10 127

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+4 t^2-6 t+7-6 t^{-1} +4 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 { 29, -4 }
Jones polynomial [math]\displaystyle{ 2 q^{-2} -2 q^{-3} +4 q^{-4} -5 q^{-5} +5 q^{-6} -5 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^8+3 z^2 a^8+2 a^8-z^6 a^6-5 z^4 a^6-9 z^2 a^6-6 a^6+2 z^4 a^4+7 z^2 a^4+5 a^4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+2 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+2 z^7 a^9-5 z^5 a^9+7 z^3 a^9-2 z a^9+z^8 a^8-2 z^6 a^8+4 z^4 a^8-2 z^2 a^8+2 a^8+3 z^7 a^7-10 z^5 a^7+16 z^3 a^7-8 z a^7+z^8 a^6-4 z^6 a^6+11 z^4 a^6-14 z^2 a^6+6 a^6+z^7 a^5-3 z^5 a^5+5 z^3 a^5-5 z a^5+3 z^4 a^4-9 z^2 a^4+5 a^4 }[/math]
The A2 invariant [math]\displaystyle{ q^{30}+q^{26}-2 q^{22}-q^{20}-3 q^{18}+q^{12}+3 q^{10}+q^8+2 q^6 }[/math]
The G2 invariant [math]\displaystyle{ q^{162}-q^{160}+2 q^{158}-3 q^{156}+q^{154}-3 q^{150}+5 q^{148}-6 q^{146}+6 q^{144}-5 q^{142}+q^{140}+3 q^{138}-8 q^{136}+12 q^{134}-11 q^{132}+9 q^{130}-3 q^{128}-5 q^{126}+13 q^{124}-13 q^{122}+12 q^{120}-3 q^{118}-5 q^{116}+11 q^{114}-8 q^{112}+2 q^{110}+9 q^{108}-14 q^{106}+16 q^{104}-8 q^{102}-5 q^{100}+15 q^{98}-22 q^{96}+21 q^{94}-16 q^{92}+2 q^{90}+7 q^{88}-17 q^{86}+17 q^{84}-17 q^{82}+5 q^{80}-11 q^{76}+9 q^{74}-9 q^{72}+7 q^{68}-13 q^{66}+10 q^{64}-4 q^{62}-7 q^{60}+17 q^{58}-17 q^{56}+14 q^{54}-3 q^{52}-4 q^{50}+14 q^{48}-12 q^{46}+13 q^{44}-4 q^{42}+q^{40}+6 q^{38}-5 q^{36}+5 q^{34}+2 q^{30}+q^{28} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_127.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{21}-q^{19}+q^{17}-2 q^{15}-q^9+2 q^7+2 q^3 }[/math]
2 [math]\displaystyle{ q^{58}-q^{56}-q^{54}+2 q^{52}-2 q^{50}-2 q^{48}+5 q^{46}-q^{44}-5 q^{42}+6 q^{40}+q^{38}-4 q^{36}+2 q^{34}+2 q^{32}-q^{30}-4 q^{28}+2 q^{26}+2 q^{24}-6 q^{22}+q^{20}+4 q^{18}-5 q^{16}+5 q^{12}-q^{10}-q^8+3 q^6+q^4 }[/math]
3 [math]\displaystyle{ q^{111}-q^{109}-q^{107}+q^{103}-q^{99}+2 q^{97}+2 q^{95}-3 q^{93}-5 q^{91}+4 q^{89}+9 q^{87}-2 q^{85}-15 q^{83}-q^{81}+17 q^{79}+7 q^{77}-19 q^{75}-10 q^{73}+14 q^{71}+12 q^{69}-8 q^{67}-12 q^{65}+4 q^{63}+9 q^{61}+3 q^{59}-7 q^{57}-7 q^{55}+5 q^{53}+12 q^{51}-4 q^{49}-13 q^{47}+q^{45}+17 q^{43}-q^{41}-17 q^{39}-5 q^{37}+17 q^{35}+7 q^{33}-14 q^{31}-12 q^{29}+7 q^{27}+13 q^{25}-3 q^{23}-10 q^{21}-2 q^{19}+8 q^{17}+5 q^{15}-2 q^{13}-4 q^{11}+2 q^9+2 q^7+2 q^5 }[/math]
