10 74

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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 -4 t^2+16 t-23+16 t^{-1} -4 t^{-2} }
Conway 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 1-4 z^4}
2nd Alexander ideal (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 \{3,t+1\}}
Determinant and Signature { 63, -2 }
Jones 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 q-3+6 q^{-1} -8 q^{-2} +11 q^{-3} -10 q^{-4} +9 q^{-5} -8 q^{-6} +4 q^{-7} -2 q^{-8} + q^{-9} }
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^8+a^8-z^4 a^6-z^2 a^6-2 a^6-2 z^4 a^4-2 z^2 a^4-z^4 a^2+z^2 a^2+2 a^2+z^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^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-7 z^5 a^9+8 z^3 a^9-4 z a^9+2 z^8 a^8-4 z^6 a^8+z^2 a^8+a^8+z^9 a^7+2 z^7 a^7-10 z^5 a^7+11 z^3 a^7-8 z a^7+5 z^8 a^6-9 z^6 a^6+3 z^4 a^6-z^2 a^6+2 a^6+z^9 a^5+5 z^7 a^5-12 z^5 a^5+9 z^3 a^5-4 z a^5+3 z^8 a^4+z^6 a^4-9 z^4 a^4+8 z^2 a^4+5 z^7 a^3-6 z^5 a^3+3 z^3 a^3+5 z^6 a^2-7 z^4 a^2+5 z^2 a^2-2 a^2+3 z^5 a-3 z^3 a+z^4-z^2}
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^{28}+2 q^{22}-3 q^{20}-2 q^{18}-2 q^{14}+2 q^{12}+2 q^8+2 q^6-q^4+3 q^2-1- q^{-2} + q^{-4} }
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^{142}-q^{140}+3 q^{138}-5 q^{136}+4 q^{134}-5 q^{132}-q^{130}+9 q^{128}-18 q^{126}+27 q^{124}-29 q^{122}+20 q^{120}+3 q^{118}-30 q^{116}+57 q^{114}-75 q^{112}+68 q^{110}-34 q^{108}-17 q^{106}+69 q^{104}-100 q^{102}+106 q^{100}-58 q^{98}+7 q^{96}+44 q^{94}-85 q^{92}+83 q^{90}-41 q^{88}-16 q^{86}+55 q^{84}-70 q^{82}+48 q^{80}+10 q^{78}-72 q^{76}+98 q^{74}-107 q^{72}+67 q^{70}-3 q^{68}-85 q^{66}+135 q^{64}-145 q^{62}+115 q^{60}-41 q^{58}-41 q^{56}+98 q^{54}-119 q^{52}+99 q^{50}-45 q^{48}-17 q^{46}+65 q^{44}-65 q^{42}+38 q^{40}+19 q^{38}-59 q^{36}+75 q^{34}-56 q^{32}+10 q^{30}+39 q^{28}-80 q^{26}+98 q^{24}-78 q^{22}+42 q^{20}+9 q^{18}-48 q^{16}+66 q^{14}-66 q^{12}+51 q^{10}-26 q^8-q^6+19 q^4-29 q^2+28-19 q^{-2} +12 q^{-4} -2 q^{-6} -3 q^{-8} +5 q^{-10} -6 q^{-12} +4 q^{-14} -2 q^{-16} + q^{-18} }
Further Quantum Invariants
Further quantum knot invariants for 10_74.

