10 14

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^3+8 t^2-12 t+13-12 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -2 z^6-4 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 57, -4 }
Jones polynomial [math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -6 q^{-3} +9 q^{-4} -9 q^{-5} +9 q^{-6} -8 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^8+2 z^2 a^8-z^6 a^6-3 z^4 a^6-2 z^2 a^6-a^6-z^6 a^4-3 z^4 a^4-z^2 a^4+a^4+z^4 a^2+3 z^2 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+4 z^6 a^{10}-4 z^4 a^{10}+4 z^7 a^9-4 z^5 a^9+2 z a^9+3 z^8 a^8-4 z^6 a^8+5 z^4 a^8-3 z^2 a^8+z^9 a^7+3 z^7 a^7-9 z^5 a^7+8 z^3 a^7-2 z a^7+5 z^8 a^6-14 z^6 a^6+16 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5+z^7 a^5-9 z^5 a^5+10 z^3 a^5-4 z a^5+2 z^8 a^4-5 z^6 a^4+2 z^4 a^4-z^2 a^4+a^4+2 z^7 a^3-7 z^5 a^3+6 z^3 a^3-z a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-a^2 }[/math]
The A2 invariant [math]\displaystyle{ q^{30}-q^{28}-2 q^{22}+q^{20}-2 q^{18}+q^{16}+q^{14}+3 q^{10}-q^8+q^6+1 }[/math]
The G2 invariant [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+12 q^{148}-18 q^{146}+22 q^{144}-20 q^{142}+10 q^{140}+5 q^{138}-22 q^{136}+40 q^{134}-46 q^{132}+41 q^{130}-27 q^{128}+29 q^{124}-52 q^{122}+63 q^{120}-50 q^{118}+25 q^{116}+10 q^{114}-37 q^{112}+44 q^{110}-33 q^{108}+8 q^{106}+21 q^{104}-42 q^{102}+36 q^{100}-7 q^{98}-31 q^{96}+66 q^{94}-81 q^{92}+59 q^{90}-21 q^{88}-33 q^{86}+73 q^{84}-97 q^{82}+88 q^{80}-51 q^{78}+q^{76}+45 q^{74}-72 q^{72}+69 q^{70}-43 q^{68}+6 q^{66}+27 q^{64}-42 q^{62}+39 q^{60}-7 q^{58}-22 q^{56}+51 q^{54}-54 q^{52}+31 q^{50}+5 q^{48}-43 q^{46}+67 q^{44}-66 q^{42}+47 q^{40}-12 q^{38}-24 q^{36}+48 q^{34}-53 q^{32}+43 q^{30}-23 q^{28}+q^{26}+14 q^{24}-21 q^{22}+22 q^{20}-15 q^{18}+9 q^{16}-3 q^{12}+4 q^{10}-4 q^8+3 q^6-q^4+q^2 }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_14.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{21}-2 q^{19}+2 q^{17}-3 q^{15}+q^{13}+3 q^7-2 q^5+2 q^3-q+ q^{-1} }[/math]
2 [math]\displaystyle{ q^{58}-2 q^{56}-q^{54}+5 q^{52}-5 q^{50}-2 q^{48}+12 q^{46}-8 q^{44}-7 q^{42}+16 q^{40}-6 q^{38}-10 q^{36}+10 q^{34}-8 q^{30}-q^{28}+8 q^{26}-q^{24}-10 q^{22}+10 q^{20}+6 q^{18}-15 q^{16}+6 q^{14}+10 q^{12}-11 q^{10}+q^8+9 q^6-5 q^4-2 q^2+4- q^{-2} - q^{-4} + q^{-6} }[/math]
