10 89

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-8 t^2+24 t-33+24 t^{-1} -8 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6-2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 99, -2 }
Jones polynomial	[math]\displaystyle{ -q^2+5 q-9+13 q^{-1} -16 q^{-2} +17 q^{-3} -15 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^8+3 z^2 a^6+2 a^6-3 z^4 a^4-4 z^2 a^4-a^4+z^6 a^2+2 z^4 a^2+2 z^2 a^2-z^4+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-5 z^4 a^8+3 z^2 a^8-a^8+5 z^7 a^7-7 z^5 a^7+4 z^3 a^7-z a^7+5 z^8 a^6-3 z^6 a^6-4 z^4 a^6+6 z^2 a^6-2 a^6+2 z^9 a^5+11 z^7 a^5-27 z^5 a^5+20 z^3 a^5-4 z a^5+12 z^8 a^4-15 z^6 a^4-2 z^4 a^4+6 z^2 a^4-a^4+2 z^9 a^3+15 z^7 a^3-35 z^5 a^3+19 z^3 a^3-2 z a^3+7 z^8 a^2-4 z^6 a^2-9 z^4 a^2+3 z^2 a^2+9 z^7 a-15 z^5 a+5 z^3 a+5 z^6-6 z^4+1+z^5 a^{-1} }[/math]
The A2 invariant	[math]\displaystyle{ -q^{26}-q^{24}+2 q^{22}-q^{20}-q^{18}+4 q^{16}-2 q^{14}+2 q^{12}-2 q^8+2 q^6-4 q^4+4 q^2- q^{-2} +3 q^{-4} - q^{-6} }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+9 q^{120}-9 q^{118}+q^{116}+13 q^{114}-32 q^{112}+52 q^{110}-67 q^{108}+62 q^{106}-34 q^{104}-26 q^{102}+111 q^{100}-190 q^{98}+234 q^{96}-208 q^{94}+89 q^{92}+87 q^{90}-276 q^{88}+405 q^{86}-402 q^{84}+261 q^{82}-14 q^{80}-243 q^{78}+405 q^{76}-399 q^{74}+229 q^{72}+28 q^{70}-252 q^{68}+332 q^{66}-238 q^{64}+7 q^{62}+270 q^{60}-447 q^{58}+447 q^{56}-250 q^{54}-76 q^{52}+406 q^{50}-617 q^{48}+623 q^{46}-423 q^{44}+92 q^{42}+264 q^{40}-515 q^{38}+577 q^{36}-433 q^{34}+148 q^{32}+148 q^{30}-350 q^{28}+361 q^{26}-198 q^{24}-53 q^{22}+282 q^{20}-371 q^{18}+283 q^{16}-53 q^{14}-226 q^{12}+424 q^{10}-463 q^8+336 q^6-99 q^4-147 q^2+319-360 q^{-2} +293 q^{-4} -149 q^{-6} - q^{-8} +105 q^{-10} -152 q^{-12} +134 q^{-14} -84 q^{-16} +37 q^{-18} +4 q^{-20} -22 q^{-22} +25 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_89.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{17}+2 q^{15}-4 q^{13}+5 q^{11}-3 q^9+2 q^7+q^5-3 q^3+4 q-4 q^{-1} +4 q^{-3} - q^{-5} }[/math]
2	[math]\displaystyle{ q^{48}-2 q^{46}+7 q^{42}-11 q^{40}-4 q^{38}+28 q^{36}-22 q^{34}-24 q^{32}+52 q^{30}-13 q^{28}-46 q^{26}+45 q^{24}+10 q^{22}-41 q^{20}+10 q^{18}+26 q^{16}-11 q^{14}-28 q^{12}+28 q^{10}+23 q^8-51 q^6+13 q^4+46 q^2-45-9 q^{-2} +41 q^{-4} -18 q^{-6} -15 q^{-8} +16 q^{-10} -4 q^{-14} + q^{-16} }[/math]
