10 27

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-8 t^2+16 t-19+16 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 { 71, -2 }
Jones polynomial [math]\displaystyle{ -q^2+3 q-5+9 q^{-1} -11 q^{-2} +12 q^{-3} -11 q^{-4} +9 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^4 a^6-2 z^2 a^6-a^6+z^6 a^4+3 z^4 a^4+3 z^2 a^4+a^4+z^6 a^2+3 z^4 a^2+3 z^2 a^2+a^2-z^4-2 z^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-6 z^4 a^8+3 z^2 a^8+4 z^7 a^7-6 z^5 a^7+z^3 a^7+3 z^8 a^6-z^6 a^6-3 z^4 a^6-z^2 a^6+a^6+z^9 a^5+6 z^7 a^5-12 z^5 a^5+7 z^3 a^5-2 z a^5+6 z^8 a^4-9 z^6 a^4+7 z^4 a^4-4 z^2 a^4+a^4+z^9 a^3+6 z^7 a^3-14 z^5 a^3+11 z^3 a^3-2 z a^3+3 z^8 a^2-2 z^6 a^2-3 z^4 a^2+4 z^2 a^2-a^2+4 z^7 a-8 z^5 a+5 z^3 a-z a+3 z^6-7 z^4+4 z^2+z^5 a^{-1} -2 z^3 a^{-1} }[/math]
The A2 invariant [math]\displaystyle{ -q^{24}+q^{22}-q^{20}-q^{18}+2 q^{16}-2 q^{14}+2 q^{12}+2 q^6-2 q^4+3 q^2+ q^{-4} - q^{-6} }[/math]
The G2 invariant [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+8 q^{120}-6 q^{118}-2 q^{116}+17 q^{114}-31 q^{112}+44 q^{110}-45 q^{108}+26 q^{106}+5 q^{104}-49 q^{102}+90 q^{100}-111 q^{98}+98 q^{96}-52 q^{94}-22 q^{92}+93 q^{90}-138 q^{88}+142 q^{86}-96 q^{84}+21 q^{82}+56 q^{80}-107 q^{78}+104 q^{76}-55 q^{74}-16 q^{72}+76 q^{70}-96 q^{68}+62 q^{66}+14 q^{64}-99 q^{62}+161 q^{60}-167 q^{58}+109 q^{56}-6 q^{54}-110 q^{52}+193 q^{50}-214 q^{48}+172 q^{46}-73 q^{44}-38 q^{42}+126 q^{40}-160 q^{38}+133 q^{36}-60 q^{34}-24 q^{32}+80 q^{30}-87 q^{28}+47 q^{26}+25 q^{24}-86 q^{22}+116 q^{20}-96 q^{18}+31 q^{16}+47 q^{14}-115 q^{12}+145 q^{10}-121 q^8+70 q^6-q^4-59 q^2+94-96 q^{-2} +73 q^{-4} -36 q^{-6} -2 q^{-8} +27 q^{-10} -38 q^{-12} +36 q^{-14} -25 q^{-16} +14 q^{-18} - q^{-20} -6 q^{-22} +6 q^{-24} -7 q^{-26} +4 q^{-28} -2 q^{-30} + q^{-32} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_27.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{17}+2 q^{15}-3 q^{13}+3 q^{11}-2 q^9+q^7+q^5-2 q^3+4 q-2 q^{-1} +2 q^{-3} - q^{-5} }[/math]
2 [math]\displaystyle{ q^{48}-2 q^{46}-q^{44}+7 q^{42}-6 q^{40}-8 q^{38}+18 q^{36}-5 q^{34}-19 q^{32}+23 q^{30}+2 q^{28}-24 q^{26}+15 q^{24}+9 q^{22}-16 q^{20}-q^{18}+11 q^{16}+2 q^{14}-17 q^{12}+7 q^{10}+19 q^8-23 q^6-q^4+25 q^2-15-6 q^{-2} +16 q^{-4} -6 q^{-6} -6 q^{-8} +6 q^{-10} - q^{-12} -2 q^{-14} + q^{-16} }[/math]
