}+q^{41}+q^{39}+q^{37}+q^{35} }[/math]
6	[math]\displaystyle{ q^{462}-q^{460}-2 q^{458}+q^{454}+3 q^{452}+q^{450}+3 q^{448}-q^{446}-8 q^{444}-7 q^{442}-3 q^{440}+5 q^{438}+9 q^{436}+17 q^{434}+11 q^{432}-5 q^{430}-23 q^{428}-32 q^{426}-25 q^{424}-9 q^{422}+36 q^{420}+67 q^{418}+67 q^{416}+27 q^{414}-38 q^{412}-103 q^{410}-141 q^{408}-89 q^{406}+29 q^{404}+161 q^{402}+225 q^{400}+189 q^{398}+23 q^{396}-209 q^{394}-356 q^{392}-325 q^{390}-92 q^{388}+226 q^{386}+487 q^{384}+490 q^{382}+194 q^{380}-253 q^{378}-600 q^{376}-627 q^{374}-291 q^{372}+270 q^{370}+694 q^{368}+726 q^{366}+319 q^{364}-287 q^{362}-728 q^{360}-743 q^{358}-288 q^{356}+318 q^{354}+709 q^{352}+646 q^{350}+200 q^{348}-332 q^{346}-619 q^{344}-485 q^{342}-81 q^{340}+319 q^{338}+457 q^{336}+303 q^{334}-11 q^{332}-264 q^{330}-296 q^{328}-148 q^{326}+60 q^{324}+174 q^{322}+159 q^{320}+50 q^{318}-62 q^{316}-103 q^{314}-71 q^{312}+3 q^{310}+50 q^{308}+60 q^{306}+27 q^{304}-20 q^{302}-53 q^{300}-47 q^{298}+3 q^{296}+52 q^{294}+78 q^{292}+49 q^{290}-27 q^{288}-114 q^{286}-131 q^{284}-39 q^{282}+100 q^{280}+207 q^{278}+176 q^{276}+4 q^{274}-228 q^{272}-331 q^{270}-193 q^{268}+106 q^{266}+384 q^{264}+424 q^{262}+168 q^{260}-265 q^{258}-551 q^{256}-476 q^{254}-74 q^{252}+413 q^{250}+660 q^{248}+493 q^{246}-33 q^{244}-539 q^{242}-707 q^{240}-443 q^{238}+98 q^{236}+582 q^{234}+718 q^{232}+401 q^{230}-131 q^{228}-560 q^{226}-652 q^{224}-383 q^{222}+88 q^{220}+484 q^{218}+572 q^{216}+358 q^{214}-25 q^{212}-353 q^{210}-476 q^{208}-348 q^{206}-57 q^{204}+220 q^{202}+359 q^{200}+311 q^{198}+130 q^{196}-93 q^{194}-237 q^{192}-257 q^{190}-159 q^{188}-10 q^{186}+120 q^{184}+186 q^{182}+158 q^{180}+75 q^{178}-26 q^{176}-96 q^{174}-119 q^{172}-93 q^{170}-32 q^{168}+30 q^{166}+70 q^{164}+77 q^{162}+59 q^{160}+21 q^{158}-18 q^{156}-41 q^{154}-46 q^{152}-38 q^{150}-16 q^{148}+8 q^{146}+24 q^{144}+27 q^{142}+23 q^{140}+14 q^{138}-4 q^{136}-15 q^{134}-18 q^{132}-16 q^{130}-14 q^{128}-5 q^{126}+4 q^{124}+7 q^{122}+7 q^{120}+6 q^{118}+5 q^{116}-2 q^{114}-6 q^{112}-6 q^{110}-6 q^{108}-6 q^{106}-4 q^{104}+q^{102}+2 q^{100}+2 q^{98}+2 q^{96}+2 q^{94}+q^{92}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}+2 q^{78}+2 q^{76}+2 q^{74}+2 q^{72}+2 q^{70}+2 q^{68}+q^{66}+q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ 2 q^{40}-q^{34}-3 q^{32}-2 q^{30}-3 q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+q^{16}+q^{14} }[/math]
