10 78

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Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+7 t^2-16 t+21-16 t^{-1} +7 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6+z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 69, -4 }
Jones polynomial [math]\displaystyle{ 1-3 q^{-1} +6 q^{-2} -8 q^{-3} +11 q^{-4} -11 q^{-5} +11 q^{-6} -9 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^{10}-3 z^2 a^8-4 a^8+3 z^4 a^6+7 z^2 a^6+4 a^6-z^6 a^4-3 z^4 a^4-3 z^2 a^4-a^4+z^4 a^2+2 z^2 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+2 z a^{11}+4 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}-a^{10}+4 z^7 a^9-7 z^3 a^9+6 z a^9+3 z^8 a^8+2 z^6 a^8-10 z^4 a^8+10 z^2 a^8-4 a^8+z^9 a^7+7 z^7 a^7-15 z^5 a^7+5 z^3 a^7+2 z a^7+6 z^8 a^6-8 z^6 a^6-7 z^4 a^6+11 z^2 a^6-4 a^6+z^9 a^5+6 z^7 a^5-21 z^5 a^5+15 z^3 a^5-3 z a^5+3 z^8 a^4-5 z^6 a^4-4 z^4 a^4+6 z^2 a^4-a^4+3 z^7 a^3-9 z^5 a^3+7 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2 }[/math]
The A2 invariant [math]\displaystyle{ q^{32}+q^{30}-2 q^{28}-q^{26}-q^{24}-3 q^{22}+2 q^{20}+2 q^{16}+2 q^{14}-q^{12}+3 q^{10}-2 q^8+q^6+q^4-q^2+1 }[/math]
The G2 invariant [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-3 q^{152}-4 q^{150}+12 q^{148}-18 q^{146}+25 q^{144}-24 q^{142}+17 q^{140}-19 q^{136}+43 q^{134}-58 q^{132}+62 q^{130}-53 q^{128}+24 q^{126}+19 q^{124}-62 q^{122}+98 q^{120}-102 q^{118}+79 q^{116}-33 q^{114}-31 q^{112}+75 q^{110}-98 q^{108}+78 q^{106}-31 q^{104}-31 q^{102}+66 q^{100}-68 q^{98}+24 q^{96}+39 q^{94}-100 q^{92}+120 q^{90}-93 q^{88}+18 q^{86}+76 q^{84}-150 q^{82}+184 q^{80}-149 q^{78}+68 q^{76}+34 q^{74}-116 q^{72}+160 q^{70}-143 q^{68}+84 q^{66}-3 q^{64}-64 q^{62}+99 q^{60}-81 q^{58}+29 q^{56}+38 q^{54}-84 q^{52}+89 q^{50}-52 q^{48}-18 q^{46}+89 q^{44}-129 q^{42}+125 q^{40}-73 q^{38}-4 q^{36}+76 q^{34}-115 q^{32}+116 q^{30}-79 q^{28}+25 q^{26}+22 q^{24}-54 q^{22}+58 q^{20}-43 q^{18}+24 q^{16}-2 q^{14}-9 q^{12}+12 q^{10}-10 q^8+6 q^6-2 q^4+q^2 }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_78.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{21}-2 q^{19}+2 q^{17}-4 q^{15}+2 q^{13}+3 q^7-2 q^5+3 q^3-2 q+ q^{-1} }[/math]
2 [math]\displaystyle{ q^{58}-2 q^{56}-q^{54}+5 q^{52}-6 q^{50}+13 q^{46}-14 q^{44}-3 q^{42}+22 q^{40}-16 q^{38}-10 q^{36}+20 q^{34}-5 q^{32}-13 q^{30}+5 q^{28}+9 q^{26}-8 q^{24}-10 q^{22}+17 q^{20}+2 q^{18}-20 q^{16}+16 q^{14}+11 q^{12}-21 q^{10}+6 q^8+14 q^6-12 q^4-2 q^2+8-2 q^{-2} -2 q^{-4} + q^{-6} }[/math]