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^{19}-q^{17}+2 q^{15}-4 q^{13}+q^{11}-q^9+q^7+3 q^5-2 q^3+3 q-2 q^{-1} + q^{-3} }
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^{54}-q^{52}-q^{50}+4 q^{48}-3 q^{46}-7 q^{44}+10 q^{42}-15 q^{38}+15 q^{36}+9 q^{34}-19 q^{32}+7 q^{30}+13 q^{28}-15 q^{26}-4 q^{24}+9 q^{22}-2 q^{20}-10 q^{18}+2 q^{16}+16 q^{14}-12 q^{12}-7 q^{10}+22 q^8-8 q^6-12 q^4+15 q^2-2-7 q^{-2} +6 q^{-4} -2 q^{-8} + q^{-10} }
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^{172}-q^{170}-q^{168}+q^{166}+3 q^{162}-5 q^{160}-5 q^{158}+5 q^{156}+4 q^{154}+14 q^{152}-11 q^{150}-25 q^{148}-3 q^{146}+14 q^{144}+58 q^{142}+4 q^{140}-63 q^{138}-64 q^{136}-17 q^{134}+134 q^{132}+102 q^{130}-43 q^{128}-170 q^{126}-176 q^{124}+130 q^{122}+259 q^{120}+146 q^{118}-161 q^{116}-404 q^{114}-75 q^{112}+277 q^{110}+411 q^{108}+81 q^{106}-467 q^{104}-374 q^{102}+51 q^{100}+517 q^{98}+392 q^{96}-291 q^{94}-518 q^{92}-239 q^{90}+397 q^{88}+538 q^{86}-38 q^{84}-462 q^{82}-397 q^{80}+212 q^{78}+508 q^{76}+138 q^{74}-318 q^{72}-409 q^{70}+40 q^{68}+390 q^{66}+261 q^{64}-144 q^{62}-374 q^{60}-166 q^{58}+203 q^{56}+385 q^{54}+100 q^{52}-279 q^{50}-404 q^{48}-79 q^{46}+446 q^{44}+387 q^{42}-52 q^{40}-530 q^{38}-401 q^{36}+308 q^{34}+537 q^{32}+241 q^{30}-411 q^{28}-548 q^{26}+48 q^{24}+407 q^{22}+377 q^{20}-144 q^{18}-418 q^{16}-111 q^{14}+149 q^{12}+283 q^{10}+23 q^8-188 q^6-84 q^4+q^2+121+35 q^{-2} -57 q^{-4} -20 q^{-6} -19 q^{-8} +38 q^{-10} +10 q^{-12} -19 q^{-14} +2 q^{-16} -8 q^{-18} +12 q^{-20} +2 q^{-22} -7 q^{-24} +2 q^{-26} -2 q^{-28} +3 q^{-30} -2 q^{-34} + q^{-36} }
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^{255}-q^{253}-q^{251}+q^{249}+q^{243}-3 q^{241}-4 q^{239}+5 q^{237}+8 q^{235}+2 q^{233}-2 q^{231}-13 q^{229}-19 q^{227}+2 q^{225}+30 q^{223}+37 q^{221}+15 q^{219}-35 q^{217}-82 q^{215}-63 q^{213}+29 q^{211}+134 q^{209}+151 q^{207}+32 q^{205}-172 q^{203}-290 q^{201}-181 q^{199}+139 q^{197}+441 q^{195}+434 q^{193}+32 q^{191}-520 q^{189}-758 q^{187}-391 q^{185}+414 q^{183}+1059 q^{181}+923 q^{179}-46 q^{177}-1159 q^{175}-1491 q^{173}-631 q^{171}+930 q^{169}+1967 q^{167}+1453 q^{165}-321 q^{163}-2062 q^{161}-2313 q^{159}-635 q^{157}+1782 q^{155}+2910 q^{153}+1681 q^{151}-1044 q^{149}-3120 q^{147}-2686 q^{145}+80 q^{143}+2905 q^{141}+3357 q^{139}+952 q^{137}-2321 q^{135}-3650 q^{133}-1827 q^{131}+1588 q^{129}+3577 q^{127}+2409 q^{125}-841 q^{123}-3222 q^{121}-2686 q^{119}+227 q^{117}+2763 q^{115}+2664 q^{113}+215 q^{111}-2276 q^{109}-2552 q^{107}-470 q^{105}+1853 q^{103}+2322 q^{101}+698 q^{99}-1469 q^{97}-2210 q^{95}-902 q^{93}+1125 q^{91}+2072 q^{89}+1258 q^{87}-648 q^{85}-2030 q^{83}-1699 q^{81}+69 q^{79}+1857 q^{77}+2240 q^{75}+741 q^{73}-1547 q^{71}-2742 q^{69}-1673 q^{67}+964 q^{65}+3033 q^{63}+2636 q^{61}-112 q^{59}-3014 q^{57}-3457 q^{55}-861 q^{53}+2573 q^{51}+3889 q^{49}+1862 q^{47}-1802 q^{45}-3874 q^{43}-2586 q^{41}+840 q^{39}+3373 q^{37}+2925 q^{35}+82 q^{33}-2553 q^{31}-2804 q^{29}-770 q^{27}+1624 q^{25}+2348 q^{23}+1081 q^{21}-797 q^{19}-1682 q^{17}-1092 q^{15}+214 q^{13}+1063 q^{11}+870 q^9+77 q^7-545 q^5-586 q^3-178 q+235 q^{-1} +342 q^{-3} +149 q^{-5} -81 q^{-7} -161 q^{-9} -95 q^{-11} +14 q^{-13} +72 q^{-15} +53 q^{-17} -6 q^{-19} -31 q^{-21} -16 q^{-23} +2 q^{-25} +10 q^{-27} +7 q^{-29} -10 q^{-33} -2 q^{-35} +7 q^{-37} + q^{-39} -2 q^{-41} + q^{-43} - q^{-45} -2 q^{-47} +3 q^{-49} -2 q^{-53} + q^{-55} }