1,1	[math]\displaystyle{ q^{108}-2 q^{106}+4 q^{104}-8 q^{102}+9 q^{100}-10 q^{98}+10 q^{96}-4 q^{94}-4 q^{92}+12 q^{90}-20 q^{88}+24 q^{86}-25 q^{84}+22 q^{82}-16 q^{80}+8 q^{78}-q^{76}-8 q^{74}+14 q^{72}-16 q^{70}+28 q^{68}-18 q^{66}+28 q^{64}-10 q^{62}+9 q^{60}-4 q^{58}-16 q^{56}-2 q^{54}-24 q^{52}-4 q^{50}-14 q^{48}-2 q^{46}+q^{44}+6 q^{42}+8 q^{40}+10 q^{38}+7 q^{36}+8 q^{34}+4 q^{32}+2 q^{30}+q^{28} }[/math]
2,0	[math]\displaystyle{ 3 q^{100}-q^{96}-2 q^{94}-3 q^{92}-3 q^{90}-6 q^{88}+3 q^{84}+5 q^{82}+5 q^{80}+6 q^{78}+5 q^{76}+5 q^{74}+3 q^{72}+q^{70}-3 q^{66}-5 q^{64}-7 q^{62}-10 q^{60}-8 q^{58}-6 q^{56}-5 q^{54}-3 q^{52}-q^{50}+4 q^{48}+3 q^{46}+3 q^{44}+4 q^{42}+6 q^{40}+4 q^{38}+3 q^{36}+2 q^{34}+2 q^{32}+q^{30}+q^{28} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{88}-q^{86}-3 q^{80}+q^{78}-q^{74}+2 q^{72}+q^{70}+q^{68}+3 q^{66}+3 q^{64}+6 q^{62}+4 q^{60}+q^{58}-2 q^{56}-8 q^{54}-13 q^{52}-10 q^{50}-11 q^{48}-6 q^{46}+2 q^{44}+3 q^{42}+9 q^{40}+7 q^{38}+7 q^{36}+5 q^{34}+3 q^{32}+q^{30}+q^{28} }[/math]
1,0,0	[math]\displaystyle{ 2 q^{53}+q^{51}+2 q^{49}-2 q^{45}-4 q^{43}-5 q^{41}-4 q^{39}-3 q^{37}+2 q^{33}+4 q^{31}+3 q^{29}+4 q^{27}+2 q^{25}+q^{23}+q^{21} }[/math]
1,0,1	[math]\displaystyle{ q^{142}-2 q^{140}+3 q^{138}-3 q^{136}-q^{134}+5 q^{132}-8 q^{130}+10 q^{128}-8 q^{126}+4 q^{124}-6 q^{120}+12 q^{118}-16 q^{116}+13 q^{114}-9 q^{112}+8 q^{110}-q^{108}+3 q^{106}+7 q^{104}-16 q^{102}+11 q^{100}-31 q^{98}+2 q^{96}-21 q^{94}-q^{92}+13 q^{90}+9 q^{88}+49 q^{86}+26 q^{84}+47 q^{82}+22 q^{80}+10 q^{78}-12 q^{76}-36 q^{74}-46 q^{72}-63 q^{70}-47 q^{68}-44 q^{66}-21 q^{64}-2 q^{62}+12 q^{60}+27 q^{58}+27 q^{56}+30 q^{54}+23 q^{52}+16 q^{50}+11 q^{48}+5 q^{46}+2 q^{44}+q^{42} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{116}+2 q^{114}-q^{112}-q^{110}+q^{108}-2 q^{106}-6 q^{104}-4 q^{102}-2 q^{100}-4 q^{98}-3 q^{96}+3 q^{94}+7 q^{92}+6 q^{90}+14 q^{88}+18 q^{86}+15 q^{84}+13 q^{82}+12 q^{80}-2 q^{78}-13 q^{76}-21 q^{74}-27 q^{72}-32 q^{70}-30 q^{68}-18 q^{66}-9 q^{64}+q^{62}+10 q^{60}+17 q^{58}+16 q^{56}+17 q^{54}+12 q^{52}+9 q^{50}+6 q^{48}+3 q^{46}+q^{44}+q^{42} }[/math]
1,0,0,0	[math]\displaystyle{ 2 q^{66}+q^{64}+3 q^{62}+2 q^{60}-2 q^{56}-5 q^{54}-6 q^{52}-7 q^{50}-5 q^{48}-3 q^{46}+q^{44}+2 q^{42}+5 q^{40}+5 q^{38}+4 q^{36}+4 q^{34}+2 q^{32}+q^{30}+q^{28} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{88}-q^{86}+2 q^{84}-2 q^{82}+q^{80}-q^{78}+q^{74}-2 q^{72}+3 q^{70}-3 q^{68}+3 q^{66}-3 q^{64}+2 q^{62}-2 q^{60}-q^{58}-4 q^{54}+q^{52}-4 q^{50}+q^{48}-2 q^{46}+2 q^{44}+q^{42}+q^{40}+3 q^{38}+q^{36}+3 q^{34}+q^{32}+q^{30}+q^{28} }[/math]
