10 49

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +3 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 3 z^6+10 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 59, -6 }
Jones polynomial [math]\displaystyle{ q^{-3} -2 q^{-4} +5 q^{-5} -6 q^{-6} +9 q^{-7} -10 q^{-8} +9 q^{-9} -8 q^{-10} +5 q^{-11} -3 q^{-12} + q^{-13} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^{12}+2 a^{12}-3 z^4 a^{10}-10 z^2 a^{10}-7 a^{10}+2 z^6 a^8+9 z^4 a^8+12 z^2 a^8+5 a^8+z^6 a^6+4 z^4 a^6+4 z^2 a^6+a^6 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{16}-z^2 a^{16}+3 z^5 a^{15}-4 z^3 a^{15}+z a^{15}+4 z^6 a^{14}-4 z^4 a^{14}+4 z^7 a^{13}-4 z^5 a^{13}+z^3 a^{13}+3 z^8 a^{12}-3 z^6 a^{12}+2 z^4 a^{12}-2 z^2 a^{12}+2 a^{12}+z^9 a^{11}+5 z^7 a^{11}-19 z^5 a^{11}+24 z^3 a^{11}-10 z a^{11}+6 z^8 a^{10}-19 z^6 a^{10}+26 z^4 a^{10}-20 z^2 a^{10}+7 a^{10}+z^9 a^9+3 z^7 a^9-18 z^5 a^9+22 z^3 a^9-9 z a^9+3 z^8 a^8-11 z^6 a^8+15 z^4 a^8-13 z^2 a^8+5 a^8+2 z^7 a^7-6 z^5 a^7+3 z^3 a^7+z^6 a^6-4 z^4 a^6+4 z^2 a^6-a^6 }[/math]
The A2 invariant [math]\displaystyle{ q^{40}+q^{38}-q^{36}-3 q^{32}-2 q^{30}-q^{28}-2 q^{26}+3 q^{24}+3 q^{20}+2 q^{18}+2 q^{14}-q^{12}+q^{10} }[/math]
The G2 invariant [math]\displaystyle{ q^{210}-2 q^{208}+4 q^{206}-6 q^{204}+4 q^{202}-2 q^{200}-4 q^{198}+12 q^{196}-18 q^{194}+22 q^{192}-20 q^{190}+9 q^{188}+5 q^{186}-22 q^{184}+38 q^{182}-43 q^{180}+41 q^{178}-26 q^{176}+2 q^{174}+26 q^{172}-46 q^{170}+60 q^{168}-54 q^{166}+33 q^{164}-30 q^{160}+49 q^{158}-44 q^{156}+24 q^{154}+8 q^{152}-36 q^{150}+40 q^{148}-26 q^{146}-13 q^{144}+53 q^{142}-80 q^{140}+72 q^{138}-36 q^{136}-23 q^{134}+71 q^{132}-105 q^{130}+97 q^{128}-66 q^{126}+8 q^{124}+41 q^{122}-78 q^{120}+84 q^{118}-61 q^{116}+18 q^{114}+20 q^{112}-48 q^{110}+48 q^{108}-26 q^{106}-6 q^{104}+43 q^{102}-56 q^{100}+48 q^{98}-10 q^{96}-32 q^{94}+72 q^{92}-79 q^{90}+62 q^{88}-20 q^{86}-22 q^{84}+58 q^{82}-66 q^{80}+58 q^{78}-29 q^{76}+q^{74}+20 q^{72}-30 q^{70}+27 q^{68}-17 q^{66}+9 q^{64}+q^{62}-4 q^{60}+5 q^{58}-4 q^{56}+3 q^{54}-q^{52}+q^{50} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_49.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{27}-2 q^{25}+2 q^{23}-3 q^{21}+q^{19}-q^{17}-q^{15}+3 q^{13}-q^{11}+3 q^9-q^7+q^5 }[/math]