3 [math]\displaystyle{ q^{111}-2 q^{109}-q^{107}+2 q^{105}+3 q^{103}-2 q^{101}-5 q^{99}+6 q^{97}+4 q^{95}-9 q^{93}-7 q^{91}+16 q^{89}+11 q^{87}-25 q^{85}-22 q^{83}+33 q^{81}+32 q^{79}-33 q^{77}-45 q^{75}+31 q^{73}+55 q^{71}-20 q^{69}-54 q^{67}+5 q^{65}+51 q^{63}+8 q^{61}-36 q^{59}-26 q^{57}+23 q^{55}+32 q^{53}-7 q^{51}-43 q^{49}-8 q^{47}+45 q^{45}+22 q^{43}-47 q^{41}-32 q^{39}+43 q^{37}+43 q^{35}-33 q^{33}-54 q^{31}+22 q^{29}+55 q^{27}-4 q^{25}-52 q^{23}-9 q^{21}+44 q^{19}+21 q^{17}-30 q^{15}-25 q^{13}+17 q^{11}+24 q^9-6 q^7-17 q^5+12 q+2 q^{-1} -6 q^{-3} -2 q^{-5} +3 q^{-7} + q^{-9} - q^{-11} - q^{-13} + q^{-15} }[/math]
4 [math]\displaystyle{ q^{180}-2 q^{178}-q^{176}+2 q^{174}+6 q^{170}-5 q^{168}-4 q^{166}+q^{164}-6 q^{162}+15 q^{160}-5 q^{158}+q^{156}+7 q^{154}-24 q^{152}+5 q^{150}-14 q^{148}+26 q^{146}+50 q^{144}-26 q^{142}-35 q^{140}-82 q^{138}+30 q^{136}+141 q^{134}+44 q^{132}-61 q^{130}-208 q^{128}-45 q^{126}+204 q^{124}+178 q^{122}+9 q^{120}-289 q^{118}-185 q^{116}+147 q^{114}+259 q^{112}+140 q^{110}-223 q^{108}-255 q^{106}-q^{104}+196 q^{102}+216 q^{100}-54 q^{98}-195 q^{96}-124 q^{94}+51 q^{92}+188 q^{90}+94 q^{88}-78 q^{86}-182 q^{84}-71 q^{82}+126 q^{80}+188 q^{78}+14 q^{76}-207 q^{74}-151 q^{72}+70 q^{70}+260 q^{68}+93 q^{66}-213 q^{64}-221 q^{62}-9 q^{60}+288 q^{58}+187 q^{56}-141 q^{54}-253 q^{52}-141 q^{50}+213 q^{48}+254 q^{46}+10 q^{44}-184 q^{42}-237 q^{40}+50 q^{38}+204 q^{36}+135 q^{34}-29 q^{32}-211 q^{30}-84 q^{28}+66 q^{26}+137 q^{24}+88 q^{22}-92 q^{20}-95 q^{18}-38 q^{16}+54 q^{14}+92 q^{12}-4 q^{10}-36 q^8-48 q^6-4 q^4+42 q^2+12-20 q^{-4} -10 q^{-6} +12 q^{-8} +3 q^{-10} +4 q^{-12} -4 q^{-14} -4 q^{-16} +3 q^{-18} + q^{-22} - q^{-24} - q^{-26} + q^{-28} }[/math]
5 [math]\displaystyle{ q^{265}-2 q^{263}-q^{261}+2 q^{259}+3 q^{255}+3 q^{253}-4 q^{251}-9 q^{249}-2 q^{247}+q^{245}+8 q^{243}+16 q^{241}+6 q^{239}-13 q^{237}-27 q^{235}-19 q^{233}+q^{231}+33 q^{229}+59 q^{227}+30 q^{225}-40 q^{223}-93 q^{221}-87 q^{219}-4 q^{217}+132 q^{215}+197 q^{213}+75 q^{211}-147 q^{209}-315 q^{207}-240 q^{205}+102 q^{203}+458 q^{201}+474 q^{199}+29 q^{197}-562 q^{195}-766 q^{193}-275 q^{191}+583 q^{189}+1062 q^{187}+637 q^{185}-469 q^{183}-1304 q^{181}-1034 q^{179}+207 q^{177}+1382 q^{175}+1416 q^{173}+181 q^{171}-1290 q^{169}-1665 q^{167}-593 q^{165}+999 q^{163}+1711 q^{161}+952 q^{159}-581 q^{157}-1560 q^{155}-1178 q^{153}+151 q^{151}+1217 q^{149}+1216 q^{147}+261 q^{145}-813 q^{143}-1122 q^{141}-523 q^{139}+392 q^{137}+916 q^{135}+710 q^{133}-41 q^{131}-714 q^{129}-778 q^{127}-225 q^{125}+525 q^{123}+849 q^{121}+411 q^{119}-410 q^{117}-903 q^{115}-575 q^{113}+339 q^{111}+1006 q^{109}+728 q^{107}-286 q^{105}-1107 q^{103}-926 q^{101}+186 q^{99}+1212 q^{97}+1142 q^{95}-27 q^{93}-1236 q^{91}-1350 q^{89}-235 q^{87}+1143 q^{85}+1524 q^{83}+541 q^{81}-920 q^{79}-1563 q^{77}-862 q^{75}+544 