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^{28}+2 q^{22}-3 q^{20}-2 q^{18}-2 q^{14}+2 q^{12}+2 q^8+2 q^6-q^4+3 q^2-1- q^{-2} + q^{-4} }
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^{72}+2 q^{64}-q^{62}-6 q^{60}-2 q^{58}+5 q^{56}+q^{54}-10 q^{52}-2 q^{50}+13 q^{48}+11 q^{46}-9 q^{44}-3 q^{42}+11 q^{40}+4 q^{38}-9 q^{36}-6 q^{34}+5 q^{32}-4 q^{30}-7 q^{28}-q^{26}-3 q^{24}-5 q^{22}+9 q^{20}+7 q^{18}-7 q^{16}+14 q^{12}+4 q^{10}-14 q^8-3 q^6+14 q^4+2 q^2-9- q^{-2} +5 q^{-4} +2 q^{-6} -2 q^{-8} - q^{-10} + q^{-12} }

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^{60}-q^{58}+q^{56}+q^{54}-6 q^{52}+q^{50}+2 q^{48}-10 q^{46}+9 q^{44}+11 q^{42}-10 q^{40}+14 q^{38}+9 q^{36}-17 q^{34}-2 q^{32}+q^{30}-13 q^{28}-6 q^{26}+4 q^{24}+6 q^{22}-3 q^{20}-2 q^{18}+16 q^{16}-7 q^{14}-8 q^{12}+18 q^{10}-4 q^8-9 q^6+13 q^4-q^2-6+5 q^{-2} -2 q^{-6} + q^{-8} }
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^{37}+q^{33}+2 q^{29}-3 q^{27}-q^{25}-3 q^{23}-2 q^{19}+q^{17}+q^{15}+2 q^{11}+q^9+3 q^7-q^5+3 q^3-q- q^{-3} + q^{-5} }

