10 24

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -4 t^2+14 t-19+14 t^{-1} -4 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ -4 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 55, -2 }
Jones polynomial [math]\displaystyle{ q-2+5 q^{-1} -7 q^{-2} +9 q^{-3} -9 q^{-4} +8 q^{-5} -7 q^{-6} +4 q^{-7} -2 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-z^2 a^6-a^6-2 z^4 a^4-3 z^2 a^4-a^4-z^4 a^2+a^2+z^2+1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-7 z^5 a^9+7 z^3 a^9-2 z a^9+2 z^8 a^8-5 z^6 a^8+3 z^4 a^8-2 z^2 a^8+a^8+z^9 a^7-2 z^5 a^7-2 z^3 a^7+4 z^8 a^6-8 z^6 a^6+6 z^4 a^6-5 z^2 a^6+a^6+z^9 a^5+z^7 a^5+z^5 a^5-7 z^3 a^5+4 z a^5+2 z^8 a^4+z^6 a^4-5 z^4 a^4+5 z^2 a^4-a^4+3 z^7 a^3-2 z^5 a^3+2 z a^3+3 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+1 }[/math]
The A2 invariant [math]\displaystyle{ q^{28}+2 q^{22}-2 q^{20}-q^{18}-2 q^{14}+q^{12}-q^{10}+q^8+q^6-q^4+3 q^2+ q^{-4} }[/math]
The G2 invariant [math]\displaystyle{ q^{142}-q^{140}+3 q^{138}-5 q^{136}+4 q^{134}-4 q^{132}-q^{130}+9 q^{128}-18 q^{126}+25 q^{124}-24 q^{122}+13 q^{120}+7 q^{118}-31 q^{116}+51 q^{114}-56 q^{112}+44 q^{110}-14 q^{108}-26 q^{106}+59 q^{104}-68 q^{102}+59 q^{100}-24 q^{98}-12 q^{96}+40 q^{94}-50 q^{92}+35 q^{90}-4 q^{88}-27 q^{86}+46 q^{84}-39 q^{82}+11 q^{80}+27 q^{78}-62 q^{76}+75 q^{74}-67 q^{72}+29 q^{70}+17 q^{68}-67 q^{66}+94 q^{64}-90 q^{62}+58 q^{60}-10 q^{58}-39 q^{56}+64 q^{54}-67 q^{52}+42 q^{50}-7 q^{48}-24 q^{46}+38 q^{44}-27 q^{42}+q^{40}+29 q^{38}-46 q^{36}+44 q^{34}-24 q^{32}-7 q^{30}+35 q^{28}-53 q^{26}+59 q^{24}-41 q^{22}+19 q^{20}+7 q^{18}-28 q^{16}+38 q^{14}-37 q^{12}+30 q^{10}-14 q^8+q^6+10 q^4-15 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_24.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{19}-q^{17}+2 q^{15}-3 q^{13}+q^{11}-q^9+2 q^5-2 q^3+3 q- q^{-1} + q^{-3} }[/math]
2 [math]\displaystyle{ q^{54}-q^{52}-q^{50}+4 q^{48}-2 q^{46}-7 q^{44}+8 q^{42}+2 q^{40}-13 q^{38}+9 q^{36}+8 q^{34}-14 q^{32}+3 q^{30}+11 q^{28}-8 q^{26}-3 q^{24}+7 q^{22}+q^{20}-8 q^{18}-2 q^{16}+12 q^{14}-8 q^{12}-8 q^{10}+16 q^8-4 q^6-9 q^4+10 q^2-1-4 q^{-2} +4 q^{-4} - q^{-8} + q^{-10} }[/math]
3 [math]\displaystyle{ q^{105}-q^{103}-q^{101}+q^{99}+3 q^{97}-2 q^{95}-7 q^{93}+13 q^{89}+5 q^{87}-18 q^{85}-16 q^{83}+20 q^{81}+30 q^{79}-15 q^{77}-41 q^{75}+3 q^{73}+52 q^{71}+9 q^{69}-51 q^{67}-27 q^{65}+48 q^{63}+40 q^{61}-39 q^{59}-50 q^{57}+27 q^{55}+52 q^{53}-15 q^{51}-52 q^{49}+2 q^{47}+49 q^{45}+8 q^{43}-36 q^{41}-23 q^{39}+26 q^{37}+34 q^{35}-7 q^{33}-46 q^{31}-10 q^{29}+50 q^{27}+29 q^{25}-48 q^{23}-42 q^{21}+40 q^{19}+47 q^{17}-30 q^{15}-43 q^{13}+17 q^{11}+36 q^9-9 q^7-26 q^5+7 q^3+15 q-2 q^{-1} -9 q^{-3} +3 q^{-5} +5 q^{-7} -2 q^{-9} -3 q^{-11} +2 q^{-13} + q^{-15} - q^{-19} + q^{-21} }[/math]