5 [math]\displaystyle{ q^{265}-q^{263}-q^{261}-q^{257}+q^{255}+3 q^{253}+2 q^{251}-q^{249}-q^{247}-4 q^{245}-5 q^{243}+q^{241}+6 q^{239}+6 q^{237}+4 q^{235}-2 q^{233}-12 q^{231}-14 q^{229}-2 q^{227}+14 q^{225}+24 q^{223}+18 q^{221}-6 q^{219}-38 q^{217}-44 q^{215}-10 q^{213}+40 q^{211}+75 q^{209}+54 q^{207}-26 q^{205}-107 q^{203}-112 q^{201}-19 q^{199}+119 q^{197}+179 q^{195}+89 q^{193}-96 q^{191}-231 q^{189}-182 q^{187}+38 q^{185}+250 q^{183}+257 q^{181}+46 q^{179}-214 q^{177}-308 q^{175}-128 q^{173}+156 q^{171}+297 q^{169}+187 q^{167}-71 q^{165}-248 q^{163}-207 q^{161}+178 q^{157}+187 q^{155}+48 q^{153}-99 q^{151}-148 q^{149}-78 q^{147}+42 q^{145}+101 q^{143}+76 q^{141}+4 q^{139}-67 q^{137}-79 q^{135}-29 q^{133}+49 q^{131}+83 q^{129}+46 q^{127}-44 q^{125}-104 q^{123}-61 q^{121}+51 q^{119}+132 q^{117}+93 q^{115}-58 q^{113}-171 q^{111}-122 q^{109}+50 q^{107}+201 q^{105}+177 q^{103}-29 q^{101}-224 q^{99}-217 q^{97}-17 q^{95}+208 q^{93}+262 q^{91}+79 q^{89}-174 q^{87}-270 q^{85}-143 q^{83}+98 q^{81}+250 q^{79}+196 q^{77}-14 q^{75}-194 q^{73}-214 q^{71}-69 q^{69}+109 q^{67}+190 q^{65}+129 q^{63}-21 q^{61}-134 q^{59}-139 q^{57}-55 q^{55}+59 q^{53}+114 q^{51}+88 q^{49}+9 q^{47}-65 q^{45}-88 q^{43}-47 q^{41}+15 q^{39}+52 q^{37}+58 q^{35}+21 q^{33}-23 q^{31}-38 q^{29}-26 q^{27}-5 q^{25}+17 q^{23}+25 q^{21}+11 q^{19}-2 q^{17}-6 q^{15}-10 q^{13}-4 q^{11}+2 q^9+4 q^7+2 q^5+2 q^3 }[/math]
6 [math]\displaystyle{ q^{366}-q^{364}-q^{362}-q^{358}+q^{356}+q^{354}+5 q^{352}-3 q^{348}-q^{346}-5 q^{344}-4 q^{342}-2 q^{340}+10 q^{338}+6 q^{336}+2 q^{334}+4 q^{332}-6 q^{330}-13 q^{328}-15 q^{326}+7 q^{324}+11 q^{322}+13 q^{320}+22 q^{318}+4 q^{316}-24 q^{314}-45 q^{312}-17 q^{310}+2 q^{308}+30 q^{306}+71 q^{304}+56 q^{302}-7 q^{300}-90 q^{298}-104 q^{296}-90 q^{294}-3 q^{292}+146 q^{290}+216 q^{288}+154 q^{286}-40 q^{284}-218 q^{282}-351 q^{280}-265 q^{278}+60 q^{276}+396 q^{274}+542 q^{272}+336 q^{270}-83 q^{268}-596 q^{266}-783 q^{264}-445 q^{262}+219 q^{260}+834 q^{258}+957 q^{256}+530 q^{254}-374 q^{252}-1085 q^{250}-1118 q^{248}-441 q^{246}+556 q^{244}+1206 q^{242}+1176 q^{240}+301 q^{238}-737 q^{236}-1274 q^{234}-1003 q^{232}-114 q^{230}+778 q^{228}+1199 q^{226}+772 q^{224}-70 q^{222}-778 q^{220}-934 q^{218}-517 q^{216}+144 q^{214}+678 q^{212}+677 q^{210}+301 q^{208}-193 q^{206}-479 