10 63

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 5 t^2-14 t+19-14 t^{-1} +5 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 5 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature { 57, -4 }
Jones polynomial [math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -7 q^{-5} +9 q^{-6} -9 q^{-7} +9 q^{-8} -7 q^{-9} +4 q^{-10} -3 q^{-11} + q^{-12} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^{12}-3 z^2 a^{10}-4 a^{10}+2 z^4 a^8+4 z^2 a^8+3 a^8+2 z^4 a^6+3 z^2 a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{14}-3 z^4 a^{14}+z^2 a^{14}+3 z^7 a^{13}-11 z^5 a^{13}+10 z^3 a^{13}-2 z a^{13}+3 z^8 a^{12}-9 z^6 a^{12}+6 z^4 a^{12}-2 z^2 a^{12}+a^{12}+z^9 a^{11}+4 z^7 a^{11}-23 z^5 a^{11}+28 z^3 a^{11}-10 z a^{11}+6 z^8 a^{10}-19 z^6 a^{10}+24 z^4 a^{10}-16 z^2 a^{10}+4 a^{10}+z^9 a^9+4 z^7 a^9-16 z^5 a^9+20 z^3 a^9-8 z a^9+3 z^8 a^8-6 z^6 a^8+11 z^4 a^8-10 z^2 a^8+3 a^8+3 z^7 a^7-2 z^5 a^7+3 z^6 a^6-3 z^4 a^6+z^2 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^{38}+q^{36}-2 q^{34}-q^{32}-2 q^{30}-3 q^{28}+2 q^{26}+q^{24}+2 q^{22}+q^{20}-q^{18}+2 q^{16}-q^{14}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
The G2 invariant [math]\displaystyle{ q^{190}-2 q^{188}+4 q^{186}-7 q^{184}+6 q^{182}-6 q^{180}-q^{178}+14 q^{176}-25 q^{174}+34 q^{172}-31 q^{170}+16 q^{168}+9 q^{166}-37 q^{164}+62 q^{162}-65 q^{160}+47 q^{158}-6 q^{156}-32 q^{154}+62 q^{152}-67 q^{150}+46 q^{148}-10 q^{146}-26 q^{144}+43 q^{142}-49 q^{140}+19 q^{138}+22 q^{136}-49 q^{134}+52 q^{132}-40 q^{130}-2 q^{128}+41 q^{126}-76 q^{124}+77 q^{122}-66 q^{120}+26 q^{118}+29 q^{116}-71 q^{114}+89 q^{112}-76 q^{110}+43 q^{108}+4 q^{106}-43 q^{104}+59 q^{102}-49 q^{100}+26 q^{98}+15 q^{96}-35 q^{94}+40 q^{92}-17 q^{90}-15 q^{88}+41 q^{86}-50 q^{84}+40 q^{82}-16 q^{80}-13 q^{78}+38 q^{76}-48 q^{74}+47 q^{72}-31 q^{70}+11 q^{68}+5 q^{66}-22 q^{64}+29 q^{62}-30 q^{60}+27 q^{58}-13 q^{56}+4 q^{54}+7 q^{52}-13 q^{50}+15 q^{48}-12 q^{46}+8 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 10_63.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{25}-2 q^{23}+q^{21}-3 q^{19}+2 q^{17}+2 q^{11}-2 q^9+3 q^7-q^5+q^3 }[/math]
2 [math]\displaystyle{ q^{70}-2 q^{68}-2 q^{66}+6 q^{64}-2 q^{62}-7 q^{60}+10 q^{58}+4 q^{56}-13 q^{54}+6 q^{52}+8 q^{50}-14 q^{48}+10 q^{44}-7 q^{42}-5 q^{40}+7 q^{38}+4 q^{36}-8 q^{34}-3 q^{32}+12 q^{30}-7 q^{28}-9 q^{26}+14 q^{24}-2 q^{22}-8 q^{20}+9 q^{18}-3 q^{14}+4 q^{12}-q^8+q^6 }[/math]