2 [math]\displaystyle{ q^{74}-2 q^{72}-q^{70}+5 q^{68}-5 q^{66}-2 q^{64}+11 q^{62}-8 q^{60}-5 q^{58}+16 q^{56}-8 q^{54}-9 q^{52}+13 q^{50}-9 q^{46}+7 q^{42}-4 q^{40}-11 q^{38}+10 q^{36}+4 q^{34}-15 q^{32}+8 q^{30}+9 q^{28}-13 q^{26}+2 q^{24}+11 q^{22}-6 q^{20}-2 q^{18}+6 q^{16}-q^{12}+q^{10} }[/math]
3 [math]\displaystyle{ q^{141}-2 q^{139}-q^{137}+2 q^{135}+3 q^{133}-2 q^{131}-5 q^{129}+5 q^{127}+4 q^{125}-6 q^{123}-5 q^{121}+12 q^{119}+4 q^{117}-21 q^{115}-9 q^{113}+33 q^{111}+15 q^{109}-42 q^{107}-30 q^{105}+48 q^{103}+43 q^{101}-40 q^{99}-53 q^{97}+25 q^{95}+58 q^{93}-5 q^{91}-48 q^{89}-17 q^{87}+37 q^{85}+33 q^{83}-19 q^{81}-46 q^{79}+8 q^{77}+49 q^{75}+9 q^{73}-58 q^{71}-19 q^{69}+52 q^{67}+31 q^{65}-51 q^{63}-43 q^{61}+40 q^{59}+53 q^{57}-24 q^{55}-61 q^{53}+5 q^{51}+55 q^{49}+14 q^{47}-45 q^{45}-27 q^{43}+28 q^{41}+33 q^{39}-13 q^{37}-25 q^{35}+20 q^{31}+6 q^{29}-8 q^{27}-4 q^{25}+4 q^{23}+3 q^{21}-q^{17}+q^{15} }[/math]
4 [math]\displaystyle{ q^{228}-2 q^{226}-q^{224}+2 q^{222}+6 q^{218}-5 q^{216}-4 q^{214}-6 q^{210}+18 q^{208}-2 q^{206}-q^{204}-3 q^{202}-30 q^{200}+15 q^{198}+7 q^{196}+35 q^{194}+22 q^{192}-68 q^{190}-39 q^{188}-24 q^{186}+96 q^{184}+124 q^{182}-55 q^{180}-138 q^{178}-155 q^{176}+101 q^{174}+276 q^{172}+77 q^{170}-168 q^{168}-334 q^{166}-34 q^{164}+324 q^{162}+258 q^{160}-35 q^{158}-373 q^{156}-216 q^{154}+168 q^{152}+300 q^{150}+158 q^{148}-203 q^{146}-265 q^{144}-56 q^{142}+165 q^{140}+234 q^{138}+23 q^{136}-175 q^{134}-196 q^{132}+8 q^{130}+212 q^{128}+169 q^{126}-88 q^{124}-256 q^{122}-79 q^{120}+183 q^{118}+260 q^{116}-30 q^{114}-304 q^{112}-157 q^{110}+144 q^{108}+338 q^{106}+66 q^{104}-295 q^{102}-255 q^{100}+16 q^{98}+343 q^{96}+212 q^{94}-151 q^{92}-284 q^{90}-175 q^{88}+195 q^{86}+275 q^{84}+70 q^{82}-152 q^{80}-271 q^{78}-26 q^{76}+165 q^{74}+182 q^{72}+48 q^{70}-179 q^{68}-137 q^{66}-12 q^{64}+117 q^{62}+133 q^{60}-27 q^{58}-83 q^{56}-78 q^{54}+7 q^{52}+78 q^{50}+32 q^{48}-4 q^{46}-40 q^{44}-22 q^{42}+17 q^{40}+13 q^{38}+11 q^{36}-5 q^{34}-7 q^{32}+2 q^{30}+q^{28}+3 q^{26}-q^{22}+q^{20} }[/math]
5 [math]\displaystyle{ q^{335}-2 q^{333}-q^{331}+2 q^{329}+3 q^{325}+3 q^{323}-4 q^{321}-9 q^{319}-3 q^{317}+q^{315}+11 q^{313}+19 q^{311}+5 q^{309}-21 q^{307}-36 q^{305}-20 q^{303}+17 q^{301}+59 q^{299}+66 q^{297}+q^{295}-93 q^{293}-123 q^{291}-49 q^{289}+96 q^{287}+218 q^{285}+162 q^{283}-91 q^{281}-324 q^{279}-321 q^{277}-5 q^{275}+425 q^{273}+573 