10 55

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ 5 t^2-15 t+21-15 t^{-1} +5 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 5 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 61, -4 }
Jones polynomial [math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -7 q^{-5} +10 q^{-6} -10 q^{-7} +9 q^{-8} -8 q^{-9} +5 q^{-10} -3 q^{-11} + q^{-12} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^{12}-3 z^2 a^{10}-3 a^{10}+2 z^4 a^8+3 z^2 a^8+a^8+2 z^4 a^6+3 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{14}-3 z^4 a^{14}+2 z^2 a^{14}+3 z^7 a^{13}-10 z^5 a^{13}+9 z^3 a^{13}-3 z a^{13}+3 z^8 a^{12}-7 z^6 a^{12}+z^4 a^{12}+z^2 a^{12}+a^{12}+z^9 a^{11}+5 z^7 a^{11}-23 z^5 a^{11}+24 z^3 a^{11}-9 z a^{11}+6 z^8 a^{10}-15 z^6 a^{10}+13 z^4 a^{10}-8 z^2 a^{10}+3 a^{10}+z^9 a^9+5 z^7 a^9-16 z^5 a^9+15 z^3 a^9-4 z a^9+3 z^8 a^8-4 z^6 a^8+5 z^4 a^8-3 z^2 a^8+a^8+3 z^7 a^7-z^5 a^7-2 z^3 a^7+2 z a^7+3 z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]
The A2 invariant [math]\displaystyle{ q^{38}+q^{36}-2 q^{34}-q^{30}-3 q^{28}+q^{26}-q^{24}+q^{22}+q^{20}+3 q^{16}-q^{14}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
The G2 invariant [math]\displaystyle{ q^{190}-2 q^{188}+5 q^{186}-9 q^{184}+9 q^{182}-8 q^{180}-3 q^{178}+19 q^{176}-34 q^{174}+45 q^{172}-41 q^{170}+18 q^{168}+20 q^{166}-58 q^{164}+86 q^{162}-82 q^{160}+52 q^{158}+q^{156}-55 q^{154}+88 q^{152}-84 q^{150}+52 q^{148}+2 q^{146}-47 q^{144}+64 q^{142}-51 q^{140}+8 q^{138}+38 q^{136}-74 q^{134}+73 q^{132}-42 q^{130}-16 q^{128}+70 q^{126}-109 q^{124}+108 q^{122}-75 q^{120}+12 q^{118}+50 q^{116}-103 q^{114}+117 q^{112}-91 q^{110}+38 q^{108}+24 q^{106}-67 q^{104}+76 q^{102}-52 q^{100}+8 q^{98}+35 q^{96}-54 q^{94}+45 q^{92}-9 q^{90}-33 q^{88}+68 q^{86}-70 q^{84}+49 q^{82}-10 q^{80}-31 q^{78}+56 q^{76}-62 q^{74}+55 q^{72}-31 q^{70}+8 q^{68}+15 q^{66}-29 q^{64}+34 q^{62}-31 q^{60}+25 q^{58}-12 q^{56}+2 q^{54}+8 q^{52}-14 q^{50}+15 q^{48}-11 q^{46}+8 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_55.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{25}-2 q^{23}+2 q^{21}-3 q^{19}+q^{17}-q^{15}+3 q^{11}-2 q^9+3 q^7-q^5+q^3 }[/math]
2 [math]\displaystyle{ q^{70}-2 q^{68}-2 q^{66}+7 q^{64}-2 q^{62}-10 q^{60}+11 q^{58}+5 q^{56}-17 q^{54}+7 q^{52}+13 q^{50}-14 q^{48}-q^{46}+13 q^{44}-7 q^{42}-8 q^{40}+6 q^{38}+5 q^{36}-10 q^{34}-5 q^{32}+16 q^{30}-7 q^{28}-13 q^{26}+16 q^{24}-10 q^{20}+9 q^{18}+q^{16}-3 q^{14}+4 q^{12}-q^8+q^6 }[/math]