3	[math]\displaystyle{ -q^{93}+2 q^{91}-3 q^{87}-q^{85}+10 q^{83}+2 q^{81}-26 q^{79}-7 q^{77}+54 q^{75}+29 q^{73}-93 q^{71}-85 q^{69}+132 q^{67}+173 q^{65}-141 q^{63}-283 q^{61}+98 q^{59}+399 q^{57}-15 q^{55}-463 q^{53}-110 q^{51}+468 q^{49}+232 q^{47}-410 q^{45}-322 q^{43}+300 q^{41}+364 q^{39}-162 q^{37}-366 q^{35}+23 q^{33}+335 q^{31}+108 q^{29}-278 q^{27}-235 q^{25}+212 q^{23}+342 q^{21}-125 q^{19}-431 q^{17}+22 q^{15}+481 q^{13}+99 q^{11}-472 q^9-220 q^7+412 q^5+305 q^3-296 q-339 q^{-1} +161 q^{-3} +316 q^{-5} -44 q^{-7} -245 q^{-9} -27 q^{-11} +153 q^{-13} +55 q^{-15} -76 q^{-17} -52 q^{-19} +33 q^{-21} +26 q^{-23} -5 q^{-25} -12 q^{-27} +4 q^{-31} - q^{-33} }[/math]
4	[math]\displaystyle{ q^{152}-2 q^{150}+3 q^{146}-3 q^{144}+2 q^{142}-8 q^{140}+6 q^{138}+23 q^{136}-16 q^{134}-22 q^{132}-49 q^{130}+41 q^{128}+151 q^{126}+15 q^{124}-146 q^{122}-324 q^{120}-11 q^{118}+554 q^{116}+467 q^{114}-146 q^{112}-1114 q^{110}-782 q^{108}+849 q^{106}+1711 q^{104}+901 q^{102}-1772 q^{100}-2651 q^{98}-221 q^{96}+2762 q^{94}+3300 q^{92}-765 q^{90}-4226 q^{88}-2817 q^{86}+1903 q^{84}+5239 q^{82}+1857 q^{80}-3629 q^{78}-4881 q^{76}-610 q^{74}+4849 q^{72}+3947 q^{70}-1251 q^{68}-4730 q^{66}-2716 q^{64}+2638 q^{62}+4147 q^{60}+1049 q^{58}-3021 q^{56}-3414 q^{54}+250 q^{52}+3176 q^{50}+2520 q^{48}-1089 q^{46}-3367 q^{44}-1789 q^{42}+2012 q^{40}+3655 q^{38}+828 q^{36}-3093 q^{34}-3776 q^{32}+481 q^{30}+4413 q^{28}+3046 q^{26}-1959 q^{24}-5266 q^{22}-1814 q^{20}+3744 q^{18}+4824 q^{16}+447 q^{14}-4886 q^{12}-3842 q^{10}+1286 q^8+4600 q^6+2754 q^4-2443 q^2-3854-1200 q^{-2} +2355 q^{-4} +3049 q^{-6} +33 q^{-8} -2001 q^{-10} -1792 q^{-12} +214 q^{-14} +1649 q^{-16} +787 q^{-18} -325 q^{-20} -942 q^{-22} -424 q^{-24} +407 q^{-26} +404 q^{-28} +159 q^{-30} -217 q^{-32} -218 q^{-34} +20 q^{-36} +70 q^{-38} +80 q^{-40} -12 q^{-42} -41 q^{-44} -6 q^{-46} + q^{-48} +12 q^{-50} -4 q^{-54} + q^{-56} }[/math]
5	[math]\displaystyle{ -q^{225}+2 q^{223}-3 q^{219}+3 q^{217}+2 q^{215}-4 q^{213}-3 q^{209}-10 q^{207}+16 q^{205}+35 q^{203}+4 q^{201}-43 q^{199}-87 q^{197}-65 q^{195}+86 q^{193}+270 q^{191}+241 q^{189}-123 q^{187}-596 q^{185}-715 q^{183}-71 q^{181}+1086 q^{179}+1754 q^{177}+853 q^{175}-1496 q^{173}-3471 q^{171}-2798 q^{169}+1107 q^{167}+5650 q^{165}+6458 q^{163}+1096 q^{161}-7421 q^{159}-11722 q^{157}-6171 q^{155}+7094 q^{153}+17634 q^{151}+14573 q^{149}-2987 q^{147}-22168 q^{145}-25315 q^{143}-5950 q^{141}+22718 q^{139}+36151 q^{137}+19352 q^{135}-17569 q^{133}-44044 q^{131}-34713 q^{129}+6358 q^{127}+46080 q^{125}+48917 q^{123}+9119 q^{121}-41311 q^{119}-58405 q^{117}-25564 q^{115}+30330 q^{113}+61169 q^{111}+39626 q^{109}-15812 