10 159

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-4 t^2+9 t-11+9 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 39, -2 }
Jones polynomial	[math]\displaystyle{ -1+4 q^{-1} -5 q^{-2} +7 q^{-3} -7 q^{-4} +6 q^{-5} -5 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^6-2 z^2 a^6-a^6+z^6 a^4+4 z^4 a^4+5 z^2 a^4+a^4-z^4 a^2-z^2 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+3 z^2 a^8+3 z^7 a^7-5 z^5 a^7-z^3 a^7+z a^7+z^8 a^6+3 z^6 a^6-8 z^4 a^6+z^2 a^6+a^6+4 z^7 a^5-5 z^5 a^5+z a^5+z^8 a^4+3 z^4 a^4-4 z^2 a^4+a^4+z^7 a^3+z^5 a^3+z a^3+4 z^4 a^2-2 z^2 a^2-a^2+z^3 a }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}+q^{22}-q^{20}+q^{16}-2 q^{14}+q^{12}-q^{10}+2 q^8+2 q^6+2 q^2-1 }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+7 q^{120}-3 q^{118}-6 q^{116}+20 q^{114}-28 q^{112}+31 q^{110}-22 q^{108}-4 q^{106}+28 q^{104}-49 q^{102}+51 q^{100}-33 q^{98}+2 q^{96}+30 q^{94}-46 q^{92}+39 q^{90}-14 q^{88}-16 q^{86}+37 q^{84}-41 q^{82}+19 q^{80}+16 q^{78}-43 q^{76}+58 q^{74}-48 q^{72}+22 q^{70}+12 q^{68}-44 q^{66}+59 q^{64}-62 q^{62}+43 q^{60}-10 q^{58}-25 q^{56}+47 q^{54}-51 q^{52}+35 q^{50}-6 q^{48}-24 q^{46}+37 q^{44}-31 q^{42}+8 q^{40}+26 q^{38}-46 q^{36}+50 q^{34}-24 q^{32}-8 q^{30}+37 q^{28}-49 q^{26}+44 q^{24}-22 q^{22}+2 q^{20}+16 q^{18}-24 q^{16}+22 q^{14}-12 q^{12}+6 q^{10}+2 q^8-4 q^6+q^4-2 q^2+2-2 q^{-2} + q^{-4} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_159.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{17}+2 q^{15}-2 q^{13}+q^{11}-q^9+2 q^5-q^3+3 q- q^{-1} }[/math]
2	[math]\displaystyle{ q^{48}-2 q^{46}-2 q^{44}+7 q^{42}-q^{40}-10 q^{38}+8 q^{36}+6 q^{34}-12 q^{32}+2 q^{30}+9 q^{28}-7 q^{26}-3 q^{24}+6 q^{22}-7 q^{18}+10 q^{14}-7 q^{12}-6 q^{10}+14 q^8-2 q^6-8 q^4+8 q^2+2-3 q^{-2} }[/math]
3	[math]\displaystyle{ -q^{93}+2 q^{91}+2 q^{89}-3 q^{87}-7 q^{85}+q^{83}+17 q^{81}+5 q^{79}-23 q^{77}-21 q^{75}+21 q^{73}+41 q^{71}-10 q^{69}-52 q^{67}-11 q^{65}+54 q^{63}+33 q^{61}-47 q^{59}-48 q^{57}+32 q^{55}+53 q^{53}-16 q^{51}-52 q^{49}+5 q^{47}+47 q^{45}+5 q^{43}-36 q^{41}-16 q^{39}+28 q^{37}+24 q^{35}-17 q^{33}-40 q^{31}+4 q^{29}+48 q^{27}+14 q^{25}-55 q^{23}-33 q^{21}+52 q^{19}+48 q^{17}-37 q^{15}-52 q^{13}+18 q^{11}+49 q^9+2 q^7-35 q^5-7 q^3+15 q+13 q^{-1} -4 q^{-3} -6 q^{-5} - q^{-7} + q^{-11} }[/math]