3 [math]\displaystyle{ -q^{93}+2 q^{91}+q^{89}-3 q^{87}-4 q^{85}+6 q^{83}+11 q^{81}-11 q^{79}-22 q^{77}+12 q^{75}+39 q^{73}-7 q^{71}-62 q^{69}-4 q^{67}+84 q^{65}+26 q^{63}-98 q^{61}-58 q^{59}+104 q^{57}+87 q^{55}-95 q^{53}-112 q^{51}+74 q^{49}+126 q^{47}-45 q^{45}-126 q^{43}+14 q^{41}+112 q^{39}+18 q^{37}-88 q^{35}-48 q^{33}+56 q^{31}+77 q^{29}-22 q^{27}-98 q^{25}-17 q^{23}+111 q^{21}+52 q^{19}-113 q^{17}-87 q^{15}+106 q^{13}+107 q^{11}-80 q^9-117 q^7+51 q^5+117 q^3-23 q-95 q^{-1} - q^{-3} +76 q^{-5} +13 q^{-7} -49 q^{-9} -19 q^{-11} +30 q^{-13} +15 q^{-15} -16 q^{-17} -12 q^{-19} +8 q^{-21} +6 q^{-23} -3 q^{-25} -3 q^{-27} + q^{-29} +2 q^{-31} - q^{-33} }[/math]
4 [math]\displaystyle{ q^{152}-2 q^{150}-q^{148}+3 q^{146}+4 q^{142}-9 q^{140}-6 q^{138}+12 q^{136}+6 q^{134}+17 q^{132}-33 q^{130}-36 q^{128}+25 q^{126}+41 q^{124}+72 q^{122}-71 q^{120}-130 q^{118}-13 q^{116}+103 q^{114}+240 q^{112}-42 q^{110}-288 q^{108}-204 q^{106}+82 q^{104}+505 q^{102}+186 q^{100}-348 q^{98}-529 q^{96}-174 q^{94}+662 q^{92}+566 q^{90}-145 q^{88}-743 q^{86}-572 q^{84}+520 q^{82}+819 q^{80}+232 q^{78}-654 q^{76}-828 q^{74}+169 q^{72}+759 q^{70}+510 q^{68}-342 q^{66}-780 q^{64}-172 q^{62}+465 q^{60}+584 q^{58}+13 q^{56}-528 q^{54}-418 q^{52}+103 q^{50}+531 q^{48}+344 q^{46}-202 q^{44}-609 q^{42}-272 q^{40}+412 q^{38}+637 q^{36}+167 q^{34}-698 q^{32}-627 q^{30}+170 q^{28}+796 q^{26}+559 q^{24}-566 q^{22}-826 q^{20}-182 q^{18}+667 q^{16}+798 q^{14}-209 q^{12}-711 q^{10}-459 q^8+296 q^6+723 q^4+121 q^2-360-459 q^{-2} -35 q^{-4} +416 q^{-6} +208 q^{-8} -54 q^{-10} -266 q^{-12} -138 q^{-14} +146 q^{-16} +118 q^{-18} +53 q^{-20} -93 q^{-22} -90 q^{-24} +32 q^{-26} +33 q^{-28} +40 q^{-30} -21 q^{-32} -33 q^{-34} +8 q^{-36} +3 q^{-38} +14 q^{-40} -3 q^{-42} -9 q^{-44} +3 q^{-46} +3 q^{-50} - q^{-52} -2 q^{-54} + q^{-56} }[/math]