q^{107}-57053 q^{105}-48626 q^{103}+919 q^{101}+47596 q^{99}+51649 q^{97}+11772 q^{95}-35438 q^{93}-49490 q^{91}-20768 q^{89}+23129 q^{87}+43925 q^{85}+26084 q^{83}-12255 q^{81}-37181 q^{79}-28848 q^{77}+3427 q^{75}+30902 q^{73}+30564 q^{71}+3847 q^{69}-25727 q^{67}-32663 q^{65}-10767 q^{63}+21497 q^{61}+35822 q^{59}+18379 q^{57}-17048 q^{55}-39615 q^{53}-27604 q^{51}+11020 q^{49}+42957 q^{47}+38068 q^{45}-2269 q^{43}-43836 q^{41}-48586 q^{39}-9591 q^{37}+40532 q^{35}+57018 q^{33}+23457 q^{31}-31987 q^{29}-60809 q^{27}-37193 q^{25}+18594 q^{23}+58265 q^{21}+47857 q^{19}-2576 q^{17}-48963 q^{15}-52714 q^{13}-12990 q^{11}+34586 q^9+50561 q^7+24670 q^5-18247 q^3-42087 q-30233 q^{-1} +3516 q^{-3} +29774 q^{-5} +29448 q^{-7} +6868 q^{-9} -17054 q^{-11} -23886 q^{-13} -11740 q^{-15} +6613 q^{-17} +16218 q^{-19} +11977 q^{-21} +54 q^{-23} -9063 q^{-25} -9378 q^{-27} -2951 q^{-29} +3795 q^{-31} +5934 q^{-33} +3362 q^{-35} -813 q^{-37} -3146 q^{-39} -2475 q^{-41} -336 q^{-43} +1258 q^{-45} +1433 q^{-47} +592 q^{-49} -389 q^{-51} -698 q^{-53} -372 q^{-55} +38 q^{-57} +252 q^{-59} +202 q^{-61} +41 q^{-63} -88 q^{-65} -84 q^{-67} -16 q^{-69} +20 q^{-71} +21 q^{-73} +10 q^{-75} - q^{-77} -12 q^{-79} +4 q^{-83} - q^{-85} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{26}-q^{24}+2 q^{22}-q^{20}-q^{18}+4 q^{16}-2 q^{14}+2 q^{12}-2 q^8+2 q^6-4 q^4+4 q^2- q^{-2} +3 q^{-4} - q^{-6} }[/math]
2,0	[math]\displaystyle{ q^{66}+q^{64}-q^{62}-3 q^{60}+q^{58}+6 q^{56}-3 q^{54}-12 q^{52}+18 q^{48}+3 q^{46}-22 q^{44}-2 q^{42}+27 q^{40}+2 q^{38}-27 q^{36}+21 q^{32}-4 q^{30}-21 q^{28}+8 q^{26}+9 q^{24}-11 q^{22}+8 q^{20}+9 q^{18}-11 q^{16}-3 q^{14}+23 q^{12}-q^{10}-28 q^8+4 q^6+28 q^4-7 q^2-27+11 q^{-2} +19 q^{-4} -6 q^{-6} -11 q^{-8} +2 q^{-10} +9 q^{-12} -2 q^{-14} -3 q^{-16} + q^{-18} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{54}-2 q^{52}+q^{50}+5 q^{48}-10 q^{46}+q^{44}+17 q^{42}-25 q^{40}-q^{38}+34 q^{36}-35 q^{34}-3 q^{32}+39 q^{30}-28 q^{28}-7 q^{26}+26 q^{24}-7 q^{22}-11 q^{20}+16 q^{16}-5 q^{14}-25 q^{12}+30 q^{10}+7 q^8-40 q^6+30 q^4+11 q^2-34+22 q^{-2} +7 q^{-4} -17 q^{-6} +10 q^{-8} +2 q^{-10} -4 q^{-12} + q^{-14} }[/math]
1,0,0	[math]\displaystyle{ -q^{35}-q^{33}-q^{31}+2 q^{29}-q^{27}+2 q^{25}-q^{23}+4 q^{21}-3 q^{19}+3 q^{17}-q^{15}-q^{11}-q^9+q^7-3 q^5+4 q^3-q+3 q^{-1} -2 q^{-3} +3 q^{-5} - q^{-7} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{54}+2 q^{52}-5 q^{50}+9 q^{48}-16 q^{46}+23 q^{44}-33 q^{42}+41 q^{40}-45 q^{38}+46 q^{36}-39 q^{34}+27 q^{32}-7 q^{30}-14 q^{28}+39 q^{26}-60 q^{24}+79 q^{22}-89 q^{20}+90 q^{18}-84 q^{16}+67 q^{14}-47 q^{12}+22 q^{10}+q^8-22 q^6+38 q^4-45 q^2+48-44 q^{-2} +39 q^{-4} -27 q^{-6} +18 q^{-8} -10 q^{-10} +4 q^{-12} - q^{-14} }[/math]
