9 40

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Polynomial invariants

Alexander polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3} }
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 75, -2 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^2 a^6-2 z^4 a^4-2 z^2 a^4+a^4+z^6 a^2+2 z^4 a^2-2 a^2-z^4+2}
Kauffman polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^8+4 z^5 a^7-2 z^3 a^7+8 z^6 a^6-9 z^4 a^6+4 z^2 a^6+9 z^7 a^5-12 z^5 a^5+6 z^3 a^5-z a^5+4 z^8 a^4+7 z^6 a^4-20 z^4 a^4+7 z^2 a^4+a^4+17 z^7 a^3-32 z^5 a^3+14 z^3 a^3-z a^3+4 z^8 a^2+4 z^6 a^2-17 z^4 a^2+3 z^2 a^2+2 a^2+8 z^7 a-15 z^5 a+6 z^3 a+5 z^6-7 z^4+2+z^5 a^{-1} }
The A2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-q^{20}-2 q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^{10}-3 q^8+q^6-4 q^4+3 q^2+1+3 q^{-4} - q^{-6} }
The G2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+10 q^{106}-8 q^{104}+q^{102}+17 q^{100}-35 q^{98}+57 q^{96}-69 q^{94}+58 q^{92}-26 q^{90}-41 q^{88}+121 q^{86}-182 q^{84}+197 q^{82}-139 q^{80}+14 q^{78}+135 q^{76}-248 q^{74}+274 q^{72}-196 q^{70}+38 q^{68}+122 q^{66}-223 q^{64}+212 q^{62}-79 q^{60}-87 q^{58}+218 q^{56}-237 q^{54}+135 q^{52}+47 q^{50}-232 q^{48}+337 q^{46}-328 q^{44}+209 q^{42}-7 q^{40}-197 q^{38}+334 q^{36}-361 q^{34}+269 q^{32}-104 q^{30}-93 q^{28}+225 q^{26}-263 q^{24}+194 q^{22}-39 q^{20}-123 q^{18}+217 q^{16}-194 q^{14}+58 q^{12}+116 q^{10}-252 q^8+282 q^6-192 q^4+31 q^2+141-245 q^{-2} +261 q^{-4} -179 q^{-6} +57 q^{-8} +53 q^{-10} -121 q^{-12} +126 q^{-14} -87 q^{-16} +44 q^{-18} -2 q^{-20} -19 q^{-22} +24 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }
Further Quantum Invariants
Further quantum knot invariants for 9_40.

A1 Invariants.