1,0	[math]\displaystyle{ q^{142}-q^{138}-q^{136}+q^{134}+q^{132}-2 q^{130}-2 q^{128}+2 q^{124}-2 q^{120}+2 q^{116}+q^{114}-q^{112}+3 q^{108}+3 q^{106}+q^{104}+3 q^{100}+3 q^{98}+2 q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-5 q^{86}-6 q^{84}-5 q^{82}-2 q^{80}-4 q^{78}-6 q^{76}-4 q^{74}+q^{70}-q^{68}+2 q^{66}+3 q^{64}+4 q^{62}+2 q^{60}+3 q^{58}+3 q^{56}+4 q^{54}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{42} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{122}-q^{120}+q^{118}-2 q^{116}+q^{114}-2 q^{112}-q^{108}+q^{104}-q^{102}+3 q^{100}-q^{98}+3 q^{96}-2 q^{94}+4 q^{92}+q^{90}+6 q^{88}+4 q^{86}+6 q^{84}+4 q^{82}+2 q^{80}-q^{78}-9 q^{76}-10 q^{74}-16 q^{72}-12 q^{70}-15 q^{68}-8 q^{66}-6 q^{64}+2 q^{62}+6 q^{60}+8 q^{58}+10 q^{56}+9 q^{54}+9 q^{52}+5 q^{50}+5 q^{48}+2 q^{46}+q^{44}+q^{42} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{210}-q^{208}+2 q^{206}-3 q^{204}+q^{200}-4 q^{198}+5 q^{196}-5 q^{194}+2 q^{192}+2 q^{190}-6 q^{188}+6 q^{186}-2 q^{184}-q^{182}+8 q^{180}-7 q^{178}+5 q^{176}+3 q^{174}-6 q^{172}+11 q^{170}-8 q^{168}+3 q^{166}+4 q^{164}-6 q^{162}+9 q^{160}-6 q^{158}+2 q^{156}+2 q^{154}-4 q^{152}+3 q^{150}-5 q^{148}+q^{146}+q^{144}-5 q^{142}+3 q^{140}-6 q^{138}+2 q^{134}-11 q^{132}+6 q^{130}-8 q^{128}-q^{126}+3 q^{124}-12 q^{122}+7 q^{120}-4 q^{118}-3 q^{116}+3 q^{114}-7 q^{112}+2 q^{110}+3 q^{108}-3 q^{106}+5 q^{104}-q^{102}+q^{100}+5 q^{98}-2 q^{96}+4 q^{94}+2 q^{92}+2 q^{90}+2 q^{88}+2 q^{86}+q^{84}+3 q^{82}+2 q^{80}+2 q^{76}+q^{74}+q^{72}+q^{70} }[/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["10 152"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-t^3-t^2+4 t-5+4 t^{-1} - t^{-2} - t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+7 z^6+13 z^4+7 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]=

{ 11, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-4} + q^{-6} + q^{-7} -2 q^{-8} +2 q^{-9} -3 q^{-10} +2 q^{-11} -2 q^{-12} + q^{-13} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ 2 z^2 a^{12}+3 a^{12}-z^6 a^{10}-8 z^4 a^{10}-17 z^2 a^{10}-10 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+22 z^2 a^8+8 a^8 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^4 a^{16}-2 z^2 a^{16}+2 z^5 a^{15}-5 z^3 a^{15}+2 z a^{15}+z^6 a^{14}-z^4 a^{14}-z^2 a^{14}+2 z^5 a^{13}-3 z^3 a^{13}+z a^{13}+2 z^4 a^{12}-3 z^2 a^{12}+3 a^{12}+z^7 a^{11}-8 z^5 a^{11}+19 z^3 a^{11}-11 z a^{11}+z^8 a^{10}-9 z^6 a^{10}+25 z^4 a^{10}-26 z^2 a^{10}+10 a^{10}+z^7 a^9-8 z^5 a^9+17 z^3 a^9-10 z a^9+z^8 a^8-8 z^6 a^8+21 z^4 a^8-22 z^2 a^8+8 a^8 }[/math]