5 [math]\displaystyle{ -q^{225}+2 q^{223}+q^{221}-3 q^{219}-q^{213}+4 q^{211}+5 q^{209}-8 q^{207}-9 q^{205}+2 q^{203}+8 q^{201}+18 q^{199}+11 q^{197}-22 q^{195}-52 q^{193}-27 q^{191}+46 q^{189}+100 q^{187}+81 q^{185}-47 q^{183}-197 q^{181}-209 q^{179}+14 q^{177}+323 q^{175}+420 q^{173}+134 q^{171}-413 q^{169}-756 q^{167}-468 q^{165}+403 q^{163}+1157 q^{161}+1012 q^{159}-140 q^{157}-1494 q^{155}-1785 q^{153}-463 q^{151}+1622 q^{149}+2652 q^{147}+1415 q^{145}-1357 q^{143}-3394 q^{141}-2665 q^{139}+621 q^{137}+3830 q^{135}+3964 q^{133}+516 q^{131}-3746 q^{129}-5056 q^{127}-1930 q^{125}+3156 q^{123}+5727 q^{121}+3304 q^{119}-2140 q^{117}-5816 q^{115}-4414 q^{113}+895 q^{111}+5379 q^{109}+5066 q^{107}+328 q^{105}-4528 q^{103}-5210 q^{101}-1351 q^{99}+3432 q^{97}+4936 q^{95}+2093 q^{93}-2306 q^{91}-4367 q^{89}-2550 q^{87}+1242 q^{85}+3658 q^{83}+2838 q^{81}-284 q^{79}-2964 q^{77}-3032 q^{75}-594 q^{73}+2291 q^{71}+3251 q^{69}+1472 q^{67}-1657 q^{65}-3511 q^{63}-2387 q^{61}+967 q^{59}+3766 q^{57}+3378 q^{55}-154 q^{53}-3890 q^{51}-4390 q^{49}-844 q^{47}+3782 q^{45}+5257 q^{43}+1987 q^{41}-3270 q^{39}-5835 q^{37}-3212 q^{35}+2406 q^{33}+5936 q^{31}+4234 q^{29}-1168 q^{27}-5475 q^{25}-4935 q^{23}-182 q^{21}+4512 q^{19}+5098 q^{17}+1384 q^{15}-3165 q^{13}-4701 q^{11}-2265 q^9+1751 q^7+3884 q^5+2613 q^3-514 q-2766 q^{-1} -2529 q^{-3} -364 q^{-5} +1708 q^{-7} +2072 q^{-9} +809 q^{-11} -805 q^{-13} -1487 q^{-15} -902 q^{-17} +225 q^{-19} +909 q^{-21} +769 q^{-23} +92 q^{-25} -486 q^{-27} -543 q^{-29} -187 q^{-31} +195 q^{-33} +335 q^{-35} +184 q^{-37} -66 q^{-39} -181 q^{-41} -121 q^{-43} +3 q^{-45} +81 q^{-47} +78 q^{-49} +11 q^{-51} -40 q^{-53} -36 q^{-55} -5 q^{-57} +13 q^{-59} +14 q^{-61} +8 q^{-63} -7 q^{-65} -10 q^{-67} +2 q^{-69} +4 q^{-71} -3 q^{-79} + q^{-81} +2 q^{-83} - q^{-85} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{24}+q^{22}-q^{20}-q^{18}+2 q^{16}-2 q^{14}+2 q^{12}+2 q^6-2 q^4+3 q^2+ q^{-4} - q^{-6} }[/math]
1,1 [math]\displaystyle{ q^{68}-4 q^{66}+12 q^{64}-28 q^{62}+56 q^{60}-102 q^{58}+168 q^{56}-252 q^{54}+351 q^{52}-462 q^{50}+560 q^{48}-622 q^{46}+639 q^{44}-592 q^{42}+466 q^{40}-252 q^{38}-18 q^{36}+318 q^{34}-636 q^{32}+920 q^{30}-1141 q^{28}+1266 q^{26}-1286 q^{24}+1198 q^{22}-1011 q^{20}+748 q^{18}-446 q^{16}+132 q^{14}+156 q^{12}-374 q^{10}+534 q^8-612 q^6+626 q^4-570 q^2+488-394 q^{-2} +296 q^{-4} -204 q^{-6} +132 q^{-8} -84 q^{-10} +46 q^{-12} -22 q^{-14} +10 q^{-16} -4 q^{-18} + q^{-20} }[/math]
