10 80

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t^3-9 t^2+15 t-17+15 t^{-1} -9 t^{-2} +3 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 3 z^6+9 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 71, -6 }
Jones polynomial [math]\displaystyle{ q^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -12 q^{-8} +11 q^{-9} -10 q^{-10} +6 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}-9 z^2 a^{10}-6 a^{10}+2 z^6 a^8+8 z^4 a^8+9 z^2 a^8+3 a^8+z^6 a^6+4 z^4 a^6+5 z^2 a^6+2 a^6 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{16}-z^2 a^{16}+3 z^5 a^{15}-3 z^3 a^{15}+z a^{15}+5 z^6 a^{14}-5 z^4 a^{14}+2 z^2 a^{14}+6 z^7 a^{13}-8 z^5 a^{13}+6 z^3 a^{13}-2 z a^{13}+4 z^8 a^{12}-z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+2 a^{12}+z^9 a^{11}+10 z^7 a^{11}-29 z^5 a^{11}+29 z^3 a^{11}-12 z a^{11}+7 z^8 a^{10}-15 z^6 a^{10}+13 z^4 a^{10}-13 z^2 a^{10}+6 a^{10}+z^9 a^9+6 z^7 a^9-23 z^5 a^9+22 z^3 a^9-8 z a^9+3 z^8 a^8-8 z^6 a^8+8 z^4 a^8-7 z^2 a^8+3 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 a^6 }[/math]
The A2 invariant [math]\displaystyle{ q^{40}+q^{38}-q^{36}+q^{34}-3 q^{32}-2 q^{30}-q^{28}-3 q^{26}+3 q^{24}-q^{22}+3 q^{20}+2 q^{18}+3 q^{14}-q^{12}+q^{10} }[/math]
The G2 invariant [math]\displaystyle{ q^{210}-2 q^{208}+4 q^{206}-6 q^{204}+5 q^{202}-4 q^{200}-2 q^{198}+11 q^{196}-20 q^{194}+28 q^{192}-32 q^{190}+25 q^{188}-8 q^{186}-19 q^{184}+54 q^{182}-78 q^{180}+88 q^{178}-74 q^{176}+29 q^{174}+32 q^{172}-94 q^{170}+139 q^{168}-135 q^{166}+89 q^{164}-9 q^{162}-71 q^{160}+123 q^{158}-119 q^{156}+67 q^{154}+16 q^{152}-87 q^{150}+111 q^{148}-74 q^{146}-12 q^{144}+110 q^{142}-174 q^{140}+165 q^{138}-94 q^{136}-29 q^{134}+143 q^{132}-219 q^{130}+216 q^{128}-150 q^{126}+31 q^{124}+83 q^{122}-168 q^{120}+182 q^{118}-135 q^{116}+42 q^{114}+50 q^{112}-112 q^{110}+114 q^{108}-58 q^{106}-25 q^{104}+104 q^{102}-135 q^{100}+105 q^{98}-22 q^{96}-75 q^{94}+152 q^{92}-166 q^{90}+126 q^{88}-41 q^{86}-48 q^{84}+111 q^{82}-123 q^{80}+102 q^{78}-49 q^{76}+36 q^{72}-49 q^{70}+43 q^{68}-25 q^{66}+11 q^{64}+2 q^{62}-6 q^{60}+7 q^{58}-5 q^{56}+4 q^{54}-q^{52}+q^{50} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_80.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{27}-2 q^{25}+3 q^{23}-4 q^{21}+q^{19}-q^{17}-q^{15}+3 q^{13}-2 q^{11}+4 q^9-q^7+q^5 }[/math]