q^{204}-449 q^{202}-176 q^{200}+178 q^{198}+338 q^{196}+295 q^{194}+79 q^{192}-135 q^{190}-255 q^{188}-205 q^{186}-29 q^{184}+159 q^{182}+245 q^{180}+135 q^{178}-60 q^{176}-248 q^{174}-246 q^{172}-71 q^{170}+219 q^{168}+383 q^{166}+233 q^{164}-103 q^{162}-440 q^{160}-466 q^{158}-173 q^{156}+355 q^{154}+683 q^{152}+500 q^{150}-49 q^{148}-645 q^{146}-812 q^{144}-461 q^{142}+322 q^{140}+926 q^{138}+888 q^{136}+259 q^{134}-593 q^{132}-1055 q^{130}-888 q^{128}-48 q^{126}+821 q^{124}+1135 q^{122}+742 q^{120}-130 q^{118}-884 q^{116}-1141 q^{114}-614 q^{112}+250 q^{110}+909 q^{108}+1006 q^{106}+507 q^{104}-240 q^{102}-863 q^{100}-877 q^{98}-423 q^{96}+222 q^{94}+688 q^{92}+745 q^{90}+406 q^{88}-170 q^{86}-528 q^{84}-598 q^{82}-356 q^{80}+34 q^{78}+376 q^{76}+487 q^{74}+305 q^{72}+48 q^{70}-220 q^{68}-345 q^{66}-303 q^{64}-96 q^{62}+121 q^{60}+223 q^{58}+241 q^{56}+128 q^{54}-24 q^{52}-158 q^{50}-175 q^{48}-111 q^{46}-24 q^{44}+76 q^{42}+118 q^{40}+100 q^{38}+26 q^{36}-28 q^{34}-59 q^{32}-64 q^{30}-33 q^{28}+5 q^{26}+34 q^{24}+31 q^{22}+21 q^{20}+8 q^{18}-8 q^{16}-15 q^{14}-11 q^{12}-3 q^{10}+q^8+3 q^6+3 q^4+3 q^2+1 }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{30}+q^{26}-2 q^{22}-q^{20}-3 q^{18}+q^{12}+3 q^{10}+q^8+2 q^6 }[/math]
1,1 [math]\displaystyle{ q^{84}-2 q^{82}+4 q^{80}-8 q^{78}+11 q^{76}-14 q^{74}+18 q^{72}-24 q^{70}+29 q^{68}-30 q^{66}+34 q^{64}-36 q^{62}+28 q^{60}-16 q^{58}-2 q^{56}+20 q^{54}-44 q^{52}+62 q^{50}-72 q^{48}+82 q^{46}-74 q^{44}+72 q^{42}-50 q^{40}+38 q^{38}-17 q^{36}-10 q^{34}+18 q^{32}-44 q^{30}+38 q^{28}-52 q^{26}+40 q^{24}-30 q^{22}+26 q^{20}-10 q^{18}+12 q^{16}+4 q^{14}+4 q^{12}+2 q^{10} }[/math]
2,0 [math]\displaystyle{ q^{76}-q^{68}-q^{66}-q^{64}+q^{62}-q^{60}-3 q^{58}+2 q^{54}+2 q^{50}+6 q^{48}+5 q^{46}+q^{44}-4 q^{38}-6 q^{36}-2 q^{34}-4 q^{32}-5 q^{30}-q^{28}-q^{24}+5 q^{20}+4 q^{18}+2 q^{16}+2 q^{14}+4 q^{12}+q^{10} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{68}-q^{66}+q^{62}-3 q^{60}+3 q^{56}-4 q^{54}-q^{52}+5 q^{50}-3 q^{48}+q^{46}+5 q^{44}+3 q^{42}+3 q^{40}+2 q^{38}+q^{36}-5 q^{34}-10 q^{32}-5 q^{30}-6 q^{28}-8 q^{26}+4 q^{24}+5 q^{22}+2 q^{20}+7 q^{18}+5 q^{16}+q^{14}+3 q^{12} }[/math]
1,0,0 [math]\displaystyle{ q^{39}+2 q^{35}+q^{31}-2 q^{29}-2 q^{27}-4 q^{25}-3 q^{23}-q^{21}+3 q^{17}+2 q^{15}+4 q^{13}+q^{11}+2 q^9 }[/math]