3 [math]\displaystyle{ q^{135}-2 q^{133}-2 q^{131}+3 q^{129}+6 q^{127}-2 q^{125}-13 q^{123}+q^{121}+19 q^{119}+7 q^{117}-25 q^{115}-20 q^{113}+22 q^{111}+35 q^{109}-16 q^{107}-45 q^{105}-3 q^{103}+52 q^{101}+24 q^{99}-44 q^{97}-39 q^{95}+32 q^{93}+53 q^{91}-20 q^{89}-55 q^{87}+2 q^{85}+56 q^{83}+6 q^{81}-50 q^{79}-16 q^{77}+43 q^{75}+24 q^{73}-30 q^{71}-34 q^{69}+16 q^{67}+40 q^{65}+3 q^{63}-45 q^{61}-25 q^{59}+41 q^{57}+41 q^{55}-33 q^{53}-51 q^{51}+18 q^{49}+51 q^{47}-5 q^{45}-41 q^{43}-6 q^{41}+28 q^{39}+9 q^{37}-15 q^{35}-4 q^{33}+4 q^{31}+3 q^{29}-q^{27}+3 q^{25}+2 q^{23}-2 q^{21}-q^{19}+3 q^{17}+q^{15}-q^{11}+q^9 }[/math]
4 [math]\displaystyle{ q^{220}-2 q^{218}-2 q^{216}+3 q^{214}+3 q^{212}+6 q^{210}-9 q^{208}-13 q^{206}+2 q^{204}+10 q^{202}+31 q^{200}-8 q^{198}-41 q^{196}-25 q^{194}+4 q^{192}+79 q^{190}+38 q^{188}-44 q^{186}-82 q^{184}-71 q^{182}+87 q^{180}+124 q^{178}+49 q^{176}-80 q^{174}-189 q^{172}-36 q^{170}+114 q^{168}+190 q^{166}+78 q^{164}-186 q^{162}-204 q^{160}-69 q^{158}+197 q^{156}+279 q^{154}+q^{152}-230 q^{150}-279 q^{148}+33 q^{146}+329 q^{144}+201 q^{142}-106 q^{140}-342 q^{138}-135 q^{136}+235 q^{134}+271 q^{132}+15 q^{130}-277 q^{128}-185 q^{126}+123 q^{124}+239 q^{122}+68 q^{120}-183 q^{118}-181 q^{116}+31 q^{114}+203 q^{112}+119 q^{110}-82 q^{108}-194 q^{106}-105 q^{104}+135 q^{102}+205 q^{100}+92 q^{98}-174 q^{96}-283 q^{94}-23 q^{92}+233 q^{90}+297 q^{88}-28 q^{86}-350 q^{84}-218 q^{82}+99 q^{80}+363 q^{78}+165 q^{76}-216 q^{74}-272 q^{72}-95 q^{70}+223 q^{68}+223 q^{66}-21 q^{64}-149 q^{62}-157 q^{60}+41 q^{58}+123 q^{56}+55 q^{54}-10 q^{52}-90 q^{50}-27 q^{48}+26 q^{46}+31 q^{44}+29 q^{42}-23 q^{40}-14 q^{38}-5 q^{36}+q^{34}+17 q^{32}-q^{30}-3 q^{26}-3 q^{24}+5 q^{22}+q^{18}-q^{14}+q^{12} }[/math]
5 [math]\displaystyle{ q^{325}-2 q^{323}-2 q^{321}+3 q^{319}+3 q^{317}+3 q^{315}-q^{313}-9 q^{311}-13 q^{309}+2 q^{307}+20 q^{305}+22 q^{303}+7 q^{301}-24 q^{299}-49 q^{297}-36 q^{295}+31 q^{293}+86 q^{291}+74 q^{289}-q^{287}-110 q^{285}-156 q^{283}-65 q^{281}+116 q^{279}+231 q^{277}+176 q^{275}-33 q^{273}-279 q^{271}-336 q^{269}-124 q^{267}+226 q^{265}+449 q^{263}+360 q^{261}-26 q^{259}-450 q^{257}-582 q^{255}-302 q^{253}+249 q^{251}+690 q^{249}+668 q^{247}+157 q^{245}-543 q^{243}-948 q^{241}-691 q^{239}+149 q^{237}+989 q^{235}+1174 q^{233}+472 q^{231}-729 q^{229}-1501 q^{227}-1138 q^{225}+214 q^{223}+1519 q^{221}+1686 q^{219}+464 q^{217}-1253 q^{215}-2024 q^{213}-1096 q^{211}+794 q^{209}+2050 q^{207}+1591 q^{205}-241 q^{203}-1872 q^{201}-1846 q^{199}-220 q^{197}+1517 q^{195}+1864 