Weight Invariant
1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{15}-3 q^{13}+4 q^{11}-3 q^9+2 q^7-2 q^3+3 q-3 q^{-1} +4 q^{-3} - q^{-5} }
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{42}-3 q^{40}+q^{38}+9 q^{36}-16 q^{34}-3 q^{32}+33 q^{30}-22 q^{28}-22 q^{26}+40 q^{24}-8 q^{22}-30 q^{20}+20 q^{18}+12 q^{16}-17 q^{14}-8 q^{12}+23 q^{10}+2 q^8-31 q^6+22 q^4+22 q^2-39+6 q^{-2} +30 q^{-4} -23 q^{-6} -9 q^{-8} +17 q^{-10} - q^{-12} -4 q^{-14} + q^{-16} }
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{81}-3 q^{79}+q^{77}+6 q^{75}-4 q^{73}-15 q^{71}+6 q^{69}+47 q^{67}-7 q^{65}-96 q^{63}-23 q^{61}+150 q^{59}+98 q^{57}-188 q^{55}-190 q^{53}+175 q^{51}+278 q^{49}-113 q^{47}-333 q^{45}+25 q^{43}+330 q^{41}+63 q^{39}-274 q^{37}-133 q^{35}+194 q^{33}+176 q^{31}-106 q^{29}-195 q^{27}+19 q^{25}+205 q^{23}+57 q^{21}-209 q^{19}-133 q^{17}+203 q^{15}+208 q^{13}-176 q^{11}-271 q^9+115 q^7+321 q^5-34 q^3-324 q-59 q^{-1} +279 q^{-3} +139 q^{-5} -192 q^{-7} -173 q^{-9} +96 q^{-11} +153 q^{-13} -16 q^{-15} -102 q^{-17} -28 q^{-19} +52 q^{-21} +24 q^{-23} -11 q^{-25} -13 q^{-27} + q^{-29} +4 q^{-31} - q^{-33} }
4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{132}-3 q^{130}+q^{128}+6 q^{126}-7 q^{124}-3 q^{122}-6 q^{120}+26 q^{118}+38 q^{116}-57 q^{114}-83 q^{112}-54 q^{110}+169 q^{108}+307 q^{106}-58 q^{104}-449 q^{102}-561 q^{100}+209 q^{98}+1103 q^{96}+687 q^{94}-602 q^{92}-1796 q^{90}-834 q^{88}+1520 q^{86}+2270 q^{84}+609 q^{82}-2404 q^{80}-2683 q^{78}+308 q^{76}+2944 q^{74}+2520 q^{72}-1207 q^{70}-3340 q^{68}-1547 q^{66}+1770 q^{64}+3142 q^{62}+602 q^{60}-2211 q^{58}-2299 q^{56}+53 q^{54}+2248 q^{52}+1548 q^{50}-641 q^{48}-1962 q^{46}-1023 q^{44}+1075 q^{42}+1812 q^{40}+483 q^{38}-1542 q^{36}-1702 q^{34}+205 q^{32}+2090 q^{30}+1450 q^{28}-1214 q^{26}-2450 q^{24}-829 q^{22}+2208 q^{20}+2605 q^{18}-303 q^{16}-2836 q^{14}-2279 q^{12}+1320 q^{10}+3252 q^8+1350 q^6-1886 q^4-3163 q^2-544+2305 q^{-2} +2419 q^{-4} +110 q^{-6} -2297 q^{-8} -1724 q^{-10} +331 q^{-12} +1714 q^{-14} +1253 q^{-16} -522 q^{-18} -1171 q^{-20} -683 q^{-22} +324 q^{-24} +800 q^{-26} +287 q^{-28} -187 q^{-30} -393 q^{-32} -164 q^{-34} +136 q^{-36} +134 q^{-38} +65 q^{-40} -44 q^{-42} -53 q^{-44} -4 q^{-46} +7 q^{-48} +13 q^{-50} - q^{-52} -4 q^{-54} + q^{-56} }
5 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{195}-3 q^{193}+q^{191}+6 q^{189}-7 q^{187}-6 q^{185}+6 q^{183}+14 q^{181}+17 q^{179}-6 q^{177}-68 q^{175}-90 q^{173}+31 q^{171}+223 q^{169}+281 q^{167}+9 q^{165}-504 q^{163}-829 q^{161}-385 q^{159}+910 q^{157}+1976 q^{155}+1479 q^{153}-919 q^{151}-3669 q^{149}-3993 q^{147}-320 q^{145}+5365 q^{143}+7993 q^{141}+3795 q^{139}-5578 q^{137}-12701 q^{135}-10047 q^{133}+2784 q^{131}+16391 q^{129}+18101 q^{127}+3797 q^{125}-16705 q^{123}-25866 q^{121}-13539 q^{119}+12444 q^{117}+30646 q^{115}+24001 q^{113}-3906 q^{111}-30412 q^{109}-32341 q^{107}-6868 q^{105}+25059 q^{103}+36221 q^{101}+16940 q^{99}-16089 q^{97}-34824 q^{95}-23986 q^{93}+5996 q^{91}+29197 q^{89}+26869 q^{87}+2813 q^{85}-21320 q^{83}-25896 q^{81}-9071 q^{79}+13300 q^{77}+22554 q^{75}+12559 q^{73}-6644 q^{71}-18493 q^{69}-13997 q^{67}+1817 q^{65}+15043 q^{63}+14672 q^{61}+1434 q^{59}-12928 q^{57}-15598 q^{55}-3914 q^{53}+11985 q^{51}+17517 q^{49}+6674 q^{47}-11643 q^{45}-20517 q^{43}-10461 q^{41}+10796 q^{39}+23989 q^{37}+15718 q^{35}-8296 q^{33}-26901 q^{31}-22122 q^{29}+3419 q^{27}+27710 q^{25}+28570 q^{23}+4041 q^{21}-25170 q^{19}-33454 q^{17}-13019 q^{15}+18728 q^{13}+34795 q^{11}+21734 q^9-9014 q^7-31506 q^5-27744 q^3-2006 q+23734 q^{-1} +29144 q^{-3} +11595 q^{-5} -13164 q^{-7} -25450 q^{-9} -17354 q^{-11} +2643 q^{-13} +17992 q^{-15} +18081 q^{-17} +5151 q^{-19} -9254 q^{-21} -14599 q^{-23} -8790 q^{-25} +1983 q^{-27} +9097 q^{-29} +8440 q^{-31} +2256 q^{-33} -3848 q^{-35} -5856 q^{-37} -3509 q^{-39} +517 q^{-41} +2960 q^{-43} +2723 q^{-45} +860 q^{-47} -888 q^{-49} -1496 q^{-51} -918 q^{-53} +30 q^{-55} +527 q^{-57} +501 q^{-59} +179 q^{-61} -102 q^{-63} -195 q^{-65} -107 q^{-67} +7 q^{-69} +45 q^{-71} +33 q^{-73} +8 q^{-75} -7 q^{-77} -13 q^{-79} + q^{-81} +4 q^{-83} - q^{-85} }