2 [math]\displaystyle{ q^{74}-2 q^{72}+5 q^{68}-9 q^{66}+17 q^{62}-17 q^{60}-7 q^{58}+28 q^{56}-14 q^{54}-15 q^{52}+22 q^{50}+q^{48}-15 q^{46}+2 q^{44}+13 q^{42}-8 q^{40}-17 q^{38}+18 q^{36}+5 q^{34}-27 q^{32}+13 q^{30}+16 q^{28}-22 q^{26}+2 q^{24}+17 q^{22}-8 q^{20}-3 q^{18}+7 q^{16}-q^{12}+q^{10} }[/math]
3 [math]\displaystyle{ q^{141}-2 q^{139}+2 q^{135}-5 q^{131}+q^{129}+14 q^{127}-5 q^{125}-26 q^{123}+6 q^{121}+48 q^{119}-78 q^{115}-17 q^{113}+109 q^{111}+42 q^{109}-123 q^{107}-87 q^{105}+125 q^{103}+120 q^{101}-98 q^{99}-141 q^{97}+53 q^{95}+142 q^{93}-3 q^{91}-122 q^{89}-45 q^{87}+95 q^{85}+81 q^{83}-54 q^{81}-112 q^{79}+23 q^{77}+129 q^{75}+17 q^{73}-142 q^{71}-46 q^{69}+136 q^{67}+86 q^{65}-126 q^{63}-119 q^{61}+93 q^{59}+137 q^{57}-53 q^{55}-145 q^{53}+4 q^{51}+128 q^{49}+33 q^{47}-96 q^{45}-55 q^{43}+53 q^{41}+61 q^{39}-22 q^{37}-42 q^{35}+28 q^{31}+8 q^{29}-10 q^{27}-5 q^{25}+5 q^{23}+3 q^{21}-q^{17}+q^{15} }[/math]
4 [math]\displaystyle{ q^{228}-2 q^{226}+2 q^{222}-3 q^{220}+4 q^{218}-4 q^{216}+4 q^{214}+7 q^{212}-18 q^{210}+2 q^{208}-5 q^{206}+31 q^{204}+34 q^{202}-66 q^{200}-54 q^{198}-24 q^{196}+133 q^{194}+167 q^{192}-119 q^{190}-252 q^{188}-200 q^{186}+271 q^{184}+538 q^{182}+29 q^{180}-523 q^{178}-688 q^{176}+154 q^{174}+975 q^{172}+555 q^{170}-454 q^{168}-1205 q^{166}-400 q^{164}+950 q^{162}+1086 q^{160}+105 q^{158}-1161 q^{156}-935 q^{154}+337 q^{152}+1054 q^{150}+681 q^{148}-532 q^{146}-964 q^{144}-341 q^{142}+539 q^{140}+850 q^{138}+153 q^{136}-622 q^{134}-723 q^{132}+13 q^{130}+757 q^{128}+604 q^{126}-291 q^{124}-906 q^{122}-350 q^{120}+636 q^{118}+932 q^{116}-1020 q^{112}-699 q^{110}+413 q^{108}+1180 q^{106}+424 q^{104}-886 q^{102}-1034 q^{100}-100 q^{98}+1108 q^{96}+897 q^{94}-336 q^{92}-1032 q^{90}-702 q^{88}+545 q^{86}+985 q^{84}+342 q^{82}-509 q^{80}-876 q^{78}-141 q^{76}+527 q^{74}+573 q^{72}+105 q^{70}-491 q^{68}-377 q^{66}-q^{64}+302 q^{62}+286 q^{60}-65 q^{58}-186 q^{56}-145 q^{54}+24 q^{52}+136 q^{50}+51 q^{48}-12 q^{46}-60 q^{44}-29 q^{42}+24 q^{40}+17 q^{38}+13 q^{36}-7 q^{34}-8 q^{32}+3 q^{30}+q^{28}+3 q^{26}-q^{22}+q^{20} }[/math]