1,0,1 [math]\displaystyle{ q^{110}-2 q^{108}+3 q^{106}-3 q^{104}-q^{102}+6 q^{100}-9 q^{98}+9 q^{96}-4 q^{94}-6 q^{92}+16 q^{90}-19 q^{88}+13 q^{86}+6 q^{84}-28 q^{82}+38 q^{80}-30 q^{78}+5 q^{76}+24 q^{74}-41 q^{72}+39 q^{70}-30 q^{68}+q^{66}+3 q^{64}-16 q^{62}+6 q^{60}+7 q^{58}+11 q^{56}+4 q^{54}+50 q^{52}-29 q^{50}+48 q^{48}-15 q^{46}-26 q^{44}+18 q^{42}-70 q^{40}+8 q^{38}-37 q^{36}-15 q^{34}+11 q^{32}-11 q^{30}+25 q^{28}+11 q^{26}+14 q^{24}+16 q^{22}+7 q^{20}+5 q^{18}+2 q^{16} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{94}-q^{92}+q^{90}-2 q^{88}+2 q^{86}-3 q^{84}+2 q^{82}-3 q^{80}+4 q^{78}-4 q^{76}+2 q^{74}-2 q^{72}+2 q^{70}-2 q^{66}+3 q^{64}-q^{62}+9 q^{60}-2 q^{58}+10 q^{56}-3 q^{54}+9 q^{52}-6 q^{50}+q^{48}-12 q^{46}-6 q^{44}-10 q^{42}-8 q^{40}-6 q^{38}-5 q^{36}+4 q^{34}+q^{32}+9 q^{30}+3 q^{28}+10 q^{26}+2 q^{24}+6 q^{22}+3 q^{18} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{162}-q^{160}+2 q^{158}-3 q^{156}+q^{154}-3 q^{150}+5 q^{148}-6 q^{146}+6 q^{144}-5 q^{142}+q^{140}+3 q^{138}-8 q^{136}+12 q^{134}-11 q^{132}+9 q^{130}-3 q^{128}-5 q^{126}+13 q^{124}-13 q^{122}+12 q^{120}-3 q^{118}-5 q^{116}+11 q^{114}-8 q^{112}+2 q^{110}+9 q^{108}-14 q^{106}+16 q^{104}-8 q^{102}-5 q^{100}+15 q^{98}-22 q^{96}+21 q^{94}-16 q^{92}+2 q^{90}+7 q^{88}-17 q^{86}+17 q^{84}-17 q^{82}+5 q^{80}-11 q^{76}+9 q^{74}-9 q^{72}+7 q^{68}-13 q^{66}+10 q^{64}-4 q^{62}-7 q^{60}+17 q^{58}-17 q^{56}+14 q^{54}-3 q^{52}-4 q^{50}+14 q^{48}-12 q^{46}+13 q^{44}-4 q^{42}+q^{40}+6 q^{38}-5 q^{36}+5 q^{34}+2 q^{30}+q^{28} }[/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["10 127"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+4 t^2-6 t+7-6 t^{-1} +4 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]= { 29, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 2 q^{-2} -2 q^{-3} +4 q^{-4} -5 q^{-5} +5 q^{-6} -5 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4 a^8+3 z^2 a^8+2 a^8-z^6 a^6-5 z^4 a^6-9 z^2 a^6-6 a^6+2 z^4 a^4+7 z^2 a^4+5 a^4 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+2 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+2 z^7 a^9-5 z^5 a^9+7 z^3 a^9-2 z a^9+z^8 a^8-2 z^6 a^8+4 z^4 a^8-2 z^2 a^8+2 a^8+3 z^7 a^7-10 z^5 a^7+16 z^3 a^7-8 z a^7+z^8 a^6-4 z^6 a^6+11 z^4 a^6-14 z^2 a^6+6 a^6+z^7 a^5-3 z^5 a^5+5 z^3 a^5-5 z a^5+3 z^4 a^4-9 z^2 a^4+5 a^4 }[/math]
Retrieved from "https://katlas.org/index.php?title=10_127&oldid=4369"