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^{60}-q^{58}+3 q^{56}-5 q^{54}+8 q^{52}-11 q^{50}+14 q^{48}-16 q^{46}+17 q^{44}-17 q^{42}+12 q^{40}-8 q^{38}-q^{36}+9 q^{34}-18 q^{32}+25 q^{30}-31 q^{28}+34 q^{26}-34 q^{24}+30 q^{22}-23 q^{20}+16 q^{18}-6 q^{16}-q^{14}+10 q^{12}-14 q^{10}+18 q^8-17 q^6+17 q^4-13 q^2+10-7 q^{-2} +4 q^{-4} -2 q^{-6} + q^{-8} }
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^{98}-q^{94}-q^{92}+2 q^{90}+3 q^{88}-2 q^{86}-7 q^{84}-3 q^{82}+7 q^{80}+7 q^{78}-7 q^{76}-14 q^{74}+18 q^{70}+13 q^{68}-11 q^{66}-14 q^{64}+7 q^{62}+22 q^{60}+5 q^{58}-16 q^{56}-11 q^{54}+7 q^{52}+7 q^{50}-9 q^{48}-14 q^{46}+9 q^{42}-2 q^{40}-12 q^{38}-q^{36}+14 q^{34}+7 q^{32}-10 q^{30}-9 q^{28}+12 q^{26}+15 q^{24}-5 q^{22}-18 q^{20}-q^{18}+18 q^{16}+11 q^{14}-11 q^{12}-15 q^{10}+2 q^8+15 q^6+6 q^4-7 q^2-8+ q^{-2} +6 q^{-4} +2 q^{-6} -2 q^{-8} -2 q^{-10} + q^{-14} }

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^{142}-q^{140}+3 q^{138}-5 q^{136}+4 q^{134}-5 q^{132}-q^{130}+9 q^{128}-18 q^{126}+27 q^{124}-29 q^{122}+20 q^{120}+3 q^{118}-30 q^{116}+57 q^{114}-75 q^{112}+68 q^{110}-34 q^{108}-17 q^{106}+69 q^{104}-100 q^{102}+106 q^{100}-58 q^{98}+7 q^{96}+44 q^{94}-85 q^{92}+83 q^{90}-41 q^{88}-16 q^{86}+55 q^{84}-70 q^{82}+48 q^{80}+10 q^{78}-72 q^{76}+98 q^{74}-107 q^{72}+67 q^{70}-3 q^{68}-85 q^{66}+135 q^{64}-145 q^{62}+115 q^{60}-41 q^{58}-41 q^{56}+98 q^{54}-119 q^{52}+99 q^{50}-45 q^{48}-17 q^{46}+65 q^{44}-65 q^{42}+38 q^{40}+19 q^{38}-59 q^{36}+75 q^{34}-56 q^{32}+10 q^{30}+39 q^{28}-80 q^{26}+98 q^{24}-78 q^{22}+42 q^{20}+9 q^{18}-48 q^{16}+66 q^{14}-66 q^{12}+51 q^{10}-26 q^8-q^6+19 q^4-29 q^2+28-19 q^{-2} +12 q^{-4} -2 q^{-6} -3 q^{-8} +5 q^{-10} -6 q^{-12} +4 q^{-14} -2 q^{-16} + q^{-18} }

.

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 74"];
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 -4 t^2+16 t-23+16 t^{-1} -4 t^{-2} }
In[5]:= Conway[K][z]
Out[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 1-4 z^4}
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]= 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 \{3,t+1\}}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 63, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= 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-3+6 q^{-1} -8 q^{-2} +11 q^{-3} -10 q^{-4} +9 q^{-5} -8 q^{-6} +4 q^{-7} -2 q^{-8} + q^{-9} }
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^8+a^8-z^4 a^6-z^2 a^6-2 a^6-2 z^4 a^4-2 z^2 a^4-z^4 a^2+z^2 a^2+2 a^2+z^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^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-7 z^5 a^9+8 z^3 a^9-4 z a^9+2 z^8 a^8-4 z^6 a^8+z^2 a^8+a^8+z^9 a^7+2 z^7 a^7-10 z^5 a^7+11 z^3 a^7-8 z a^7+5 z^8 a^6-9 z^6 a^6+3 z^4 a^6-z^2 a^6+2 a^6+z^9 a^5+5 z^7 a^5-12 z^5 a^5+9 z^3 a^5-4 z a^5+3 z^8 a^4+z^6 a^4-9 z^4 a^4+8 z^2 a^4+5 z^7 a^3-6 z^5 a^3+3 z^3 a^3+5 z^6 a^2-7 z^4 a^2+5 z^2 a^2-2 a^2+3 z^5 a-3 z^3 a+z^4-z^2}
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