3 [math]\displaystyle{ q^{135}-2 q^{133}-2 q^{131}+3 q^{129}+7 q^{127}-2 q^{125}-16 q^{123}-2 q^{121}+23 q^{119}+15 q^{117}-27 q^{115}-34 q^{113}+22 q^{111}+53 q^{109}-5 q^{107}-63 q^{105}-23 q^{103}+66 q^{101}+46 q^{99}-53 q^{97}-69 q^{95}+32 q^{93}+81 q^{91}-10 q^{89}-82 q^{87}-8 q^{85}+77 q^{83}+24 q^{81}-64 q^{79}-37 q^{77}+53 q^{75}+44 q^{73}-31 q^{71}-56 q^{69}+10 q^{67}+58 q^{65}+23 q^{63}-61 q^{61}-48 q^{59}+49 q^{57}+69 q^{55}-32 q^{53}-83 q^{51}+9 q^{49}+76 q^{47}+7 q^{45}-60 q^{43}-19 q^{41}+38 q^{39}+20 q^{37}-21 q^{35}-10 q^{33}+8 q^{31}+7 q^{29}-2 q^{27}+q^{25}+2 q^{23}-q^{21}-q^{19}+3 q^{17}+q^{15}-q^{11}+q^9 }[/math]
4 [math]\displaystyle{ q^{220}-2 q^{218}-2 q^{216}+3 q^{214}+3 q^{212}+7 q^{210}-9 q^{208}-16 q^{206}-q^{204}+10 q^{202}+41 q^{200}+2 q^{198}-44 q^{196}-45 q^{194}-21 q^{192}+92 q^{190}+81 q^{188}-8 q^{186}-104 q^{184}-156 q^{182}+41 q^{180}+167 q^{178}+165 q^{176}-13 q^{174}-279 q^{172}-180 q^{170}+54 q^{168}+314 q^{166}+268 q^{164}-164 q^{162}-361 q^{160}-262 q^{158}+205 q^{156}+490 q^{154}+160 q^{152}-289 q^{150}-511 q^{148}-85 q^{146}+458 q^{144}+410 q^{142}-56 q^{140}-519 q^{138}-303 q^{136}+270 q^{134}+458 q^{132}+120 q^{130}-387 q^{128}-353 q^{126}+102 q^{124}+389 q^{122}+196 q^{120}-232 q^{118}-342 q^{116}-52 q^{114}+297 q^{112}+274 q^{110}-33 q^{108}-320 q^{106}-274 q^{104}+125 q^{102}+361 q^{100}+272 q^{98}-191 q^{96}-492 q^{94}-176 q^{92}+295 q^{90}+542 q^{88}+100 q^{86}-489 q^{84}-443 q^{82}+29 q^{80}+541 q^{78}+347 q^{76}-233 q^{74}-429 q^{72}-210 q^{70}+280 q^{68}+337 q^{66}+14 q^{64}-204 q^{62}-222 q^{60}+48 q^{58}+162 q^{56}+70 q^{54}-24 q^{52}-107 q^{50}-21 q^{48}+37 q^{46}+31 q^{44}+23 q^{42}-28 q^{40}-11 q^{38}+2 q^{34}+16 q^{32}-3 q^{30}-2 q^{26}-3 q^{24}+5 q^{22}+q^{18}-q^{14}+q^{12} }[/math]
5 [math]\displaystyle{ q^{325}-2 q^{323}-2 q^{321}+3 q^{319}+3 q^{317}+3 q^{315}-9 q^{311}-16 q^{309}-q^{307}+20 q^{305}+28 q^{303}+19 q^{301}-16 q^{299}-58 q^{297}-65 q^{295}+q^{293}+84 q^{291}+121 q^{289}+74 q^{287}-66 q^{285}-203 q^{283}-205 q^{281}-19 q^{279}+226 q^{277}+358 q^{275}+238 q^{273}-124 q^{271}-480 q^{269}-531 q^{267}-155 q^{265}+419 q^{263}+798 q^{261}+623 q^{259}-96 q^{257}-885 q^{255}-1133 q^{253}-508 q^{251}+632 q^{249}+1476 q^{247}+1290 q^{245}+24 q^{243}-1485 q^{241}-2015 q^{239}-955 