10 36

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -3 t^2+13 t-19+13 t^{-1} -3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ -3 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 51, -2 }
Jones polynomial [math]\displaystyle{ q-2+4 q^{-1} -6 q^{-2} +8 q^{-3} -8 q^{-4} +8 q^{-5} -6 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^8-z^4 a^6-z^2 a^6-a^6-z^4 a^4+z^2 a^4+2 a^4-z^4 a^2-z^2 a^2-a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-11 z^5 a^9+9 z^3 a^9-z a^9+3 z^8 a^8-10 z^6 a^8+8 z^4 a^8-2 z^2 a^8+z^9 a^7+2 z^7 a^7-15 z^5 a^7+16 z^3 a^7-4 z a^7+5 z^8 a^6-16 z^6 a^6+18 z^4 a^6-8 z^2 a^6+a^6+z^9 a^5+z^7 a^5-6 z^5 a^5+8 z^3 a^5-3 z a^5+2 z^8 a^4-3 z^6 a^4+6 z^4 a^4-6 z^2 a^4+2 a^4+2 z^7 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-3 z^2 a^2+a^2+2 z^5 a-3 z^3 a+z a+z^4-2 z^2+1 }[/math]
The A2 invariant [math]\displaystyle{ q^{28}-q^{26}-q^{24}+q^{22}-2 q^{20}+q^{16}+2 q^{12}+q^8-2 q^4+2 q^2+ q^{-4} }[/math]
The G2 invariant [math]\displaystyle{ q^{142}-2 q^{140}+4 q^{138}-7 q^{136}+6 q^{134}-5 q^{132}-2 q^{130}+15 q^{128}-23 q^{126}+29 q^{124}-25 q^{122}+9 q^{120}+12 q^{118}-35 q^{116}+48 q^{114}-44 q^{112}+26 q^{110}+3 q^{108}-28 q^{106}+42 q^{104}-37 q^{102}+18 q^{100}+3 q^{98}-22 q^{96}+26 q^{94}-19 q^{92}-2 q^{90}+25 q^{88}-35 q^{86}+33 q^{84}-18 q^{82}-11 q^{80}+34 q^{78}-52 q^{76}+52 q^{74}-38 q^{72}+10 q^{70}+24 q^{68}-48 q^{66}+54 q^{64}-41 q^{62}+18 q^{60}+9 q^{58}-27 q^{56}+30 q^{54}-18 q^{52}+5 q^{50}+15 q^{48}-20 q^{46}+15 q^{44}+q^{42}-16 q^{40}+27 q^{38}-26 q^{36}+18 q^{34}-5 q^{32}-10 q^{30}+20 q^{28}-26 q^{26}+26 q^{24}-19 q^{22}+9 q^{20}-11 q^{16}+16 q^{14}-20 q^{12}+19 q^{10}-12 q^8+5 q^6+3 q^4-8 q^2+11-9 q^{-2} +8 q^{-4} -3 q^{-6} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_36.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{19}-2 q^{17}+q^{15}-2 q^{13}+2 q^{11}+2 q^5-2 q^3+2 q- q^{-1} + q^{-3} }[/math]
2 [math]\displaystyle{ q^{54}-2 q^{52}-2 q^{50}+6 q^{48}-q^{46}-7 q^{44}+7 q^{42}+4 q^{40}-10 q^{38}+3 q^{36}+7 q^{34}-9 q^{32}-q^{30}+7 q^{28}-4 q^{26}-4 q^{24}+4 q^{22}+4 q^{20}-5 q^{18}-3 q^{16}+10 q^{14}-3 q^{12}-7 q^{10}+9 q^8-q^6-5 q^4+6 q^2-1-2 q^{-2} +3 q^{-4} - q^{-6} - q^{-8} + q^{-10} }[/math]