q^{73}+1465 q^{71}+1128 q^{69}-126 q^{67}-1190 q^{65}-1236 q^{63}-315 q^{61}+784 q^{59}+1187 q^{57}+640 q^{55}-328 q^{53}-947 q^{51}-805 q^{49}-98 q^{47}+600 q^{45}+788 q^{43}+392 q^{41}-229 q^{39}-611 q^{37}-515 q^{35}-77 q^{33}+359 q^{31}+489 q^{29}+258 q^{27}-121 q^{25}-353 q^{23}-303 q^{21}-54 q^{19}+194 q^{17}+263 q^{15}+129 q^{13}-64 q^{11}-171 q^9-135 q^7-13 q^5+90 q^3+102 q+40 q^{-1} -34 q^{-3} -62 q^{-5} -34 q^{-7} +5 q^{-9} +28 q^{-11} +26 q^{-13} +3 q^{-15} -14 q^{-17} -11 q^{-19} -2 q^{-21} + q^{-23} +6 q^{-25} +4 q^{-27} -3 q^{-29} -2 q^{-31} + q^{-33} + q^{-39} - q^{-41} - q^{-43} + q^{-45} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{30}-q^{28}-2 q^{22}+q^{20}-2 q^{18}+q^{16}+q^{14}+3 q^{10}-q^8+q^6+1 }[/math]
1,1 [math]\displaystyle{ q^{84}-4 q^{82}+10 q^{80}-20 q^{78}+34 q^{76}-54 q^{74}+78 q^{72}-106 q^{70}+136 q^{68}-164 q^{66}+190 q^{64}-208 q^{62}+217 q^{60}-202 q^{58}+164 q^{56}-106 q^{54}+23 q^{52}+80 q^{50}-192 q^{48}+298 q^{46}-388 q^{44}+448 q^{42}-480 q^{40}+470 q^{38}-426 q^{36}+346 q^{34}-244 q^{32}+130 q^{30}-17 q^{28}-88 q^{26}+174 q^{24}-226 q^{22}+252 q^{20}-246 q^{18}+226 q^{16}-186 q^{14}+144 q^{12}-102 q^{10}+70 q^8-42 q^6+25 q^4-12 q^2+6-2 q^{-2} + q^{-4} }[/math]
2,0 [math]\displaystyle{ q^{76}-q^{74}-q^{72}+3 q^{62}-q^{60}-2 q^{58}+4 q^{56}+5 q^{54}-5 q^{52}-3 q^{50}+5 q^{48}-9 q^{44}-4 q^{42}+5 q^{40}-3 q^{38}-4 q^{36}+5 q^{34}+3 q^{32}-4 q^{30}+3 q^{28}+4 q^{26}-3 q^{24}-2 q^{22}+6 q^{20}+2 q^{18}-6 q^{16}+q^{14}+7 q^{12}-4 q^8+q^6+3 q^4-1+ q^{-4} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{68}-2 q^{66}+4 q^{62}-6 q^{60}-q^{58}+9 q^{56}-8 q^{54}-3 q^{52}+15 q^{50}-6 q^{48}-5 q^{46}+10 q^{44}-4 q^{42}-7 q^{40}+q^{38}+2 q^{36}-3 q^{34}-6 q^{32}+6 q^{30}+3 q^{28}-12 q^{26}+7 q^{24}+7 q^{22}-10 q^{20}+6 q^{18}+7 q^{16}-6 q^{14}+4 q^{12}+3 q^{10}-3 q^8+2 q^6+q^4-q^2+1 }[/math]
1,0,0 [math]\displaystyle{ q^{39}-q^{37}+q^{35}-2 q^{33}+q^{31}-2 q^{29}+q^{27}-2 q^{25}+2 q^{17}+3 q^{13}-q^{11}+2 q^9-q^7+q^5+q }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{86}-q^{84}-2 q^{82}+3 q^{80}+2 q^{78}-6 q^{76}-2 q^{74}+6 q^{72}-q^{70}-7 q^{68}+3 q^{66}+10 q^{64}-2 q^{62}-3 q^{60}+9 q^{58}+2 q^{56}-9 q^{54}+2 q^{52}+3 q^{50}-10 q^{48}-5 q^{46}+5 q^{44}-4 q^{42}-11 q^{40}+2 q^{38}+7 q^{36}-5 q^{34}-4 q^{32}+9 q^{30}+4 q^{28}-2 q^{26}+3 q^{24}+5 q^{22}+q^{20}+q^{18}+2 q^{16}+q^{14}+q^{10}+q^8+q^2 }[/math]
1,0,0,0 [math]\displaystyle{ q^{48}-q^{46}+q^{44}-q^{42}-q^{40}+q^{38}-2 q^{36}+q^{34}-2 q^{32}-q^{28}+q^{22}+2 q^{20}+3 q^{16}-q^{14}+2 q^{12}+q^6+q^2 }[/math]

B2 Invariants.