A2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-q^{20}-2 q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^{10}-3 q^8+q^6-4 q^4+3 q^2+1+3 q^{-4} - q^{-6} }
1,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-6 q^{58}+18 q^{56}-38 q^{54}+75 q^{52}-144 q^{50}+244 q^{48}-382 q^{46}+576 q^{44}-798 q^{42}+1004 q^{40}-1168 q^{38}+1223 q^{36}-1110 q^{34}+828 q^{32}-360 q^{30}-236 q^{28}+880 q^{26}-1502 q^{24}+1988 q^{22}-2316 q^{20}+2426 q^{18}-2296 q^{16}+1966 q^{14}-1452 q^{12}+850 q^{10}-198 q^8-408 q^6+885 q^4-1194 q^2+1304-1250 q^{-2} +1061 q^{-4} -814 q^{-6} +568 q^{-8} -338 q^{-10} +184 q^{-12} -88 q^{-14} +32 q^{-16} -8 q^{-18} + q^{-20} }
2,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{56}-q^{54}-3 q^{52}+2 q^{50}+6 q^{48}-q^{46}-11 q^{44}-q^{42}+16 q^{40}-3 q^{38}-13 q^{36}+6 q^{34}+16 q^{32}-3 q^{30}-19 q^{28}+7 q^{26}+4 q^{24}-12 q^{22}-q^{20}+8 q^{18}-2 q^{16}+16 q^{12}-12 q^8+2 q^6+14 q^4-10 q^2-18+12 q^{-2} +10 q^{-4} -8 q^{-6} -6 q^{-8} +6 q^{-10} +9 q^{-12} -3 q^{-14} -3 q^{-16} + q^{-18} }

A3 Invariants.

Weight Invariant
0,1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-3 q^{46}+9 q^{42}-10 q^{40}-4 q^{38}+22 q^{36}-19 q^{34}-10 q^{32}+29 q^{30}-20 q^{28}-8 q^{26}+24 q^{24}-7 q^{22}-6 q^{20}+5 q^{18}+7 q^{16}-4 q^{14}-15 q^{12}+13 q^{10}+7 q^8-28 q^6+15 q^4+15 q^2-26+16 q^{-2} +10 q^{-4} -15 q^{-6} +9 q^{-8} +2 q^{-10} -4 q^{-12} + q^{-14} }
1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{29}-q^{27}-2 q^{23}+3 q^{21}-2 q^{19}+4 q^{17}+q^{13}-2 q^{11}-2 q^9-q^7-3 q^5+3 q^3+4 q^{-1} - q^{-3} +3 q^{-5} - q^{-7} }

A4 Invariants.

Weight Invariant
0,1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{62}-q^{60}-3 q^{58}+q^{56}+6 q^{54}+q^{52}-10 q^{50}+2 q^{48}+15 q^{46}-9 q^{44}-19 q^{42}+16 q^{40}+13 q^{38}-21 q^{36}-6 q^{34}+23 q^{32}-21 q^{28}+13 q^{26}+18 q^{24}-20 q^{22}-4 q^{20}+29 q^{18}-12 q^{16}-23 q^{14}+20 q^{12}+8 q^{10}-28 q^8-8 q^6+21 q^4-15+11 q^{-2} +17 q^{-4} -6 q^{-6} -4 q^{-8} +8 q^{-10} - q^{-12} -3 q^{-14} + q^{-16} }
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-q^{34}-2 q^{28}+3 q^{26}-2 q^{24}+3 q^{22}+2 q^{20}+q^{16}-2 q^{14}-q^{12}-4 q^{10}-3 q^6+3 q^4+3+3 q^{-2} - q^{-4} +3 q^{-6} - q^{-8} }

B2 Invariants.