q^{193}+572 q^{191}-1147 q^{189}-1719 q^{187}-730 q^{185}+801 q^{183}+1476 q^{181}+762 q^{179}-556 q^{177}-1232 q^{175}-723 q^{173}+386 q^{171}+1050 q^{169}+697 q^{167}-261 q^{165}-918 q^{163}-745 q^{161}+85 q^{159}+849 q^{157}+905 q^{155}+185 q^{153}-744 q^{151}-1128 q^{149}-603 q^{147}+525 q^{145}+1361 q^{143}+1128 q^{141}-138 q^{139}-1474 q^{137}-1677 q^{135}-421 q^{133}+1358 q^{131}+2126 q^{129}+1083 q^{127}-997 q^{125}-2328 q^{123}-1689 q^{121}+413 q^{119}+2195 q^{117}+2114 q^{115}+245 q^{113}-1779 q^{111}-2216 q^{109}-808 q^{107}+1148 q^{105}+2006 q^{103}+1164 q^{101}-518 q^{99}-1572 q^{97}-1232 q^{95}+14 q^{93}+1032 q^{91}+1085 q^{89}+299 q^{87}-568 q^{85}-821 q^{83}-389 q^{81}+221 q^{79}+526 q^{77}+373 q^{75}-31 q^{73}-306 q^{71}-278 q^{69}-50 q^{67}+148 q^{65}+184 q^{63}+70 q^{61}-61 q^{59}-105 q^{57}-62 q^{55}+17 q^{53}+58 q^{51}+38 q^{49}+5 q^{47}-20 q^{45}-24 q^{43}-9 q^{41}+13 q^{39}+10 q^{37}+4 q^{35}+2 q^{33}-5 q^{31}-4 q^{29}+3 q^{27}+2 q^{25}+q^{21}-q^{17}+q^{15} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{38}+q^{36}-2 q^{34}-q^{32}-2 q^{30}-3 q^{28}+2 q^{26}+q^{24}+2 q^{22}+q^{20}-q^{18}+2 q^{16}-q^{14}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
2,0 [math]\displaystyle{ q^{96}+q^{94}-q^{92}-4 q^{90}-3 q^{88}+2 q^{86}+q^{84}-q^{82}+2 q^{80}+10 q^{78}+8 q^{76}-3 q^{74}-4 q^{72}+2 q^{70}-3 q^{68}-12 q^{66}-9 q^{64}+4 q^{62}+5 q^{60}-2 q^{58}+2 q^{56}+4 q^{54}-q^{52}-3 q^{50}-4 q^{46}-6 q^{44}+3 q^{42}+6 q^{40}-5 q^{38}-3 q^{36}+11 q^{34}+5 q^{32}-7 q^{30}+8 q^{26}+3 q^{24}-5 q^{22}+q^{20}+4 q^{18}-q^{16}-q^{14}+q^{12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{80}-2 q^{78}+2 q^{74}-5 q^{72}+4 q^{70}+3 q^{68}-6 q^{66}+8 q^{64}+5 q^{62}-9 q^{60}+7 q^{58}+3 q^{56}-13 q^{54}-q^{52}-8 q^{48}-5 q^{46}+q^{44}+5 q^{42}-3 q^{40}+q^{38}+14 q^{36}-5 q^{34}-4 q^{32}+12 q^{30}-4 q^{28}-6 q^{26}+9 q^{24}-3 q^{20}+4 q^{18}+q^{16}-q^{14}+q^{12} }[/math]
1,0,0 [math]\displaystyle{ q^{51}+q^{49}+q^{47}-2 q^{45}-q^{43}-4 q^{41}-2 q^{39}-3 q^{37}+2 q^{35}+q^{33}+3 q^{31}+2 q^{29}+q^{27}-q^{23}+2 q^{21}-q^{19}+2 q^{17}+2 q^{13}-q^{11}+q^9 }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{80}-2 q^{78}+4 q^{76}-6 q^{74}+9 q^{72}-12 q^{70}+13 q^{68}-14 q^{66}+12 q^{64}-11 q^{62}+5 q^{60}-q^{58}-7 q^{56}+13 q^{54}-19 q^{52}+24 q^{50}-26 q^{48}+27 q^{46}-23 q^{44}+19 q^{42}-13 q^{40}+7 q^{38}-5 q^{34}+10 q^{32}-12 q^{30}+14 q^{28}-12 q^{26}+11 q^{24}-8 q^{22}+7 q^{20}-4 q^{18}+3 q^{16}-q^{14}+q^{12} }[/math]