5	[math]\displaystyle{ -q^{225}+2 q^{223}+2 q^{221}-3 q^{219}-3 q^{217}-3 q^{215}+8 q^{211}+17 q^{209}+5 q^{207}-19 q^{205}-34 q^{203}-31 q^{201}+8 q^{199}+67 q^{197}+100 q^{195}+37 q^{193}-85 q^{191}-182 q^{189}-170 q^{187}+6 q^{185}+254 q^{183}+379 q^{181}+205 q^{179}-191 q^{177}-553 q^{175}-570 q^{173}-108 q^{171}+576 q^{169}+954 q^{167}+613 q^{165}-288 q^{163}-1147 q^{161}-1235 q^{159}-315 q^{157}+1018 q^{155}+1719 q^{153}+1090 q^{151}-499 q^{149}-1867 q^{147}-1834 q^{145}-276 q^{143}+1631 q^{141}+2302 q^{139}+1087 q^{137}-1068 q^{135}-2400 q^{133}-1736 q^{131}+380 q^{129}+2158 q^{127}+2085 q^{125}+241 q^{123}-1709 q^{121}-2108 q^{119}-685 q^{117}+1213 q^{115}+1904 q^{113}+897 q^{111}-784 q^{109}-1587 q^{107}-915 q^{105}+468 q^{103}+1263 q^{101}+845 q^{99}-270 q^{97}-1016 q^{95}-760 q^{93}+140 q^{91}+838 q^{89}+760 q^{87}+2 q^{85}-764 q^{83}-857 q^{81}-200 q^{79}+680 q^{77}+1070 q^{75}+548 q^{73}-557 q^{71}-1327 q^{69}-1006 q^{67}+276 q^{65}+1528 q^{63}+1572 q^{61}+188 q^{59}-1555 q^{57}-2114 q^{55}-822 q^{53}+1313 q^{51}+2453 q^{49}+1516 q^{47}-779 q^{45}-2470 q^{43}-2099 q^{41}+52 q^{39}+2098 q^{37}+2352 q^{35}+694 q^{33}-1404 q^{31}-2210 q^{29}-1225 q^{27}+604 q^{25}+1720 q^{23}+1376 q^{21}+84 q^{19}-1029 q^{17}-1198 q^{15}-479 q^{13}+431 q^{11}+793 q^9+531 q^7+2 q^5-390 q^3-399 q-145 q^{-1} +112 q^{-3} +195 q^{-5} +135 q^{-7} +16 q^{-9} -61 q^{-11} -69 q^{-13} -31 q^{-15} +7 q^{-17} +15 q^{-19} +14 q^{-21} +5 q^{-23} -3 q^{-25} -3 q^{-27} }[/math]
6	[math]\displaystyle{ q^{312}-2 q^{310}-2 q^{308}+3 q^{306}+3 q^{304}+3 q^{302}-4 q^{300}-8 q^{296}-17 q^{294}+4 q^{292}+19 q^{290}+34 q^{288}+18 q^{286}+9 q^{284}-43 q^{282}-100 q^{280}-80 q^{278}-7 q^{276}+118 q^{274}+185 q^{272}+239 q^{270}+75 q^{268}-222 q^{266}-453 q^{264}-498 q^{262}-214 q^{260}+241 q^{258}+868 q^{256}+1056 q^{254}+625 q^{252}-316 q^{250}-1349 q^{248}-1866 q^{246}-1489 q^{244}+144 q^{242}+1985 q^{240}+3123 q^{238}+2615 q^{236}+459 q^{234}-2521 q^{232}-4791 q^{230}-4348 q^{228}-1354 q^{226}+3101 q^{224}+6444 q^{222}+6618 q^{220}+2712 q^{218}-3620 q^{216}-8439 q^{214}-8989 q^{212}-4064 q^{210}+3838 q^{208}+10568 q^{206}+11424 q^{204}+5258 q^{202}-4465 q^{200}-12443 q^{198}-13299 q^{196}-6170 q^{194}+5480 q^{192}+14177 q^{190}+14421 q^{188}+5894 q^{186}-6669 q^{184}-15289 q^{182}-14629 q^{180}-4520 q^{178}+8197 q^{176}+15648 q^{174}+13209 q^{172}+2415 q^{170}-9513 q^{168}-15072 