3 [math]\displaystyle{ q^{111}-2 q^{109}-q^{107}+2 q^{105}+3 q^{103}-3 q^{101}-3 q^{99}+8 q^{97}-q^{95}-13 q^{93}+q^{91}+27 q^{89}-6 q^{87}-45 q^{85}+4 q^{83}+69 q^{81}+2 q^{79}-88 q^{77}-21 q^{75}+106 q^{73}+43 q^{71}-106 q^{69}-63 q^{67}+83 q^{65}+87 q^{63}-55 q^{61}-88 q^{59}+12 q^{57}+86 q^{55}+23 q^{53}-68 q^{51}-60 q^{49}+51 q^{47}+81 q^{45}-31 q^{43}-101 q^{41}+6 q^{39}+109 q^{37}+18 q^{35}-110 q^{33}-45 q^{31}+105 q^{29}+68 q^{27}-81 q^{25}-90 q^{23}+56 q^{21}+97 q^{19}-21 q^{17}-88 q^{15}-7 q^{13}+71 q^{11}+29 q^9-47 q^7-33 q^5+21 q^3+29 q-4 q^{-1} -19 q^{-3} -2 q^{-5} +8 q^{-7} +3 q^{-9} -2 q^{-11} -2 q^{-13} + q^{-15} }[/math]
4 [math]\displaystyle{ q^{180}-2 q^{178}-q^{176}+2 q^{174}+6 q^{170}-6 q^{168}-2 q^{166}+3 q^{164}-10 q^{162}+12 q^{160}-6 q^{158}+12 q^{156}+13 q^{154}-43 q^{152}-7 q^{150}-4 q^{148}+71 q^{146}+65 q^{144}-107 q^{142}-104 q^{140}-41 q^{138}+196 q^{136}+227 q^{134}-152 q^{132}-308 q^{130}-204 q^{128}+316 q^{126}+519 q^{124}-45 q^{122}-500 q^{120}-523 q^{118}+241 q^{116}+772 q^{114}+266 q^{112}-440 q^{110}-779 q^{108}-87 q^{106}+687 q^{104}+551 q^{102}-71 q^{100}-692 q^{98}-424 q^{96}+268 q^{94}+557 q^{92}+319 q^{90}-316 q^{88}-527 q^{86}-187 q^{84}+349 q^{82}+523 q^{80}+69 q^{78}-467 q^{76}-485 q^{74}+134 q^{72}+582 q^{70}+352 q^{68}-366 q^{66}-672 q^{64}-61 q^{62}+564 q^{60}+582 q^{58}-195 q^{56}-760 q^{54}-308 q^{52}+397 q^{50}+748 q^{48}+115 q^{46}-645 q^{44}-538 q^{42}+44 q^{40}+690 q^{38}+427 q^{36}-283 q^{34}-542 q^{32}-325 q^{30}+358 q^{28}+491 q^{26}+109 q^{24}-266 q^{22}-424 q^{20}-13 q^{18}+262 q^{16}+243 q^{14}+36 q^{12}-241 q^{10}-143 q^8+13 q^6+126 q^4+118 q^2-41-68 q^{-2} -53 q^{-4} +10 q^{-6} +53 q^{-8} +11 q^{-10} -4 q^{-12} -19 q^{-14} -9 q^{-16} +8 q^{-18} +3 q^{-20} +3 q^{-22} -2 q^{-24} -2 q^{-26} + q^{-28} }[/math]
5 [math]\displaystyle{ q^{265}-2 q^{263}-q^{261}+2 q^{259}+3 q^{255}+3 q^{253}-5 q^{251}-7 q^{249}-3 q^{245}+6 q^{243}+16 q^{241}+8 q^{239}-8 q^{237}-25 q^{235}-28 q^{233}-6 q^{231}+47 q^{229}+81 q^{227}+28 q^{225}-87 q^{223}-151 q^{221}-88 q^{219}+108 q^{217}+303 q^{215}+223 q^{213}-168 q^{211}-509 q^{209}-440 q^{207}+148 q^{205}+817 q^{203}+838 q^{201}-65 q^{199}-1195 q^{197}-1407 q^{195}-211 q^{193}+1572 q^{191}+2190 q^{189}+730 q^{187}-1806 q^{185}-3106 q^{183}-1576 q^{181}+1789 q^{179}+3974 q^{177}+2663 q^{175}-1325 q^{173}-4556 q^{171}-3897 q^{169}+436 q^{167}+4664 q^{165}+4936 q^{163}+794 q^{161}-4138 q^{159}-5525 q^{157}-2164 q^{155}+3048 q^{153}+5537 q^{151}+3261 