1,0 [math]\displaystyle{ q^{130}-2 q^{126}-2 q^{124}+2 q^{122}+4 q^{120}-2 q^{118}-7 q^{116}-q^{114}+10 q^{112}+7 q^{110}-7 q^{108}-11 q^{106}+4 q^{104}+15 q^{102}+6 q^{100}-13 q^{98}-11 q^{96}+6 q^{94}+13 q^{92}-q^{90}-13 q^{88}-5 q^{86}+7 q^{84}+4 q^{82}-8 q^{80}-7 q^{78}+4 q^{76}+6 q^{74}-6 q^{72}-10 q^{70}+2 q^{68}+11 q^{66}+q^{64}-10 q^{62}-3 q^{60}+12 q^{58}+10 q^{56}-6 q^{54}-12 q^{52}+2 q^{50}+14 q^{48}+6 q^{46}-9 q^{44}-9 q^{42}+2 q^{40}+10 q^{38}+4 q^{36}-4 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+q^{18} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{190}-2 q^{188}+4 q^{186}-7 q^{184}+6 q^{182}-6 q^{180}-q^{178}+14 q^{176}-25 q^{174}+34 q^{172}-31 q^{170}+16 q^{168}+9 q^{166}-37 q^{164}+62 q^{162}-65 q^{160}+47 q^{158}-6 q^{156}-32 q^{154}+62 q^{152}-67 q^{150}+46 q^{148}-10 q^{146}-26 q^{144}+43 q^{142}-49 q^{140}+19 q^{138}+22 q^{136}-49 q^{134}+52 q^{132}-40 q^{130}-2 q^{128}+41 q^{126}-76 q^{124}+77 q^{122}-66 q^{120}+26 q^{118}+29 q^{116}-71 q^{114}+89 q^{112}-76 q^{110}+43 q^{108}+4 q^{106}-43 q^{104}+59 q^{102}-49 q^{100}+26 q^{98}+15 q^{96}-35 q^{94}+40 q^{92}-17 q^{90}-15 q^{88}+41 q^{86}-50 q^{84}+40 q^{82}-16 q^{80}-13 q^{78}+38 q^{76}-48 q^{74}+47 q^{72}-31 q^{70}+11 q^{68}+5 q^{66}-22 q^{64}+29 q^{62}-30 q^{60}+27 q^{58}-13 q^{56}+4 q^{54}+7 q^{52}-13 q^{50}+15 q^{48}-12 q^{46}+8 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["10 63"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 5 t^2-14 t+19-14 t^{-1} +5 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 5 z^4+6 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{ \left\{2,t^2+t+1\right\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 57, -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} -7 q^{-5} +9 q^{-6} -9 q^{-7} +9 q^{-8} -7 q^{-9} +4 q^{-10} -3 q^{-11} + q^{-12} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^{12}-3 z^2 a^{10}-4 a^{10}+2 z^4 a^8+4 z^2 a^8+3 a^8+2 z^4 a^6+3 z^2 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^6 a^{14}-3 z^4 a^{14}+z^2 a^{14}+3 z^7 a^{13}-11 z^5 a^{13}+10 z^3 a^{13}-2 z a^{13}+3 z^8 a^{12}-9 z^6 a^{12}+6 z^4 a^{12}-2 z^2 a^{12}+a^{12}+z^9 a^{11}+4 z^7 a^{11}-23 z^5 a^{11}+28 z^3 a^{11}-10 z a^{11}+6 z^8 a^{10}-19 z^6 a^{10}+24 z^4 a^{10}-16 z^2 a^{10}+4 a^{10}+z^9 a^9+4 z^7 a^9-16 z^5 a^9+20 z^3 a^9-8 z a^9+3 z^8 a^8-6 z^6 a^8+11 z^4 a^8-10 z^2 a^8+3 a^8+3 z^7 a^7-2 z^5 a^7+3 z^6 a^6-3 z^4 a^6+z^2 a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]
Retrieved from "https://katlas.org/index.php?title=10_63&oldid=4655"