q^{166}-10522 q^{164}+283 q^{162}+10272 q^{160}+13020 q^{158}+7169 q^{156}-2786 q^{154}-10137 q^{152}-9989 q^{150}-3533 q^{148}+4582 q^{146}+8720 q^{144}+6635 q^{142}+416 q^{140}-5354 q^{138}-6624 q^{136}-3380 q^{134}+1833 q^{132}+5062 q^{130}+4453 q^{128}+764 q^{126}-3163 q^{124}-4462 q^{122}-2564 q^{120}+1217 q^{118}+3959 q^{116}+4007 q^{114}+1138 q^{112}-2712 q^{110}-4976 q^{108}-3971 q^{106}+56 q^{104}+4378 q^{102}+6410 q^{100}+4200 q^{98}-1200 q^{96}-6731 q^{94}-8368 q^{92}-4361 q^{90}+3030 q^{88}+9591 q^{86}+10372 q^{84}+4282 q^{82}-5692 q^{80}-12793 q^{78}-11929 q^{76}-3197 q^{74}+8601 q^{72}+15503 q^{70}+12786 q^{68}+1262 q^{66}-11398 q^{64}-17106 q^{62}-12176 q^{60}+719 q^{58}+13143 q^{56}+17409 q^{54}+10472 q^{52}-2507 q^{50}-13447 q^{48}-15869 q^{46}-8503 q^{44}+3412 q^{42}+12497 q^{40}+13179 q^{38}+6336 q^{36}-3402 q^{34}-10144 q^{32}-10240 q^{30}-4553 q^{28}+2904 q^{26}+7350 q^{24}+7190 q^{22}+3168 q^{20}-1851 q^{18}-4870 q^{16}-4657 q^{14}-2003 q^{12}+930 q^{10}+2755 q^8+2734 q^6+1306 q^4-392 q^2-1368-1357 q^{-2} -752 q^{-4} +46 q^{-6} +557 q^{-8} +618 q^{-10} +350 q^{-12} +38 q^{-14} -154 q^{-16} -218 q^{-18} -148 q^{-20} -35 q^{-22} +34 q^{-24} +49 q^{-26} +37 q^{-28} +19 q^{-30} -11 q^{-34} -5 q^{-36} - q^{-38} - q^{-40} - q^{-42} + q^{-46} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{24}+q^{22}-q^{20}+q^{16}-2 q^{14}+q^{12}-q^{10}+2 q^8+2 q^6+2 q^2-1 }[/math]
1,1	[math]\displaystyle{ q^{68}-4 q^{66}+12 q^{64}-28 q^{62}+50 q^{60}-80 q^{58}+116 q^{56}-144 q^{54}+158 q^{52}-154 q^{50}+122 q^{48}-62 q^{46}-11 q^{44}+92 q^{42}-172 q^{40}+234 q^{38}-273 q^{36}+284 q^{34}-272 q^{32}+234 q^{30}-172 q^{28}+94 q^{26}-18 q^{24}-60 q^{22}+114 q^{20}-150 q^{18}+160 q^{16}-140 q^{14}+119 q^{12}-76 q^{10}+52 q^8-28 q^6+15 q^4-4 q^2+2-4 q^{-2} + q^{-4} }[/math]
2,0	[math]\displaystyle{ q^{62}-q^{60}-2 q^{58}+2 q^{56}+2 q^{54}-2 q^{52}-2 q^{50}+2 q^{48}+5 q^{46}-4 q^{44}-3 q^{42}+4 q^{40}-q^{38}-3 q^{36}-q^{34}+3 q^{32}-q^{30}-2 q^{28}+q^{26}-2 q^{24}-5 q^{22}+2 q^{20}+5 q^{18}-3 q^{16}+q^{14}+7 q^{12}+3 q^{10}-q^8-q^6+5 q^4-3 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{54}-2 q^{52}+q^{50}+3 q^{48}-6 q^{46}+4 q^{44}+3 q^{42}-9 q^{40}+6 q^{38}+2 q^{36}-8 q^{34}+2 q^{32}+3 q^{30}-3 q^{28}-q^{26}+q^{24}+2 q^{22}-3 q^{20}-4 q^{18}+9 q^{16}-3 q^{14}-2 q^{12}+12 q^{10}-2 q^8-3 q^6+6 q^4-2 q^2-2+ q^{-2} }[/math]