q^{149}-1587 q^{147}-4837 q^{145}-3983 q^{143}+26 q^{141}+3737 q^{139}+4139 q^{137}+1286 q^{135}-2316 q^{133}-3875 q^{131}-2296 q^{129}+1003 q^{127}+3322 q^{125}+2898 q^{123}+149 q^{121}-2715 q^{119}-3268 q^{117}-994 q^{115}+2180 q^{113}+3488 q^{111}+1657 q^{109}-1799 q^{107}-3696 q^{105}-2184 q^{103}+1487 q^{101}+3935 q^{99}+2738 q^{97}-1166 q^{95}-4170 q^{93}-3343 q^{91}+707 q^{89}+4279 q^{87}+4013 q^{85}-11 q^{83}-4165 q^{81}-4642 q^{79}-907 q^{77}+3685 q^{75}+5042 q^{73}+2003 q^{71}-2793 q^{69}-5134 q^{67}-3037 q^{65}+1563 q^{63}+4686 q^{61}+3831 q^{59}-121 q^{57}-3787 q^{55}-4162 q^{53}-1207 q^{51}+2473 q^{49}+3923 q^{47}+2205 q^{45}-1027 q^{43}-3155 q^{41}-2666 q^{39}-251 q^{37}+2062 q^{35}+2544 q^{33}+1116 q^{31}-891 q^{29}-1975 q^{27}-1480 q^{25}-21 q^{23}+1199 q^{21}+1354 q^{19}+562 q^{17}-450 q^{15}-973 q^{13}-715 q^{11}-35 q^9+510 q^7+583 q^5+265 q^3-145 q-363 q^{-1} -278 q^{-3} -37 q^{-5} +156 q^{-7} +184 q^{-9} +93 q^{-11} -28 q^{-13} -99 q^{-15} -73 q^{-17} -10 q^{-19} +33 q^{-21} +35 q^{-23} +20 q^{-25} -4 q^{-27} -19 q^{-29} -9 q^{-31} + q^{-33} +3 q^{-35} +3 q^{-37} +3 q^{-39} -2 q^{-41} -2 q^{-43} + q^{-45} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{32}+q^{30}-2 q^{28}-q^{26}-q^{24}-3 q^{22}+2 q^{20}+2 q^{16}+2 q^{14}-q^{12}+3 q^{10}-2 q^8+q^6+q^4-q^2+1 }[/math]
2,0 [math]\displaystyle{ q^{80}+q^{78}-q^{76}-4 q^{74}-2 q^{72}+2 q^{70}-2 q^{66}+5 q^{64}+10 q^{62}-7 q^{58}+3 q^{56}+8 q^{54}-10 q^{52}-10 q^{50}+4 q^{48}+3 q^{46}-9 q^{44}-4 q^{42}+8 q^{40}-3 q^{38}-4 q^{36}+7 q^{34}+2 q^{32}-8 q^{30}+4 q^{28}+9 q^{26}-5 q^{24}-6 q^{22}+9 q^{20}+8 q^{18}-7 q^{16}-4 q^{14}+9 q^{12}+2 q^{10}-5 q^8-q^6+3 q^4+2 q^2-2- q^{-2} + q^{-4} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{68}-2 q^{66}+4 q^{62}-7 q^{60}-2 q^{58}+12 q^{56}-7 q^{54}-5 q^{52}+22 q^{50}-6 q^{48}-12 q^{46}+14 q^{44}-9 q^{42}-16 q^{40}+3 q^{38}+q^{36}-4 q^{34}-3 q^{32}+11 q^{30}+8 q^{28}-13 q^{26}+9 q^{24}+12 q^{22}-16 q^{20}+4 q^{18}+12 q^{16}-13 q^{14}+4 q^{12}+7 q^{10}-7 q^8+3 q^6+2 q^4-2 q^2+1 }[/math]
1,0,0 [math]\displaystyle{ q^{43}+q^{41}+q^{39}-2 q^{37}-q^{35}-4 q^{33}-q^{31}-3 q^{29}+3 q^{27}+3 q^{23}+2 q^{21}+q^{19}+q^{17}-q^{15}+2 q^{13}-2 q^{11}+2 q^9-q^7+2 q^5-q^3+q }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{68}-2 q^{66}+4 q^{64}-6 q^{62}+9 q^{60}-12 q^{58}+16 q^{56}-19 q^{54}+19 q^{52}-20 q^{50}+14 q^{48}-10 q^{46}+9 q^{42}-20 q^{40}+29 q^{38}-35 q^{36}+40 q^{34}-39 q^{32}+37 q^{30}-28 q^{28}+21 q^{26}-9 q^{24}+10 q^{20}-16 q^{18}+20 q^{16}-21 q^{14}+20 q^{12}-17 q^{10}+13 q^8-9 q^6+6 q^4-2 q^2+1 }[/math]