q^{237}+1011 q^{235}+2428 q^{233}+2015 q^{231}-143 q^{229}-2410 q^{227}-2870 q^{225}-958 q^{223}+1898 q^{221}+3362 q^{219}+2055 q^{217}-1071 q^{215}-3398 q^{213}-2890 q^{211}+108 q^{209}+3036 q^{207}+3361 q^{205}+776 q^{203}-2440 q^{201}-3443 q^{199}-1413 q^{197}+1768 q^{195}+3223 q^{193}+1764 q^{191}-1169 q^{189}-2847 q^{187}-1860 q^{185}+706 q^{183}+2437 q^{181}+1814 q^{179}-406 q^{177}-2061 q^{175}-1735 q^{173}+139 q^{171}+1794 q^{169}+1741 q^{167}+122 q^{165}-1537 q^{163}-1842 q^{161}-575 q^{159}+1243 q^{157}+2070 q^{155}+1160 q^{153}-769 q^{151}-2230 q^{149}-1959 q^{147}+52 q^{145}+2254 q^{143}+2757 q^{141}+909 q^{139}-1937 q^{137}-3398 q^{135}-2009 q^{133}+1263 q^{131}+3678 q^{129}+3029 q^{127}-304 q^{125}-3463 q^{123}-3705 q^{121}-781 q^{119}+2791 q^{117}+3905 q^{115}+1697 q^{113}-1825 q^{111}-3568 q^{109}-2238 q^{107}+792 q^{105}+2847 q^{103}+2347 q^{101}+34 q^{99}-1967 q^{97}-2065 q^{95}-523 q^{93}+1118 q^{91}+1569 q^{89}+709 q^{87}-513 q^{85}-1053 q^{83}-638 q^{81}+137 q^{79}+599 q^{77}+491 q^{75}+31 q^{73}-314 q^{71}-316 q^{69}-79 q^{67}+139 q^{65}+185 q^{63}+74 q^{61}-53 q^{59}-96 q^{57}-56 q^{55}+16 q^{53}+54 q^{51}+31 q^{49}+q^{47}-17 q^{45}-20 q^{43}-7 q^{41}+14 q^{39}+9 q^{37}+2 q^{35}+2 q^{33}-4 q^{31}-4 q^{29}+3 q^{27}+2 q^{25}+q^{21}-q^{17}+q^{15} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{80}-2 q^{78}+q^{76}+2 q^{74}-7 q^{72}+4 q^{70}+2 q^{68}-9 q^{66}+11 q^{64}+6 q^{62}-9 q^{60}+11 q^{58}+4 q^{56}-13 q^{54}-q^{52}+q^{50}-8 q^{48}-6 q^{46}+5 q^{42}-7 q^{40}-2 q^{38}+15 q^{36}-8 q^{34}-3 q^{32}+15 q^{30}-4 q^{28}-5 q^{26}+10 q^{24}-3 q^{20}+4 q^{18}+q^{16}-q^{14}+q^{12} }[/math]
1,0,0 [math]\displaystyle{ q^{51}+q^{49}+q^{47}-2 q^{45}-3 q^{41}-q^{39}-3 q^{37}+q^{35}-q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+3 q^{21}-q^{19}+2 q^{17}+2 q^{13}-q^{11}+q^9 }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+11 q^{72}-14 q^{70}+16 q^{68}-17 q^{66}+15 q^{64}-12 q^{62}+5 q^{60}+q^{58}-10 q^{56}+17 q^{54}-25 q^{52}+29 q^{50}-32 q^{48}+30 q^{46}-26 q^{44}+21 q^{42}-13 q^{40}+6 q^{38}+3 q^{36}-8 q^{34}+13 q^{32}-15 q^{30}+16 q^{28}-13 q^{26}+12 q^{24}-8 q^{22}+7 q^{20}-4 q^{18}+3 q^{16}-q^{14}+q^{12} }[/math]