5 [math]\displaystyle{ q^{335}-2 q^{333}+2 q^{329}-3 q^{327}+q^{325}+5 q^{323}-q^{321}-3 q^{319}-9 q^{315}-3 q^{313}+20 q^{311}+21 q^{309}-2 q^{307}-36 q^{305}-56 q^{303}-24 q^{301}+72 q^{299}+159 q^{297}+88 q^{295}-133 q^{293}-329 q^{291}-265 q^{289}+140 q^{287}+632 q^{285}+675 q^{283}-41 q^{281}-1034 q^{279}-1369 q^{277}-390 q^{275}+1401 q^{273}+2433 q^{271}+1337 q^{269}-1506 q^{267}-3722 q^{265}-2907 q^{263}+996 q^{261}+4904 q^{259}+5007 q^{257}+400 q^{255}-5518 q^{253}-7302 q^{251}-2595 q^{249}+5112 q^{247}+9087 q^{245}+5346 q^{243}-3498 q^{241}-9907 q^{239}-7923 q^{237}+993 q^{235}+9251 q^{233}+9705 q^{231}+1932 q^{229}-7327 q^{227}-10214 q^{225}-4512 q^{223}+4568 q^{221}+9347 q^{219}+6260 q^{217}-1590 q^{215}-7505 q^{213}-6959 q^{211}-970 q^{209}+5191 q^{207}+6729 q^{205}+2872 q^{203}-2962 q^{201}-6023 q^{199}-4022 q^{197}+1153 q^{195}+5199 q^{193}+4725 q^{191}+109 q^{189}-4615 q^{187}-5208 q^{185}-1011 q^{183}+4341 q^{181}+5830 q^{179}+1736 q^{177}-4306 q^{175}-6587 q^{173}-2667 q^{171}+4208 q^{169}+7571 q^{167}+3875 q^{165}-3791 q^{163}-8393 q^{161}-5462 q^{159}+2703 q^{157}+8856 q^{155}+7198 q^{153}-957 q^{151}-8470 q^{149}-8718 q^{147}-1408 q^{145}+7113 q^{143}+9559 q^{141}+3958 q^{139}-4788 q^{137}-9321 q^{135}-6119 q^{133}+1835 q^{131}+7869 q^{129}+7363 q^{127}+1150 q^{125}-5442 q^{123}-7313 q^{121}-3480 q^{119}+2560 q^{117}+6030 q^{115}+4683 q^{113}+87 q^{111}-3979 q^{109}-4623 q^{107}-1875 q^{105}+1754 q^{103}+3560 q^{101}+2624 q^{99}+9 q^{97}-2136 q^{95}-2386 q^{93}-982 q^{91}+768 q^{89}+1647 q^{87}+1233 q^{85}+67 q^{83}-831 q^{81}-958 q^{79}-437 q^{77}+233 q^{75}+564 q^{73}+424 q^{71}+57 q^{69}-230 q^{67}-277 q^{65}-123 q^{63}+52 q^{61}+127 q^{59}+101 q^{57}+12 q^{55}-47 q^{53}-45 q^{51}-13 q^{49}+6 q^{47}+21 q^{45}+14 q^{43}-3 q^{41}-5 q^{39}-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}+q^{34}-3 q^{32}-2 q^{30}-q^{28}-3 q^{26}+3 q^{24}-q^{22}+3 q^{20}+2 q^{18}+3 q^{14}-q^{12}+q^{10} }[/math]
1,1 [math]\displaystyle{ q^{108}-4 q^{106}+10 q^{104}-20 q^{102}+38 q^{100}-66 q^{98}+102 q^{96}-154 q^{94}+223 q^{92}-302 q^{90}+386 q^{88}-474 q^{86}+548 q^{84}-578 q^{82}+552 q^{80}-460 q^{78}+289 q^{76}-44 q^{74}-250 q^{72}+562 q^{70}-849 q^{68}+1084 q^{66}-1220 q^{64}+1260 q^{62}-1181 q^{60}+1008 q^{58}-748 q^{56}+436 q^{54}-129 q^{52}-184 q^{50}+414 q^{48}-596 q^{46}+658 q^{44}-662 q^{42}+594 q^{40}-474 q^{38}+360 q^{36}-234 q^{34}+154 q^{32}-76 q^{30}+43 q^{28}-16 q^{26}+8 q^{24}-2 q^{22}+q^{20} }[/math]