q^{271}+187 q^{269}-487 q^{267}-870 q^{265}-508 q^{263}+442 q^{261}+1181 q^{259}+953 q^{257}-230 q^{255}-1420 q^{253}-1481 q^{251}-159 q^{249}+1486 q^{247}+1960 q^{245}+726 q^{243}-1289 q^{241}-2304 q^{239}-1321 q^{237}+852 q^{235}+2338 q^{233}+1829 q^{231}-223 q^{229}-2059 q^{227}-2113 q^{225}-415 q^{223}+1516 q^{221}+2073 q^{219}+930 q^{217}-826 q^{215}-1774 q^{213}-1243 q^{211}+189 q^{209}+1293 q^{207}+1302 q^{205}+335 q^{203}-787 q^{201}-1226 q^{199}-670 q^{197}+369 q^{195}+1074 q^{193}+882 q^{191}-88 q^{189}-979 q^{187}-993 q^{185}-94 q^{183}+960 q^{181}+1147 q^{179}+188 q^{177}-1020 q^{175}-1304 q^{173}-332 q^{171}+1080 q^{169}+1565 q^{167}+523 q^{165}-1101 q^{163}-1779 q^{161}-832 q^{159}+965 q^{157}+1978 q^{155}+1200 q^{153}-686 q^{151}-1998 q^{149}-1566 q^{147}+224 q^{145}+1823 q^{143}+1848 q^{141}+321 q^{139}-1422 q^{137}-1924 q^{135}-851 q^{133}+823 q^{131}+1752 q^{129}+1254 q^{127}-168 q^{125}-1327 q^{123}-1401 q^{121}-439 q^{119}+734 q^{117}+1265 q^{115}+844 q^{113}-115 q^{111}-893 q^{109}-976 q^{107}-372 q^{105}+400 q^{103}+821 q^{101}+654 q^{99}+50 q^{97}-521 q^{95}-664 q^{93}-341 q^{91}+155 q^{89}+499 q^{87}+453 q^{85}+89 q^{83}-262 q^{81}-378 q^{79}-222 q^{77}+54 q^{75}+244 q^{73}+220 q^{71}+57 q^{69}-103 q^{67}-155 q^{65}-84 q^{63}+18 q^{61}+78 q^{59}+71 q^{57}+14 q^{55}-29 q^{53}-34 q^{51}-13 q^{49}+3 q^{47}+17 q^{45}+12 q^{43}-q^{41}-4 q^{39}-q^{37}-q^{35}+q^{33}+3 q^{31}-q^{27}+q^{25} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{40}+q^{38}-q^{36}-3 q^{32}-2 q^{30}-q^{28}-2 q^{26}+3 q^{24}+3 q^{20}+2 q^{18}+2 q^{14}-q^{12}+q^{10} }[/math]
1,1 [math]\displaystyle{ q^{108}-4 q^{106}+10 q^{104}-20 q^{102}+34 q^{100}-54 q^{98}+78 q^{96}-104 q^{94}+130 q^{92}-156 q^{90}+180 q^{88}-196 q^{86}+207 q^{84}-204 q^{82}+184 q^{80}-144 q^{78}+75 q^{76}+12 q^{74}-118 q^{72}+238 q^{70}-343 q^{68}+440 q^{66}-492 q^{64}+518 q^{62}-497 q^{60}+434 q^{58}-350 q^{56}+220 q^{54}-107 q^{52}-32 q^{50}+132 q^{48}-226 q^{46}+277 q^{44}-292 q^{42}+280 q^{40}-234 q^{38}+193 q^{36}-130 q^{34}+92 q^{32}-48 q^{30}+31 q^{28}-12 q^{26}+6 q^{24}-2 q^{22}+q^{20} }[/math]