1,0	[math]\displaystyle{ q^{88}-2 q^{84}-2 q^{82}+3 q^{80}+7 q^{78}-13 q^{74}-10 q^{72}+12 q^{70}+25 q^{68}-q^{66}-36 q^{64}-21 q^{62}+30 q^{60}+42 q^{58}-11 q^{56}-51 q^{54}-13 q^{52}+45 q^{50}+31 q^{48}-28 q^{46}-38 q^{44}+14 q^{42}+38 q^{40}-q^{38}-35 q^{36}-6 q^{34}+31 q^{32}+12 q^{30}-28 q^{28}-20 q^{26}+25 q^{24}+27 q^{22}-20 q^{20}-36 q^{18}+12 q^{16}+46 q^{14}+7 q^{12}-47 q^{10}-29 q^8+36 q^6+45 q^4-13 q^2-46-11 q^{-2} +33 q^{-4} +25 q^{-6} -15 q^{-8} -22 q^{-10} +14 q^{-14} +6 q^{-16} -4 q^{-18} -4 q^{-20} + q^{-24} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+9 q^{120}-9 q^{118}+q^{116}+13 q^{114}-32 q^{112}+52 q^{110}-67 q^{108}+62 q^{106}-34 q^{104}-26 q^{102}+111 q^{100}-190 q^{98}+234 q^{96}-208 q^{94}+89 q^{92}+87 q^{90}-276 q^{88}+405 q^{86}-402 q^{84}+261 q^{82}-14 q^{80}-243 q^{78}+405 q^{76}-399 q^{74}+229 q^{72}+28 q^{70}-252 q^{68}+332 q^{66}-238 q^{64}+7 q^{62}+270 q^{60}-447 q^{58}+447 q^{56}-250 q^{54}-76 q^{52}+406 q^{50}-617 q^{48}+623 q^{46}-423 q^{44}+92 q^{42}+264 q^{40}-515 q^{38}+577 q^{36}-433 q^{34}+148 q^{32}+148 q^{30}-350 q^{28}+361 q^{26}-198 q^{24}-53 q^{22}+282 q^{20}-371 q^{18}+283 q^{16}-53 q^{14}-226 q^{12}+424 q^{10}-463 q^8+336 q^6-99 q^4-147 q^2+319-360 q^{-2} +293 q^{-4} -149 q^{-6} - q^{-8} +105 q^{-10} -152 q^{-12} +134 q^{-14} -84 q^{-16} +37 q^{-18} +4 q^{-20} -22 q^{-22} +25 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }[/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 89"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-8 t^2+24 t-33+24 t^{-1} -8 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 99, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^2+5 q-9+13 q^{-1} -16 q^{-2} +17 q^{-3} -15 q^{-4} +12 q^{-5} -7 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{ -a^8+3 z^2 a^6+2 a^6-3 z^4 a^4-4 z^2 a^4-a^4+z^6 a^2+2 z^4 a^2+2 z^2 a^2-z^4+1 }[/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^8-5 z^4 a^8+3 z^2 a^8-a^8+5 z^7 a^7-7 z^5 a^7+4 z^3 a^7-z a^7+5 z^8 a^6-3 z^6 a^6-4 z^4 a^6+6 z^2 a^6-2 a^6+2 z^9 a^5+11 z^7 a^5-27 z^5 a^5+20 z^3 a^5-4 z a^5+12 z^8 a^4-15 z^6 a^4-2 z^4 a^4+6 z^2 a^4-a^4+2 z^9 a^3+15 z^7 a^3-35 z^5 a^3+19 z^3 a^3-2 z a^3+7 z^8 a^2-4 z^6 a^2-9 z^4 a^2+3 z^2 a^2+9 z^7 a-15 z^5 a+5 z^3 a+5 z^6-6 z^4+1+z^5 a^{-1} }[/math]