2,0 [math]\displaystyle{ q^{100}+q^{98}-q^{94}-2 q^{92}-2 q^{90}-5 q^{88}+5 q^{84}-q^{82}-q^{80}+6 q^{78}+9 q^{76}-3 q^{74}-5 q^{72}+12 q^{70}+6 q^{68}-8 q^{66}-2 q^{64}+6 q^{62}-8 q^{60}-12 q^{58}-q^{56}-3 q^{54}-11 q^{52}-2 q^{50}+8 q^{48}-7 q^{46}-5 q^{44}+12 q^{42}+6 q^{40}-6 q^{38}+3 q^{36}+11 q^{34}+3 q^{32}-5 q^{30}+3 q^{28}+5 q^{26}-q^{24}-q^{22}+q^{20} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{88}-2 q^{86}+5 q^{82}-7 q^{80}-3 q^{78}+14 q^{76}-11 q^{74}-9 q^{72}+21 q^{70}-11 q^{68}-9 q^{66}+22 q^{64}-4 q^{60}+10 q^{58}+2 q^{56}-8 q^{54}-16 q^{52}-2 q^{48}-23 q^{46}+7 q^{44}+13 q^{42}-16 q^{40}+10 q^{38}+15 q^{36}-11 q^{34}+9 q^{32}+7 q^{30}-4 q^{28}+4 q^{26}+2 q^{24}-q^{22}+q^{20} }[/math]
1,0,0 [math]\displaystyle{ q^{53}+q^{51}+2 q^{49}-q^{47}+q^{45}-5 q^{43}-q^{41}-5 q^{39}-q^{37}-2 q^{35}+q^{33}+2 q^{31}+q^{29}+4 q^{27}+4 q^{23}-q^{21}+3 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}+2 q^{106}+3 q^{104}-7 q^{102}-8 q^{100}+7 q^{98}-16 q^{94}+q^{92}+15 q^{90}-q^{88}+25 q^{84}+21 q^{82}-q^{80}+8 q^{78}+15 q^{76}-20 q^{74}-29 q^{72}-4 q^{70}-20 q^{68}-38 q^{66}-5 q^{64}+9 q^{62}-9 q^{60}-4 q^{58}+19 q^{56}+12 q^{54}-3 q^{52}+7 q^{50}+14 q^{48}+q^{46}+10 q^{42}+2 q^{40}+3 q^{36}+2 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}+q^{56}-5 q^{54}-3 q^{52}-4 q^{50}-5 q^{48}-q^{46}-2 q^{44}+2 q^{42}+4 q^{38}+q^{36}+4 q^{34}+q^{32}+2 q^{30}+3 q^{28}-q^{26}+3 q^{24}-q^{22}+q^{20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{88}-2 q^{86}+4 q^{84}-7 q^{82}+11 q^{80}-15 q^{78}+20 q^{76}-23 q^{74}+23 q^{72}-21 q^{70}+15 q^{68}-7 q^{66}-4 q^{64}+16 q^{62}-28 q^{60}+36 q^{58}-44 q^{56}+44 q^{54}-44 q^{52}+36 q^{50}-28 q^{48}+15 q^{46}-3 q^{44}-7 q^{42}+16 q^{40}-20 q^{38}+25 q^{36}-21 q^{34}+21 q^{32}-15 q^{30}+12 q^{28}-6 q^{26}+4 q^{24}-q^{22}+q^{20} }[/math]
1,0 [math]\displaystyle{ q^{142}-2 q^{138}-2 q^{136}+2 q^{134}+6 q^{132}+q^{130}-9 q^{128}-9 q^{126}+5 q^{124}+17 q^{122}+4 q^{120}-19 q^{118}-16 q^{116}+11 q^{114}+24 q^{112}-23 q^{108}-9 q^{106}+20 q^{104}+18 q^{102}-9 q^{100}-16 q^{98}+7 q^{96}+19 q^{94}+q^{92}-18 q^{90}-5 q^{88}+12 q^{86}+4 q^{84}-17 q^{82}-13 q^{80}+9 q^{78}+11 q^{76}-13 q^{74}-23 q^{72}+2 q^{70}+24 q^{68}+9 q^{66}-21 q^{64}-18 q^{62}+14 q^{60}+26 q^{58}+q^{56}-17 q^{54}-8 q^{52}+14 q^{50}+13 q^{48}-2 q^{46}-8 q^{44}+5 q^{40}+3 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}+6 q^{114}-9 q^{112}+8 q^{110}-11 q^{108}+17 q^{106}-18 