1,0,0	[math]\displaystyle{ -q^{31}+q^{29}-2 q^{27}+q^{25}-q^{23}+q^{21}-q^{19}+2 q^{11}+q^9+3 q^7-q^5+2 q^3-q }[/math]
1,0,1	[math]\displaystyle{ q^{88}-4 q^{86}+10 q^{84}-14 q^{82}+7 q^{80}+15 q^{78}-48 q^{76}+71 q^{74}-64 q^{72}+19 q^{70}+54 q^{68}-120 q^{66}+153 q^{64}-133 q^{62}+60 q^{60}+37 q^{58}-117 q^{56}+150 q^{54}-124 q^{52}+68 q^{50}-22 q^{48}+9 q^{46}-27 q^{44}+36 q^{42}-15 q^{40}-50 q^{38}+121 q^{36}-173 q^{34}+156 q^{32}-91 q^{30}-14 q^{28}+99 q^{26}-138 q^{24}+123 q^{22}-57 q^{20}+q^{18}+51 q^{16}-41 q^{14}+30 q^{12}+6 q^{10}-15 q^8+11 q^6-2 q^4-7 q^2+6-4 q^{-2} + q^{-4} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{54}+2 q^{52}-5 q^{50}+7 q^{48}-8 q^{46}+10 q^{44}-9 q^{42}+7 q^{40}-4 q^{38}+4 q^{34}-10 q^{32}+13 q^{30}-17 q^{28}+17 q^{26}-17 q^{24}+14 q^{22}-9 q^{20}+6 q^{18}+q^{16}-3 q^{14}+8 q^{12}-8 q^{10}+10 q^8-9 q^6+8 q^4-4 q^2+2- q^{-2} }[/math]
1,0	[math]\displaystyle{ q^{88}-2 q^{84}-2 q^{82}+3 q^{80}+5 q^{78}-2 q^{76}-7 q^{74}-2 q^{72}+9 q^{70}+6 q^{68}-8 q^{66}-9 q^{64}+3 q^{62}+10 q^{60}+q^{58}-9 q^{56}-4 q^{54}+6 q^{52}+4 q^{50}-4 q^{48}-5 q^{46}+3 q^{44}+6 q^{42}-q^{40}-8 q^{38}-q^{36}+7 q^{34}+2 q^{32}-7 q^{30}-6 q^{28}+7 q^{26}+8 q^{24}-3 q^{22}-9 q^{20}+3 q^{18}+11 q^{16}+5 q^{14}-6 q^{12}-6 q^{10}+3 q^8+7 q^6-3 q^2-2+ q^{-4} }[/math]

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+7 q^{120}-3 q^{118}-6 q^{116}+20 q^{114}-28 q^{112}+31 q^{110}-22 q^{108}-4 q^{106}+28 q^{104}-49 q^{102}+51 q^{100}-33 q^{98}+2 q^{96}+30 q^{94}-46 q^{92}+39 q^{90}-14 q^{88}-16 q^{86}+37 q^{84}-41 q^{82}+19 q^{80}+16 q^{78}-43 q^{76}+58 q^{74}-48 q^{72}+22 q^{70}+12 q^{68}-44 q^{66}+59 q^{64}-62 q^{62}+43 q^{60}-10 q^{58}-25 q^{56}+47 q^{54}-51 q^{52}+35 q^{50}-6 q^{48}-24 q^{46}+37 q^{44}-31 q^{42}+8 q^{40}+26 q^{38}-46 q^{36}+50 q^{34}-24 q^{32}-8 q^{30}+37 q^{28}-49 q^{26}+44 q^{24}-22 q^{22}+2 q^{20}+16 q^{18}-24 q^{16}+22 q^{14}-12 q^{12}+6 q^{10}+2 q^8-4 q^6+q^4-2 q^2+2-2 q^{-2} + q^{-4} }[/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["10 159"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-4 t^2+9 t-11+9 t^{-1} -4 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+2 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]=

{ 39, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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