3 [math]\displaystyle{ q^{105}-2 q^{103}-2 q^{101}+3 q^{99}+6 q^{97}-q^{95}-13 q^{93}-2 q^{91}+16 q^{89}+10 q^{87}-16 q^{85}-20 q^{83}+10 q^{81}+28 q^{79}-q^{77}-28 q^{75}-13 q^{73}+25 q^{71}+24 q^{69}-17 q^{67}-29 q^{65}+6 q^{63}+34 q^{61}+2 q^{59}-31 q^{57}-11 q^{55}+30 q^{53}+13 q^{51}-25 q^{49}-19 q^{47}+19 q^{45}+21 q^{43}-11 q^{41}-24 q^{39}+24 q^{35}+14 q^{33}-20 q^{31}-26 q^{29}+14 q^{27}+30 q^{25}-3 q^{23}-32 q^{21}-5 q^{19}+27 q^{17}+10 q^{15}-15 q^{13}-9 q^{11}+8 q^9+7 q^7-2 q^5-2 q^3-q- q^{-1} +2 q^{-3} +3 q^{-5} - q^{-7} -3 q^{-9} +3 q^{-13} - q^{-17} - q^{-19} + q^{-21} }[/math]
4 [math]\displaystyle{ q^{172}-2 q^{170}-2 q^{168}+3 q^{166}+3 q^{164}+6 q^{162}-8 q^{160}-13 q^{158}-q^{156}+7 q^{154}+31 q^{152}+q^{150}-30 q^{148}-27 q^{146}-13 q^{144}+54 q^{142}+41 q^{140}-6 q^{138}-45 q^{136}-73 q^{134}+20 q^{132}+62 q^{130}+59 q^{128}+8 q^{126}-92 q^{124}-58 q^{122}-6 q^{120}+80 q^{118}+103 q^{116}-18 q^{114}-86 q^{112}-115 q^{110}+6 q^{108}+142 q^{106}+94 q^{104}-25 q^{102}-165 q^{100}-93 q^{98}+102 q^{96}+151 q^{94}+53 q^{92}-141 q^{90}-142 q^{88}+40 q^{86}+144 q^{84}+85 q^{82}-95 q^{80}-134 q^{78}-q^{76}+114 q^{74}+86 q^{72}-49 q^{70}-114 q^{68}-42 q^{66}+79 q^{64}+94 q^{62}+21 q^{60}-81 q^{58}-108 q^{56}-2 q^{54}+91 q^{52}+127 q^{50}-155 q^{46}-115 q^{44}+24 q^{42}+189 q^{40}+117 q^{38}-108 q^{36}-174 q^{34}-81 q^{32}+145 q^{30}+171 q^{28}-10 q^{26}-124 q^{24}-129 q^{22}+46 q^{20}+126 q^{18}+45 q^{16}-36 q^{14}-98 q^{12}-13 q^{10}+53 q^8+39 q^6+14 q^4-48 q^2-22+11 q^{-2} +17 q^{-4} +22 q^{-6} -16 q^{-8} -12 q^{-10} -3 q^{-12} +2 q^{-14} +14 q^{-16} -3 q^{-18} -3 q^{-20} -3 q^{-22} -2 q^{-24} +5 q^{-26} - q^{-32} - q^{-34} + q^{-36} }[/math]
5 [math]\displaystyle{ q^{255}-2 q^{253}-2 q^{251}+3 q^{249}+3 q^{247}+3 q^{245}-q^{243}-8 q^{241}-13 q^{239}-q^{237}+17 q^{235}+22 q^{233}+13 q^{231}-13 q^{229}-40 q^{227}-44 q^{225}+4 q^{223}+57 q^{221}+69 q^{219}+36 q^{217}-43 q^{215}-107 q^{213}-93 q^{211}+6 q^{209}+107 q^{207}+139 q^{205}+80 q^{203}-57 q^{201}-163 q^{199}-158 q^{197}-38 q^{195}+108 q^{193}+195 q^{191}+163 q^{189}+14 q^{187}-154 q^{185}-249 q^{183}-186 q^{181}+16 q^{179}+245 q^{177}+348 q^{175}+205 q^{173}-133 q^{171}-438 q^{169}-436 q^{167}-80 q^{165}+403 q^{163}+635 q^{161}+354 q^{159}-267 q^{157}-736 q^{155}-611 q^{153}+42 q^{151}+709 q^{149}+816 q^{147}+214 q^{145}-607 q^{143}-916 q^{141}-429 q^{139}+426 q^{137}+919 q^{135}+600 q^{133}-254 q^{131}-846 q^{129}-666 q^{127}+91 q^{125}+730 q^{123}+675 q^{121}+14 q^{119}-604 q^{117}-624 q^{115}-80 q^{113}+494 q^{111}+559 q^{109}+106 q^{107}-401 q^{105}-503 q^{103}-139 q^{101}+332 