2,0 [math]\displaystyle{ q^{62}-q^{60}-q^{58}+2 q^{56}+q^{54}-3 q^{52}-3 q^{50}+5 q^{48}+3 q^{46}-8 q^{44}-q^{42}+10 q^{40}-10 q^{36}+2 q^{34}+10 q^{32}-5 q^{30}-9 q^{28}+6 q^{26}+2 q^{24}-10 q^{22}+2 q^{20}+7 q^{18}-6 q^{16}-2 q^{14}+11 q^{12}+3 q^{10}-9 q^8+3 q^6+13 q^4-4 q^2-7+6 q^{-2} +5 q^{-4} -6 q^{-6} -4 q^{-8} +3 q^{-10} +2 q^{-12} -2 q^{-14} - q^{-16} + q^{-18} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{54}-2 q^{52}+q^{50}+4 q^{48}-8 q^{46}+3 q^{44}+10 q^{42}-16 q^{40}+4 q^{38}+15 q^{36}-21 q^{34}+16 q^{30}-14 q^{28}-4 q^{26}+11 q^{24}-5 q^{20}-q^{18}+13 q^{16}-3 q^{14}-13 q^{12}+19 q^{10}-19 q^6+17 q^4+3 q^2-13+10 q^{-2} +3 q^{-4} -7 q^{-6} +3 q^{-8} -2 q^{-12} + q^{-14} }[/math]
1,0,0 [math]\displaystyle{ -q^{31}+q^{29}-2 q^{27}+q^{25}-2 q^{23}+2 q^{21}-2 q^{19}+2 q^{17}+q^{13}+q^{11}+2 q^7-2 q^5+3 q^3-q+2 q^{-1} - q^{-3} + q^{-5} - q^{-7} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{68}-q^{66}-q^{64}+4 q^{62}-6 q^{58}+4 q^{56}+8 q^{54}-7 q^{52}-8 q^{50}+9 q^{48}+5 q^{46}-16 q^{44}-6 q^{42}+13 q^{40}-5 q^{38}-15 q^{36}+10 q^{34}+9 q^{32}-11 q^{30}+3 q^{28}+15 q^{26}-3 q^{24}-9 q^{22}+12 q^{20}+8 q^{18}-14 q^{16}-2 q^{14}+15 q^{12}-2 q^{10}-12 q^8+7 q^6+8 q^4-3 q^2-3+5 q^{-2} +2 q^{-4} -3 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{38}+q^{36}-2 q^{34}-2 q^{28}+2 q^{26}-2 q^{24}+2 q^{22}+q^{18}+q^{16}+q^{14}+q^{12}+2 q^8-2 q^6+3 q^4-q^2+1+ q^{-2} - q^{-4} + q^{-6} - q^{-8} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{54}+2 q^{52}-5 q^{50}+8 q^{48}-12 q^{46}+17 q^{44}-20 q^{42}+22 q^{40}-22 q^{38}+19 q^{36}-13 q^{34}+4 q^{32}+6 q^{30}-18 q^{28}+28 q^{26}-37 q^{24}+42 q^{22}-43 q^{20}+41 q^{18}-33 q^{16}+25 q^{14}-13 q^{12}+3 q^{10}+8 q^8-15 q^6+21 q^4-21 q^2+21-18 q^{-2} +15 q^{-4} -11 q^{-6} +7 q^{-8} -4 q^{-10} +2 q^{-12} - q^{-14} }[/math]
1,0 [math]\displaystyle{ q^{88}-2 q^{84}-2 q^{82}+3 q^{80}+6 q^{78}-q^{76}-10 q^{74}-6 q^{72}+11 q^{70}+15 q^{68}-5 q^{66}-21 q^{64}-7 q^{62}+20 q^{60}+18 q^{58}-12 q^{56}-24 q^{54}-q^{52}+21 q^{50}+9 q^{48}-16 q^{46}-13 q^{44}+10 q^{42}+14 q^{40}-6 q^{38}-14 q^{36}+3 q^{34}+15 q^{32}+q^{30}-15 q^{28}-3 q^{26}+16 q^{24}+9 q^{22}-15 q^{20}-14 q^{18}+12 q^{16}+22 q^{14}-3 q^{12}-24 q^{10}-9 q^8+20 q^6+18 q^4-7 q^2-19-3 q^{-2} +14 q^{-4} +10 q^{-6} -5 q^{-8} -9 q^{-10} - q^{-12} +5 q^{-14} +2 