Weight Invariant
0,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-3 q^{46}+6 q^{44}-11 q^{42}+18 q^{40}-24 q^{38}+30 q^{36}-33 q^{34}+28 q^{32}-21 q^{30}+10 q^{28}+4 q^{26}-18 q^{24}+37 q^{22}-48 q^{20}+57 q^{18}-59 q^{16}+54 q^{14}-47 q^{12}+31 q^{10}-17 q^8-2 q^6+15 q^4-23 q^2+30-30 q^{-2} +32 q^{-4} -23 q^{-6} +17 q^{-8} -10 q^{-10} +4 q^{-12} - q^{-14} }
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-3 q^{74}-3 q^{72}+3 q^{70}+10 q^{68}+3 q^{66}-14 q^{64}-14 q^{62}+9 q^{60}+26 q^{58}+4 q^{56}-31 q^{54}-19 q^{52}+22 q^{50}+29 q^{48}-9 q^{46}-32 q^{44}-2 q^{42}+29 q^{40}+12 q^{38}-23 q^{36}-14 q^{34}+17 q^{32}+18 q^{30}-11 q^{28}-21 q^{26}+6 q^{24}+21 q^{22}-q^{20}-23 q^{18}-5 q^{16}+24 q^{14}+15 q^{12}-24 q^{10}-27 q^8+15 q^6+34 q^4-32-15 q^{-2} +23 q^{-4} +25 q^{-6} -10 q^{-8} -19 q^{-10} - q^{-12} +13 q^{-14} +6 q^{-16} -4 q^{-18} -4 q^{-20} + q^{-24} }

D4 Invariants.

Weight Invariant
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}-3 q^{64}+3 q^{62}-4 q^{60}+10 q^{58}-14 q^{56}+14 q^{54}-17 q^{52}+26 q^{50}-27 q^{48}+21 q^{46}-23 q^{44}+20 q^{42}-11 q^{40}+2 q^{38}+q^{36}-10 q^{34}+28 q^{32}-29 q^{30}+36 q^{28}-41 q^{26}+48 q^{24}-43 q^{22}+40 q^{20}-43 q^{18}+30 q^{16}-24 q^{14}+14 q^{12}-12 q^{10}-4 q^8+14 q^6-15 q^4+21 q^2-22+29 q^{-2} -22 q^{-4} +24 q^{-6} -19 q^{-8} +14 q^{-10} -8 q^{-12} +6 q^{-14} -4 q^{-16} + q^{-18} }

G2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+10 q^{106}-8 q^{104}+q^{102}+17 q^{100}-35 q^{98}+57 q^{96}-69 q^{94}+58 q^{92}-26 q^{90}-41 q^{88}+121 q^{86}-182 q^{84}+197 q^{82}-139 q^{80}+14 q^{78}+135 q^{76}-248 q^{74}+274 q^{72}-196 q^{70}+38 q^{68}+122 q^{66}-223 q^{64}+212 q^{62}-79 q^{60}-87 q^{58}+218 q^{56}-237 q^{54}+135 q^{52}+47 q^{50}-232 q^{48}+337 q^{46}-328 q^{44}+209 q^{42}-7 q^{40}-197 q^{38}+334 q^{36}-361 q^{34}+269 q^{32}-104 q^{30}-93 q^{28}+225 q^{26}-263 q^{24}+194 q^{22}-39 q^{20}-123 q^{18}+217 q^{16}-194 q^{14}+58 q^{12}+116 q^{10}-252 q^8+282 q^6-192 q^4+31 q^2+141-245 q^{-2} +261 q^{-4} -179 q^{-6} +57 q^{-8} +53 q^{-10} -121 q^{-12} +126 q^{-14} -87 q^{-16} +44 q^{-18} -2 q^{-20} -19 q^{-22} +24 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }

.

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["9 40"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3} }
In[5]:= Conway[K][z]
Out[5]=
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]=
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 75, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]=
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^2 a^6-2 z^4 a^4-2 z^2 a^4+a^4+z^6 a^2+2 z^4 a^2-2 a^2-z^4+2}
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^8+4 z^5 a^7-2 z^3 a^7+8 z^6 a^6-9 z^4 a^6+4 z^2 a^6+9 z^7 a^5-12 z^5 a^5+6 z^3 a^5-z a^5+4 z^8 a^4+7 z^6 a^4-20 z^4 a^4+7 z^2 a^4+a^4+17 z^7 a^3-32 z^5 a^3+14 z^3 a^3-z a^3+4 z^8 a^2+4 z^6 a^2-17 z^4 a^2+3 z^2 a^2+2 a^2+8 z^7 a-15 z^5 a+6 z^3 a+5 z^6-7 z^4+2+z^5 a^{-1} }
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