q^{99}+468 q^{97}+190 q^{95}-239 q^{93}-459 q^{91}-297 q^{89}+115 q^{87}+458 q^{85}+446 q^{83}+69 q^{81}-409 q^{79}-617 q^{77}-324 q^{75}+298 q^{73}+761 q^{71}+601 q^{69}-107 q^{67}-803 q^{65}-876 q^{63}-167 q^{61}+748 q^{59}+1057 q^{57}+441 q^{55}-570 q^{53}-1100 q^{51}-682 q^{49}+327 q^{47}+1018 q^{45}+808 q^{43}-81 q^{41}-825 q^{39}-805 q^{37}-122 q^{35}+590 q^{33}+716 q^{31}+232 q^{29}-369 q^{27}-563 q^{25}-260 q^{23}+192 q^{21}+405 q^{19}+244 q^{17}-83 q^{15}-272 q^{13}-195 q^{11}+25 q^9+171 q^7+147 q^5+7 q^3-106 q-107 q^{-1} -16 q^{-3} +66 q^{-5} +73 q^{-7} +19 q^{-9} -36 q^{-11} -49 q^{-13} -21 q^{-15} +19 q^{-17} +34 q^{-19} +14 q^{-21} -8 q^{-23} -16 q^{-25} -13 q^{-27} +12 q^{-31} +7 q^{-33} - q^{-35} -2 q^{-37} -5 q^{-39} -2 q^{-41} +3 q^{-43} +2 q^{-45} - q^{-51} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{28}-q^{26}-q^{24}+q^{22}-2 q^{20}+q^{16}+2 q^{12}+q^8-2 q^4+2 q^2+ q^{-4} }[/math]
1,1 [math]\displaystyle{ q^{76}-4 q^{74}+10 q^{72}-22 q^{70}+42 q^{68}-70 q^{66}+100 q^{64}-132 q^{62}+161 q^{60}-172 q^{58}+166 q^{56}-136 q^{54}+88 q^{52}-24 q^{50}-50 q^{48}+124 q^{46}-192 q^{44}+238 q^{42}-272 q^{40}+278 q^{38}-266 q^{36}+232 q^{34}-178 q^{32}+122 q^{30}-62 q^{28}+12 q^{26}+30 q^{24}-60 q^{22}+74 q^{20}-80 q^{18}+82 q^{16}-80 q^{14}+80 q^{12}-74 q^{10}+72 q^8-64 q^6+58 q^4-46 q^2+36-26 q^{-2} +17 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0 [math]\displaystyle{ q^{72}-q^{70}-2 q^{68}+3 q^{64}+2 q^{62}-5 q^{60}-q^{58}+5 q^{56}+4 q^{54}-4 q^{52}-4 q^{50}+5 q^{48}+3 q^{46}-6 q^{44}-6 q^{42}+3 q^{40}+2 q^{38}-3 q^{36}-2 q^{34}+3 q^{32}+q^{30}-q^{28}+3 q^{26}-q^{24}-3 q^{22}+6 q^{20}+4 q^{18}-6 q^{16}-4 q^{14}+7 q^{12}+5 q^{10}-7 q^8-3 q^6+7 q^4+2 q^2-3+2 q^{-4} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{60}-2 q^{58}+2 q^{54}-4 q^{52}+4 q^{50}+2 q^{48}-5 q^{46}+5 q^{44}+q^{42}-8 q^{40}+4 q^{38}+3 q^{36}-8 q^{34}+2 q^{32}+3 q^{30}-3 q^{28}-2 q^{26}+2 q^{24}+4 q^{22}-4 q^{20}+9 q^{16}-5 q^{14}-3 q^{12}+9 q^{10}-3 q^8-5 q^6+6 q^4-3+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0 [math]\displaystyle{ q^{37}-q^{35}-q^{31}+q^{29}-2 q^{27}-q^{23}+q^{21}+2 q^{17}+2 q^{15}+q^{11}-q^9-2 q^5+2 q^3+ q^{-1} + q^{-5} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{60}-2 q^{58}+4 q^{56}-6 q^{54}+8 q^{52}-10 q^{50}+10 q^{48}-11 q^{46}+9 q^{44}-7 q^{42}+2 q^{40}+2 q^{38}-7 q^{36}+12 q^{34}-16 q^{32}+19 q^{30}-19 q^{28}+20 q^{26}-16 q^{24}+14 q^{22}-8 q^{20}+4 q^{18}+q^{16}-5 q^{14}+7 q^{12}-9 q^{10}+9 q^8-9 q^6+8 q^4-6 q^2+5-3 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }[/math]