1,0 [math]\displaystyle{ q^{110}-2 q^{106}-2 q^{104}+2 q^{102}+5 q^{100}-8 q^{96}-7 q^{94}+5 q^{92}+14 q^{90}+4 q^{88}-14 q^{86}-12 q^{84}+9 q^{82}+23 q^{80}+3 q^{78}-19 q^{76}-12 q^{74}+13 q^{72}+15 q^{70}-9 q^{68}-20 q^{66}-2 q^{64}+14 q^{62}+2 q^{60}-15 q^{58}-8 q^{56}+11 q^{54}+9 q^{52}-8 q^{50}-9 q^{48}+10 q^{46}+14 q^{44}-4 q^{42}-17 q^{40}+q^{38}+21 q^{36}+11 q^{34}-17 q^{32}-19 q^{30}+8 q^{28}+23 q^{26}+4 q^{24}-18 q^{22}-12 q^{20}+11 q^{18}+14 q^{16}-2 q^{14}-10 q^{12}-3 q^{10}+6 q^8+4 q^6-2 q^4-2 q^2+ q^{-2} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-3 q^{152}-4 q^{150}+12 q^{148}-18 q^{146}+25 q^{144}-24 q^{142}+17 q^{140}-19 q^{136}+43 q^{134}-58 q^{132}+62 q^{130}-53 q^{128}+24 q^{126}+19 q^{124}-62 q^{122}+98 q^{120}-102 q^{118}+79 q^{116}-33 q^{114}-31 q^{112}+75 q^{110}-98 q^{108}+78 q^{106}-31 q^{104}-31 q^{102}+66 q^{100}-68 q^{98}+24 q^{96}+39 q^{94}-100 q^{92}+120 q^{90}-93 q^{88}+18 q^{86}+76 q^{84}-150 q^{82}+184 q^{80}-149 q^{78}+68 q^{76}+34 q^{74}-116 q^{72}+160 q^{70}-143 q^{68}+84 q^{66}-3 q^{64}-64 q^{62}+99 q^{60}-81 q^{58}+29 q^{56}+38 q^{54}-84 q^{52}+89 q^{50}-52 q^{48}-18 q^{46}+89 q^{44}-129 q^{42}+125 q^{40}-73 q^{38}-4 q^{36}+76 q^{34}-115 q^{32}+116 q^{30}-79 q^{28}+25 q^{26}+22 q^{24}-54 q^{22}+58 q^{20}-43 q^{18}+24 q^{16}-2 q^{14}-9 q^{12}+12 q^{10}-10 q^8+6 q^6-2 q^4+q^2 }[/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 78"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+7 t^2-16 t+21-16 t^{-1} +7 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6+z^4+3 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]= { 69, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 1-3 q^{-1} +6 q^{-2} -8 q^{-3} +11 q^{-4} -11 q^{-5} +11 q^{-6} -9 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^{10}-3 z^2 a^8-4 a^8+3 z^4 a^6+7 z^2 a^6+4 a^6-z^6 a^4-3 z^4 a^4-3 z^2 a^4-a^4+z^4 a^2+2 z^2 a^2+a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+2 z a^{11}+4 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}-a^{10}+4 z^7 a^9-7 z^3 a^9+6 z a^9+3 z^8 a^8+2 z^6 a^8-10 z^4 a^8+10 z^2 a^8-4 a^8+z^9 a^7+7 z^7 a^7-15 z^5 a^7+5 z^3 a^7+2 z a^7+6 z^8 a^6-8 z^6 a^6-7 z^4 a^6+11 z^2 a^6-4 a^6+z^9 a^5+6 z^7 a^5-21 z^5 a^5+15 z^3 a^5-3 z a^5+3 z^8 a^4-5 z^6 a^4-4 z^4 a^4+6 z^2 a^4-a^4+3 z^7 a^3-9 z^5 a^3+7 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2 }[/math]
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