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

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{94}-2 q^{92}+2 q^{90}-3 q^{88}+5 q^{86}-7 q^{84}+6 q^{82}-8 q^{80}+11 q^{78}-11 q^{76}+9 q^{74}-9 q^{72}+12 q^{70}-5 q^{68}+3 q^{66}-q^{64}-2 q^{62}+8 q^{60}-12 q^{58}+11 q^{56}-18 q^{54}+19 q^{52}-20 q^{50}+19 q^{48}-22 q^{46}+17 q^{44}-15 q^{42}+11 q^{40}-11 q^{38}+4 q^{36}-q^{32}+6 q^{30}-6 q^{28}+12 q^{26}-8 q^{24}+11 q^{22}-9 q^{20}+11 q^{18}-7 q^{16}+7 q^{14}-5 q^{12}+5 q^{10}-2 q^8+2 q^6-q^4+q^2 }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+12 q^{148}-18 q^{146}+22 q^{144}-20 q^{142}+10 q^{140}+5 q^{138}-22 q^{136}+40 q^{134}-46 q^{132}+41 q^{130}-27 q^{128}+29 q^{124}-52 q^{122}+63 q^{120}-50 q^{118}+25 q^{116}+10 q^{114}-37 q^{112}+44 q^{110}-33 q^{108}+8 q^{106}+21 q^{104}-42 q^{102}+36 q^{100}-7 q^{98}-31 q^{96}+66 q^{94}-81 q^{92}+59 q^{90}-21 q^{88}-33 q^{86}+73 q^{84}-97 q^{82}+88 q^{80}-51 q^{78}+q^{76}+45 q^{74}-72 q^{72}+69 q^{70}-43 q^{68}+6 q^{66}+27 q^{64}-42 q^{62}+39 q^{60}-7 q^{58}-22 q^{56}+51 q^{54}-54 q^{52}+31 q^{50}+5 q^{48}-43 q^{46}+67 q^{44}-66 q^{42}+47 q^{40}-12 q^{38}-24 q^{36}+48 q^{34}-53 q^{32}+43 q^{30}-23 q^{28}+q^{26}+14 q^{24}-21 q^{22}+22 q^{20}-15 q^{18}+9 q^{16}-3 q^{12}+4 q^{10}-4 q^8+3 q^6-q^4+q^2 }[/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 14"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^3+8 t^2-12 t+13-12 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -2 z^6-4 z^4+2 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]= { 57, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -6 q^{-3} +9 q^{-4} -9 q^{-5} +9 q^{-6} -8 q^{-7} +5 q^{-8} -3 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+2 z^2 a^8-z^6 a^6-3 z^4 a^6-2 z^2 a^6-a^6-z^6 a^4-3 z^4 a^4-z^2 a^4+a^4+z^4 a^2+3 z^2 a^2+a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+4 z^6 a^{10}-4 z^4 a^{10}+4 z^7 a^9-4 z^5 a^9+2 z a^9+3 z^8 a^8-4 z^6 a^8+5 z^4 a^8-3 z^2 a^8+z^9 a^7+3 z^7 a^7-9 z^5 a^7+8 z^3 a^7-2 z a^7+5 z^8 a^6-14 z^6 a^6+16 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5+z^7 a^5-9 z^5 a^5+10 z^3 a^5-4 z a^5+2 z^8 a^4-5 z^6 a^4+2 z^4 a^4-z^2 a^4+a^4+2 z^7 a^3-7 z^5 a^3+6 z^3 a^3-z a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-a^2 }[/math]
Retrieved from "https://katlas.org/index.php?title=10_14&oldid=4384"