1,0 [math]\displaystyle{ q^{130}-2 q^{126}-2 q^{124}+3 q^{122}+5 q^{120}-3 q^{118}-9 q^{116}-2 q^{114}+11 q^{112}+8 q^{110}-10 q^{108}-14 q^{106}+5 q^{104}+19 q^{102}+7 q^{100}-15 q^{98}-11 q^{96}+10 q^{94}+16 q^{92}-2 q^{90}-14 q^{88}-4 q^{86}+9 q^{84}+3 q^{82}-10 q^{80}-8 q^{78}+6 q^{76}+6 q^{74}-8 q^{72}-12 q^{70}+3 q^{68}+13 q^{66}-q^{64}-14 q^{62}-4 q^{60}+15 q^{58}+10 q^{56}-9 q^{54}-14 q^{52}+4 q^{50}+17 q^{48}+6 q^{46}-10 q^{44}-9 q^{42}+3 q^{40}+11 q^{38}+4 q^{36}-4 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+q^{18} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{190}-2 q^{188}+5 q^{186}-9 q^{184}+9 q^{182}-8 q^{180}-3 q^{178}+19 q^{176}-34 q^{174}+45 q^{172}-41 q^{170}+18 q^{168}+20 q^{166}-58 q^{164}+86 q^{162}-82 q^{160}+52 q^{158}+q^{156}-55 q^{154}+88 q^{152}-84 q^{150}+52 q^{148}+2 q^{146}-47 q^{144}+64 q^{142}-51 q^{140}+8 q^{138}+38 q^{136}-74 q^{134}+73 q^{132}-42 q^{130}-16 q^{128}+70 q^{126}-109 q^{124}+108 q^{122}-75 q^{120}+12 q^{118}+50 q^{116}-103 q^{114}+117 q^{112}-91 q^{110}+38 q^{108}+24 q^{106}-67 q^{104}+76 q^{102}-52 q^{100}+8 q^{98}+35 q^{96}-54 q^{94}+45 q^{92}-9 q^{90}-33 q^{88}+68 q^{86}-70 q^{84}+49 q^{82}-10 q^{80}-31 q^{78}+56 q^{76}-62 q^{74}+55 q^{72}-31 q^{70}+8 q^{68}+15 q^{66}-29 q^{64}+34 q^{62}-31 q^{60}+25 q^{58}-12 q^{56}+2 q^{54}+8 q^{52}-14 q^{50}+15 q^{48}-11 q^{46}+8 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]

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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 55"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 5 t^2-15 t+21-15 t^{-1} +5 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 5 z^4+5 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]= { 61, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -7 q^{-5} +10 q^{-6} -10 q^{-7} +9 q^{-8} -8 q^{-9} +5 q^{-10} -3 q^{-11} + q^{-12} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^{12}-3 z^2 a^{10}-3 a^{10}+2 z^4 a^8+3 z^2 a^8+a^8+2 z^4 a^6+3 z^2 a^6+a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^6 a^{14}-3 z^4 a^{14}+2 z^2 a^{14}+3 z^7 a^{13}-10 z^5 a^{13}+9 z^3 a^{13}-3 z a^{13}+3 z^8 a^{12}-7 z^6 a^{12}+z^4 a^{12}+z^2 a^{12}+a^{12}+z^9 a^{11}+5 z^7 a^{11}-23 z^5 a^{11}+24 z^3 a^{11}-9 z a^{11}+6 z^8 a^{10}-15 z^6 a^{10}+13 z^4 a^{10}-8 z^2 a^{10}+3 a^{10}+z^9 a^9+5 z^7 a^9-16 z^5 a^9+15 z^3 a^9-4 z a^9+3 z^8 a^8-4 z^6 a^8+5 z^4 a^8-3 z^2 a^8+a^8+3 z^7 a^7-z^5 a^7-2 z^3 a^7+2 z a^7+3 z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]
Retrieved from "https://katlas.org/index.php?title=10_55&oldid=4330"