2,0 [math]\displaystyle{ q^{100}+q^{98}-2 q^{94}-3 q^{92}-2 q^{90}-3 q^{88}+q^{86}+3 q^{84}+2 q^{82}+3 q^{80}+7 q^{78}+7 q^{76}-2 q^{74}-2 q^{72}+6 q^{70}-10 q^{66}-5 q^{64}+q^{62}-8 q^{60}-9 q^{58}-q^{56}-5 q^{52}+q^{50}+7 q^{48}-3 q^{46}+8 q^{42}+3 q^{40}-5 q^{38}+3 q^{36}+8 q^{34}+q^{32}-3 q^{30}+3 q^{28}+4 q^{26}-q^{24}-q^{22}+q^{20} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{88}-2 q^{86}+4 q^{82}-6 q^{80}-q^{78}+9 q^{76}-8 q^{74}-4 q^{72}+14 q^{70}-6 q^{68}-5 q^{66}+14 q^{64}+q^{62}-3 q^{60}+5 q^{58}-7 q^{54}-14 q^{52}-2 q^{50}-4 q^{48}-16 q^{46}+6 q^{44}+9 q^{42}-6 q^{40}+9 q^{38}+12 q^{36}-5 q^{34}+6 q^{32}+4 q^{30}-3 q^{28}+3 q^{26}+q^{24}-q^{22}+q^{20} }[/math]
1,0,0 [math]\displaystyle{ q^{53}+q^{51}+2 q^{49}-q^{47}-5 q^{43}-2 q^{41}-5 q^{39}-q^{37}-q^{35}+2 q^{33}+3 q^{31}+2 q^{29}+4 q^{27}+3 q^{23}-q^{21}+2 q^{19}-q^{17}+q^{15} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{114}+q^{112}-q^{110}-2 q^{108}+q^{106}+q^{104}-6 q^{102}-6 q^{100}+4 q^{98}+q^{96}-8 q^{94}+3 q^{92}+13 q^{90}+5 q^{88}+5 q^{86}+18 q^{84}+14 q^{82}+q^{78}+q^{76}-21 q^{74}-26 q^{72}-12 q^{70}-19 q^{68}-26 q^{66}-4 q^{64}+8 q^{62}+4 q^{58}+16 q^{56}+13 q^{54}+4 q^{52}+7 q^{50}+9 q^{48}+2 q^{46}+q^{44}+5 q^{42}+2 q^{36}+q^{34}-q^{32}+q^{30} }[/math]
1,0,0,0 [math]\displaystyle{ q^{66}+q^{64}+2 q^{62}+2 q^{60}-q^{58}-5 q^{54}-4 q^{52}-5 q^{50}-5 q^{48}-q^{46}-q^{44}+3 q^{42}+2 q^{40}+5 q^{38}+2 q^{36}+4 q^{34}+q^{32}+q^{30}+2 q^{28}-q^{26}+2 q^{24}-q^{22}+q^{20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{88}-2 q^{86}+4 q^{84}-6 q^{82}+8 q^{80}-11 q^{78}+13 q^{76}-14 q^{74}+14 q^{72}-12 q^{70}+8 q^{68}-3 q^{66}-4 q^{64}+11 q^{62}-19 q^{60}+23 q^{58}-28 q^{56}+27 q^{54}-28 q^{52}+22 q^{50}-18 q^{48}+10 q^{46}-2 q^{44}-3 q^{42}+10 q^{40}-11 q^{38}+16 q^{36}-13 q^{34}+14 q^{32}-10 q^{30}+9 q^{28}-5 q^{26}+3 q^{24}-q^{22}+q^{20} }[/math]
1,0 [math]\displaystyle{ q^{142}-2 q^{138}-2 q^{136}+2 q^{134}+5 q^{132}-7 q^{128}-6 q^{126}+5 q^{124}+11 q^{122}+q^{120}-12 q^{118}-9 q^{116}+8 q^{114}+15 q^{112}-13 q^{108}-5 q^{106}+12 q^{104}+11 q^{102}-5 q^{100}-10 q^{98}+4 q^{96}+11 q^{94}-12 q^{90}-4 q^{88}+6 q^{86}+q^{84}-12 q^{82}-8 q^{80}+5 q^{78}+5 q^{76}-9 q^{74}-14 q^{72}+2 q^{70}+15 q^{68}+7 q^{66}-12 q^{64}-10 q^{62}+9 q^{60}+18 q^{58}+3 q^{56}-10 q^{54}-6 q^{52}+9 q^{50}+9 q^{48}-q^{46}-6 q^{44}+4 q^{40}+2 q^{38}-q^{36}-q^{34}+q^{30} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{122}-2 q^{120}+2 q^{118}-3 q^{116}+5 q^{114}-7 q^{112}+6 q^{110}-8 q^{108}+11 q^{106}-11 q^{104}+9 q^{102}-10 q^{100}+12 q^{98}-7 q^{96}+4 