4 [math]\displaystyle{ q^{172}-q^{170}-q^{168}+q^{166}+3 q^{162}-4 q^{160}-5 q^{158}+3 q^{156}+3 q^{154}+15 q^{152}-5 q^{150}-22 q^{148}-12 q^{146}+50 q^{142}+26 q^{140}-27 q^{138}-59 q^{136}-59 q^{134}+65 q^{132}+99 q^{130}+45 q^{128}-70 q^{126}-170 q^{124}-18 q^{122}+119 q^{120}+174 q^{118}+37 q^{116}-217 q^{114}-161 q^{112}+10 q^{110}+231 q^{108}+205 q^{106}-125 q^{104}-241 q^{102}-157 q^{100}+167 q^{98}+303 q^{96}+24 q^{94}-209 q^{92}-259 q^{90}+52 q^{88}+291 q^{86}+131 q^{84}-132 q^{82}-272 q^{80}-30 q^{78}+224 q^{76}+171 q^{74}-59 q^{72}-230 q^{70}-93 q^{68}+130 q^{66}+193 q^{64}+33 q^{62}-155 q^{60}-169 q^{58}-13 q^{56}+191 q^{54}+162 q^{52}-14 q^{50}-223 q^{48}-200 q^{46}+112 q^{44}+252 q^{42}+169 q^{40}-171 q^{38}-323 q^{36}-32 q^{34}+212 q^{32}+276 q^{30}-40 q^{28}-283 q^{26}-115 q^{24}+80 q^{22}+229 q^{20}+49 q^{18}-151 q^{16}-92 q^{14}-11 q^{12}+116 q^{10}+47 q^8-53 q^6-28 q^4-25 q^2+38+17 q^{-2} -16 q^{-4} +3 q^{-6} -12 q^{-8} +11 q^{-10} + q^{-12} -8 q^{-14} +7 q^{-16} -3 q^{-18} +4 q^{-20} - q^{-22} -4 q^{-24} +3 q^{-26} - q^{-28} + q^{-30} - q^{-34} + q^{-36} }[/math]
5 [math]\displaystyle{ q^{255}-q^{253}-q^{251}+q^{249}+q^{243}-2 q^{241}-4 q^{239}+3 q^{237}+7 q^{235}+3 q^{233}+q^{231}-9 q^{229}-19 q^{227}-9 q^{225}+16 q^{223}+34 q^{221}+32 q^{219}-54 q^{215}-78 q^{213}-41 q^{211}+51 q^{209}+131 q^{207}+124 q^{205}+4 q^{203}-160 q^{201}-238 q^{199}-131 q^{197}+124 q^{195}+337 q^{193}+314 q^{191}+23 q^{189}-356 q^{187}-519 q^{185}-278 q^{183}+248 q^{181}+656 q^{179}+584 q^{177}+24 q^{175}-642 q^{173}-883 q^{171}-415 q^{169}+466 q^{167}+1042 q^{165}+835 q^{163}-83 q^{161}-1041 q^{159}-1211 q^{157}-365 q^{155}+842 q^{153}+1419 q^{151}+844 q^{149}-492 q^{147}-1474 q^{145}-1220 q^{143}+85 q^{141}+1347 q^{139}+1461 q^{137}+307 q^{135}-1120 q^{133}-1548 q^{131}-610 q^{129}+850 q^{127}+1511 q^{125}+789 q^{123}-596 q^{121}-1386 q^{119}-877 q^{117}+401 q^{115}+1226 q^{113}+887 q^{111}-251 q^{109}-1085 q^{107}-857 q^{105}+131 q^{103}+941 q^{101}+864 q^{99}+9 q^{97}-827 q^{95}-882 q^{93}-188 q^{91}+644 q^{89}+948 q^{87}+463 q^{85}-425 q^{83}-987 q^{81}-780 q^{79}+76 q^{77}+963 q^{75}+1133 q^{73}+351 q^{71}-815 q^{69}-1407 q^{67}-842 q^{65}+523 q^{63}+1541 q^{61}+1291 q^{59}-113 q^{57}-1493 q^{55}-1608 q^{53}-326 q^{51}+1241 q^{49}+1722 q^{47}+714 q^{45}-864 q^{43}-1630 q^{41}-945 q^{39}+465 q^{37}+1339 q^{35}+1011 q^{33}-112 q^{31}-985 q^{29}-917 q^{27}-106 q^{25}+632 q^{23}+713 q^{21}+214 q^{19}-340 q^{17}-498 q^{15}-225 q^{13}+163 q^{11}+300 q^9+169 q^7-41 q^5-162 q^3-117 q+2 q^{-1} +75 q^{-3} +63 q^{-5} +13 q^{-7} -27 q^{-9} -31 q^{-11} -10 q^{-13} +3 q^{-15} +12 q^{-17} +10 q^{-19} - q^{-21} - q^{-23} -5 q^{-27} -4 q^{-29} +5 q^{-31} +4 q^{-37} -2 q^{-39} -3 q^{-41} +2 q^{-43} - q^{-47} + q^{-49} - q^{-53} + q^{-55} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{28}+2 q^{22}-2 q^{20}-q^{18}-2 q^{14}+q^{12}-q^{10}+q^8+q^6-q^4+3 q^2+ q^{-4} }[/math]