1,0 [math]\displaystyle{ q^{98}-2 q^{94}-2 q^{92}+2 q^{90}+4 q^{88}-2 q^{86}-6 q^{84}+9 q^{80}+5 q^{78}-7 q^{76}-9 q^{74}+4 q^{72}+11 q^{70}+2 q^{68}-11 q^{66}-8 q^{64}+5 q^{62}+9 q^{60}-q^{58}-9 q^{56}-2 q^{54}+6 q^{52}+3 q^{50}-5 q^{48}-3 q^{46}+5 q^{44}+5 q^{42}-5 q^{40}-6 q^{38}+3 q^{36}+8 q^{34}-q^{32}-8 q^{30}-2 q^{28}+9 q^{26}+6 q^{24}-6 q^{22}-8 q^{20}+2 q^{18}+10 q^{16}+4 q^{14}-6 q^{12}-7 q^{10}+q^8+7 q^6+3 q^4-3 q^2-4+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{142}-2 q^{140}+4 q^{138}-7 q^{136}+6 q^{134}-5 q^{132}-2 q^{130}+15 q^{128}-23 q^{126}+29 q^{124}-25 q^{122}+9 q^{120}+12 q^{118}-35 q^{116}+48 q^{114}-44 q^{112}+26 q^{110}+3 q^{108}-28 q^{106}+42 q^{104}-37 q^{102}+18 q^{100}+3 q^{98}-22 q^{96}+26 q^{94}-19 q^{92}-2 q^{90}+25 q^{88}-35 q^{86}+33 q^{84}-18 q^{82}-11 q^{80}+34 q^{78}-52 q^{76}+52 q^{74}-38 q^{72}+10 q^{70}+24 q^{68}-48 q^{66}+54 q^{64}-41 q^{62}+18 q^{60}+9 q^{58}-27 q^{56}+30 q^{54}-18 q^{52}+5 q^{50}+15 q^{48}-20 q^{46}+15 q^{44}+q^{42}-16 q^{40}+27 q^{38}-26 q^{36}+18 q^{34}-5 q^{32}-10 q^{30}+20 q^{28}-26 q^{26}+26 q^{24}-19 q^{22}+9 q^{20}-11 q^{16}+16 q^{14}-20 q^{12}+19 q^{10}-12 q^8+5 q^6+3 q^4-8 q^2+11-9 q^{-2} +8 q^{-4} -3 q^{-6} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/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 36"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -3 t^2+13 t-19+13 t^{-1} -3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -3 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]= { 51, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q-2+4 q^{-1} -6 q^{-2} +8 q^{-3} -8 q^{-4} +8 q^{-5} -6 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^2 a^8-z^4 a^6-z^2 a^6-a^6-z^4 a^4+z^2 a^4+2 a^4-z^4 a^2-z^2 a^2-a^2+z^2+1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-11 z^5 a^9+9 z^3 a^9-z a^9+3 z^8 a^8-10 z^6 a^8+8 z^4 a^8-2 z^2 a^8+z^9 a^7+2 z^7 a^7-15 z^5 a^7+16 z^3 a^7-4 z a^7+5 z^8 a^6-16 z^6 a^6+18 z^4 a^6-8 z^2 a^6+a^6+z^9 a^5+z^7 a^5-6 z^5 a^5+8 z^3 a^5-3 z a^5+2 z^8 a^4-3 z^6 a^4+6 z^4 a^4-6 z^2 a^4+2 a^4+2 z^7 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-3 z^2 a^2+a^2+2 z^5 a-3 z^3 a+z a+z^4-2 z^2+1 }[/math]
Retrieved from "https://katlas.org/index.php?title=10_36&oldid=4307"