q^{94}-2 q^{92}+2 q^{90}+10 q^{88}-6 q^{86}+16 q^{84}-14 q^{82}+22 q^{80}-22 q^{78}+16 q^{76}-31 q^{74}+10 q^{72}-27 q^{70}+5 q^{68}-19 q^{66}+3 q^{64}-2 q^{62}+3 q^{60}+9 q^{58}-q^{56}+17 q^{54}-5 q^{52}+15 q^{50}-8 q^{48}+14 q^{46}-8 q^{44}+9 q^{42}-6 q^{40}+6 q^{38}-2 q^{36}+2 q^{34}-q^{32}+q^{30} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{210}-2 q^{208}+4 q^{206}-6 q^{204}+4 q^{202}-2 q^{200}-4 q^{198}+12 q^{196}-18 q^{194}+22 q^{192}-20 q^{190}+9 q^{188}+5 q^{186}-22 q^{184}+38 q^{182}-43 q^{180}+41 q^{178}-26 q^{176}+2 q^{174}+26 q^{172}-46 q^{170}+60 q^{168}-54 q^{166}+33 q^{164}-30 q^{160}+49 q^{158}-44 q^{156}+24 q^{154}+8 q^{152}-36 q^{150}+40 q^{148}-26 q^{146}-13 q^{144}+53 q^{142}-80 q^{140}+72 q^{138}-36 q^{136}-23 q^{134}+71 q^{132}-105 q^{130}+97 q^{128}-66 q^{126}+8 q^{124}+41 q^{122}-78 q^{120}+84 q^{118}-61 q^{116}+18 q^{114}+20 q^{112}-48 q^{110}+48 q^{108}-26 q^{106}-6 q^{104}+43 q^{102}-56 q^{100}+48 q^{98}-10 q^{96}-32 q^{94}+72 q^{92}-79 q^{90}+62 q^{88}-20 q^{86}-22 q^{84}+58 q^{82}-66 q^{80}+58 q^{78}-29 q^{76}+q^{74}+20 q^{72}-30 q^{70}+27 q^{68}-17 q^{66}+9 q^{64}+q^{62}-4 q^{60}+5 q^{58}-4 q^{56}+3 q^{54}-q^{52}+q^{50} }[/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 49"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +3 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 z^6+10 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]= { 59, -6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-3} -2 q^{-4} +5 q^{-5} -6 q^{-6} +9 q^{-7} -10 q^{-8} +9 q^{-9} -8 q^{-10} +5 q^{-11} -3 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{ z^2 a^{12}+2 a^{12}-3 z^4 a^{10}-10 z^2 a^{10}-7 a^{10}+2 z^6 a^8+9 z^4 a^8+12 z^2 a^8+5 a^8+z^6 a^6+4 z^4 a^6+4 z^2 a^6+a^6 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^4 a^{16}-z^2 a^{16}+3 z^5 a^{15}-4 z^3 a^{15}+z a^{15}+4 z^6 a^{14}-4 z^4 a^{14}+4 z^7 a^{13}-4 z^5 a^{13}+z^3 a^{13}+3 z^8 a^{12}-3 z^6 a^{12}+2 z^4 a^{12}-2 z^2 a^{12}+2 a^{12}+z^9 a^{11}+5 z^7 a^{11}-19 z^5 a^{11}+24 z^3 a^{11}-10 z a^{11}+6 z^8 a^{10}-19 z^6 a^{10}+26 z^4 a^{10}-20 z^2 a^{10}+7 a^{10}+z^9 a^9+3 z^7 a^9-18 z^5 a^9+22 z^3 a^9-9 z a^9+3 z^8 a^8-11 z^6 a^8+15 z^4 a^8-13 z^2 a^8+5 a^8+2 z^7 a^7-6 z^5 a^7+3 z^3 a^7+z^6 a^6-4 z^4 a^6+4 z^2 a^6-a^6 }[/math]
Retrieved from "https://katlas.org/index.php?title=10_49&oldid=4331"