q^{104}+15 q^{102}-18 q^{100}+19 q^{98}-14 q^{96}+8 q^{94}-6 q^{92}+3 q^{90}+14 q^{88}-9 q^{86}+24 q^{84}-22 q^{82}+35 q^{80}-33 q^{78}+29 q^{76}-44 q^{74}+22 q^{72}-37 q^{70}+14 q^{68}-26 q^{66}+6 q^{64}-3 q^{62}+10 q^{58}-9 q^{56}+22 q^{54}-14 q^{52}+21 q^{50}-15 q^{48}+20 q^{46}-11 q^{44}+14 q^{42}-7 q^{40}+8 q^{38}-2 q^{36}+3 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}+5 q^{202}-4 q^{200}-2 q^{198}+11 q^{196}-20 q^{194}+28 q^{192}-32 q^{190}+25 q^{188}-8 q^{186}-19 q^{184}+54 q^{182}-78 q^{180}+88 q^{178}-74 q^{176}+29 q^{174}+32 q^{172}-94 q^{170}+139 q^{168}-135 q^{166}+89 q^{164}-9 q^{162}-71 q^{160}+123 q^{158}-119 q^{156}+67 q^{154}+16 q^{152}-87 q^{150}+111 q^{148}-74 q^{146}-12 q^{144}+110 q^{142}-174 q^{140}+165 q^{138}-94 q^{136}-29 q^{134}+143 q^{132}-219 q^{130}+216 q^{128}-150 q^{126}+31 q^{124}+83 q^{122}-168 q^{120}+182 q^{118}-135 q^{116}+42 q^{114}+50 q^{112}-112 q^{110}+114 q^{108}-58 q^{106}-25 q^{104}+104 q^{102}-135 q^{100}+105 q^{98}-22 q^{96}-75 q^{94}+152 q^{92}-166 q^{90}+126 q^{88}-41 q^{86}-48 q^{84}+111 q^{82}-123 q^{80}+102 q^{78}-49 q^{76}+36 q^{72}-49 q^{70}+43 q^{68}-25 q^{66}+11 q^{64}+2 q^{62}-6 q^{60}+7 q^{58}-5 q^{56}+4 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 80"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t^3-9 t^2+15 t-17+15 t^{-1} -9 t^{-2} +3 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 z^6+9 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{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 71, -6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -12 q^{-8} +11 q^{-9} -10 q^{-10} +6 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}-9 z^2 a^{10}-6 a^{10}+2 z^6 a^8+8 z^4 a^8+9 z^2 a^8+3 a^8+z^6 a^6+4 z^4 a^6+5 z^2 a^6+2 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}-3 z^3 a^{15}+z a^{15}+5 z^6 a^{14}-5 z^4 a^{14}+2 z^2 a^{14}+6 z^7 a^{13}-8 z^5 a^{13}+6 z^3 a^{13}-2 z a^{13}+4 z^8 a^{12}-z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+2 a^{12}+z^9 a^{11}+10 z^7 a^{11}-29 z^5 a^{11}+29 z^3 a^{11}-12 z a^{11}+7 z^8 a^{10}-15 z^6 a^{10}+13 z^4 a^{10}-13 z^2 a^{10}+6 a^{10}+z^9 a^9+6 z^7 a^9-23 z^5 a^9+22 z^3 a^9-8 z a^9+3 z^8 a^8-8 z^6 a^8+8 z^4 a^8-7 z^2 a^8+3 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 a^6 }[/math]
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