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{74}-2 q^{72}+3 q^{70}-4 q^{68}+7 q^{66}-10 q^{64}+11 q^{62}-13 q^{60}+17 q^{58}-18 q^{56}+16 q^{54}-16 q^{52}+16 q^{50}-13 q^{48}+3 q^{46}-3 q^{44}-3 q^{42}+10 q^{40}-20 q^{38}+21 q^{36}-25 q^{34}+34 q^{32}-32 q^{30}+32 q^{28}-31 q^{26}+33 q^{24}-22 q^{22}+19 q^{20}-15 q^{18}+9 q^{16}+2 q^{14}-4 q^{12}+7 q^{10}-14 q^8+18 q^6-15 q^4+16 q^2-16+16 q^{-2} -11 q^{-4} +10 q^{-6} -9 q^{-8} +6 q^{-10} -4 q^{-12} +2 q^{-14} -2 q^{-16} + q^{-18} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+8 q^{120}-6 q^{118}-2 q^{116}+17 q^{114}-31 q^{112}+44 q^{110}-45 q^{108}+26 q^{106}+5 q^{104}-49 q^{102}+90 q^{100}-111 q^{98}+98 q^{96}-52 q^{94}-22 q^{92}+93 q^{90}-138 q^{88}+142 q^{86}-96 q^{84}+21 q^{82}+56 q^{80}-107 q^{78}+104 q^{76}-55 q^{74}-16 q^{72}+76 q^{70}-96 q^{68}+62 q^{66}+14 q^{64}-99 q^{62}+161 q^{60}-167 q^{58}+109 q^{56}-6 q^{54}-110 q^{52}+193 q^{50}-214 q^{48}+172 q^{46}-73 q^{44}-38 q^{42}+126 q^{40}-160 q^{38}+133 q^{36}-60 q^{34}-24 q^{32}+80 q^{30}-87 q^{28}+47 q^{26}+25 q^{24}-86 q^{22}+116 q^{20}-96 q^{18}+31 q^{16}+47 q^{14}-115 q^{12}+145 q^{10}-121 q^8+70 q^6-q^4-59 q^2+94-96 q^{-2} +73 q^{-4} -36 q^{-6} -2 q^{-8} +27 q^{-10} -38 q^{-12} +36 q^{-14} -25 q^{-16} +14 q^{-18} - q^{-20} -6 q^{-22} +6 q^{-24} -7 q^{-26} +4 q^{-28} -2 q^{-30} + q^{-32} }[/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 27"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-8 t^2+16 t-19+16 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]= { 71, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^2+3 q-5+9 q^{-1} -11 q^{-2} +12 q^{-3} -11 q^{-4} +9 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^4 a^6-2 z^2 a^6-a^6+z^6 a^4+3 z^4 a^4+3 z^2 a^4+a^4+z^6 a^2+3 z^4 a^2+3 z^2 a^2+a^2-z^4-2 z^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-6 z^4 a^8+3 z^2 a^8+4 z^7 a^7-6 z^5 a^7+z^3 a^7+3 z^8 a^6-z^6 a^6-3 z^4 a^6-z^2 a^6+a^6+z^9 a^5+6 z^7 a^5-12 z^5 a^5+7 z^3 a^5-2 z a^5+6 z^8 a^4-9 z^6 a^4+7 z^4 a^4-4 z^2 a^4+a^4+z^9 a^3+6 z^7 a^3-14 z^5 a^3+11 z^3 a^3-2 z a^3+3 z^8 a^2-2 z^6 a^2-3 z^4 a^2+4 z^2 a^2-a^2+4 z^7 a-8 z^5 a+5 z^3 a-z a+3 z^6-7 z^4+4 z^2+z^5 a^{-1} -2 z^3 a^{-1} }[/math]
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