1,1 [math]\displaystyle{ q^{76}-2 q^{74}+6 q^{72}-14 q^{70}+27 q^{68}-48 q^{66}+80 q^{64}-118 q^{62}+158 q^{60}-200 q^{58}+234 q^{56}-248 q^{54}+229 q^{52}-192 q^{50}+124 q^{48}-28 q^{46}-83 q^{44}+200 q^{42}-298 q^{40}+388 q^{38}-443 q^{36}+464 q^{34}-446 q^{32}+390 q^{30}-306 q^{28}+204 q^{26}-98 q^{24}-4 q^{22}+92 q^{20}-158 q^{18}+188 q^{16}-206 q^{14}+204 q^{12}-186 q^{10}+156 q^8-126 q^6+101 q^4-70 q^2+50-30 q^{-2} +21 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0 [math]\displaystyle{ q^{72}+2 q^{64}-5 q^{60}-2 q^{58}+4 q^{56}+2 q^{54}-7 q^{52}-4 q^{50}+7 q^{48}+6 q^{46}-7 q^{44}-3 q^{42}+8 q^{40}+4 q^{38}-4 q^{36}-q^{34}+6 q^{32}-3 q^{28}+2 q^{26}-3 q^{24}-6 q^{22}+4 q^{20}+3 q^{18}-8 q^{16}-3 q^{14}+9 q^{12}+3 q^{10}-10 q^8-3 q^6+10 q^4+2 q^2-4+ q^{-2} +4 q^{-4} + q^{-6} - q^{-8} + q^{-12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{60}-q^{58}+q^{56}+q^{54}-5 q^{52}+2 q^{50}+2 q^{48}-8 q^{46}+7 q^{44}+5 q^{42}-10 q^{40}+9 q^{38}+6 q^{36}-10 q^{34}+3 q^{32}+6 q^{30}-5 q^{28}-3 q^{26}+q^{24}+2 q^{22}-8 q^{20}-5 q^{18}+10 q^{16}-7 q^{14}-5 q^{12}+14 q^{10}-3 q^8-6 q^6+10 q^4-3+4 q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
1,0,0 [math]\displaystyle{ q^{37}+q^{33}+2 q^{29}-2 q^{27}-2 q^{23}-2 q^{19}-q^{13}+q^{11}+2 q^7-q^5+3 q^3+ q^{-1} + q^{-5} }[/math]

B2 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{142}-q^{140}+3 q^{138}-5 q^{136}+4 q^{134}-4 q^{132}-q^{130}+9 q^{128}-18 q^{126}+25 q^{124}-24 q^{122}+13 q^{120}+7 q^{118}-31 q^{116}+51 q^{114}-56 q^{112}+44 q^{110}-14 q^{108}-26 q^{106}+59 q^{104}-68 q^{102}+59 q^{100}-24 q^{98}-12 q^{96}+40 q^{94}-50 q^{92}+35 q^{90}-4 q^{88}-27 q^{86}+46 q^{84}-39 q^{82}+11 q^{80}+27 q^{78}-62 q^{76}+75 q^{74}-67 q^{72}+29 q^{70}+17 q^{68}-67 q^{66}+94 q^{64}-90 q^{62}+58 q^{60}-10 q^{58}-39 q^{56}+64 q^{54}-67 q^{52}+42 q^{50}-7 q^{48}-24 q^{46}+38 q^{44}-27 q^{42}+q^{40}+29 q^{38}-46 q^{36}+44 q^{34}-24 q^{32}-7 q^{30}+35 q^{28}-53 q^{26}+59 q^{24}-41 q^{22}+19 q^{20}+7 q^{18}-28 q^{16}+38 q^{14}-37 q^{12}+30 q^{10}-14 q^8+q^6+10 q^4-15 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 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 24"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -4 t^2+14 t-19+14 t^{-1} -4 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -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]= { 55, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q-2+5 q^{-1} -7 q^{-2} +9 q^{-3} -9 q^{-4} +8 q^{-5} -7 q^{-6} +4 q^{-7} -2 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+a^8-z^4 a^6-z^2 a^6-a^6-2 z^4 a^4-3 z^2 a^4-a^4-z^4 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}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-7 z^5 a^9+7 z^3 a^9-2 z a^9+2 z^8 a^8-5 z^6 a^8+3 z^4 a^8-2 z^2 a^8+a^8+z^9 a^7-2 z^5 a^7-2 z^3 a^7+4 z^8 a^6-8 z^6 a^6+6 z^4 a^6-5 z^2 a^6+a^6+z^9 a^5+z^7 a^5+z^5 a^5-7 z^3 a^5+4 z a^5+2 z^8 a^4+z^6 a^4-5 z^4 a^4+5 z^2 a^4-a^4+3 z^7 a^3-2 z^5 a^3+2 z a^3+3 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+1 }[/math]
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