10 143

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-3 t^2+6 t-7+6 t^{-1} -3 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+3 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 27, -2 }
Jones polynomial	[math]\displaystyle{ -1+3 q^{-1} -3 q^{-2} +5 q^{-3} -5 q^{-4} +4 q^{-5} -3 q^{-6} +2 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^6-3 z^2 a^6-2 a^6+z^6 a^4+5 z^4 a^4+8 z^2 a^4+3 a^4-z^4 a^2-2 z^2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+3 z^2 a^8+2 z^7 a^7-6 z^5 a^7+5 z^3 a^7-2 z a^7+z^8 a^6-2 z^6 a^6+2 z^4 a^6-3 z^2 a^6+2 a^6+3 z^7 a^5-10 z^5 a^5+14 z^3 a^5-5 z a^5+z^8 a^4-4 z^6 a^4+11 z^4 a^4-10 z^2 a^4+3 a^4+z^7 a^3-3 z^5 a^3+7 z^3 a^3-3 z a^3+3 z^4 a^2-4 z^2 a^2+z^3 a-z a }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}-q^{20}+q^{16}-q^{14}+q^{12}+2 q^8+2 q^6+q^2-1 }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-q^{126}+2 q^{124}-3 q^{122}+2 q^{120}-q^{118}-2 q^{116}+7 q^{114}-8 q^{112}+9 q^{110}-7 q^{108}+5 q^{104}-13 q^{102}+15 q^{100}-11 q^{98}+2 q^{96}+6 q^{94}-11 q^{92}+11 q^{90}-6 q^{88}-4 q^{86}+8 q^{84}-13 q^{82}+7 q^{80}+2 q^{78}-12 q^{76}+18 q^{74}-14 q^{72}+8 q^{70}+2 q^{68}-11 q^{66}+14 q^{64}-19 q^{62}+15 q^{60}-3 q^{58}-4 q^{56}+10 q^{54}-14 q^{52}+14 q^{50}-q^{48}-5 q^{46}+5 q^{44}-9 q^{42}+9 q^{40}+8 q^{38}-13 q^{36}+14 q^{34}-7 q^{32}+3 q^{30}+10 q^{28}-16 q^{26}+11 q^{24}-6 q^{22}+5 q^{20}+2 q^{18}-8 q^{16}+7 q^{14}-3 q^{12}+3 q^{10}-q^8-q^6-q^4-q^2+1- q^{-2} + q^{-4} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_143.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{17}+q^{15}-q^{13}+q^{11}-q^9+2 q^5+2 q- q^{-1} }[/math]
2	[math]\displaystyle{ q^{48}-q^{46}-q^{44}+3 q^{42}-q^{40}-4 q^{38}+3 q^{36}+2 q^{34}-5 q^{32}+q^{30}+4 q^{28}-3 q^{26}+3 q^{22}-q^{20}-4 q^{18}-q^{16}+4 q^{14}-3 q^{12}-q^{10}+7 q^8-2 q^4+4 q^2-2 q^{-2} }[/math]
3	[math]\displaystyle{ -q^{93}+q^{91}+q^{89}-q^{87}-2 q^{85}+q^{83}+5 q^{81}-7 q^{77}-3 q^{75}+7 q^{73}+8 q^{71}-5 q^{69}-13 q^{67}+14 q^{63}+7 q^{61}-13 q^{59}-12 q^{57}+10 q^{55}+15 q^{53}-6 q^{51}-15 q^{49}+3 q^{47}+12 q^{45}-q^{43}-10 q^{41}-2 q^{39}+10 q^{37}+4 q^{35}-4 q^{33}-10 q^{31}+q^{29}+11 q^{27}+q^{25}-16 q^{23}-7 q^{21}+15 q^{19}+11 q^{17}-10 q^{15}-12 q^{13}+9 q^{11}+14 q^9+q^7-9 q^5-q^3+4 q+4 q^{-1} -2 q^{-3} -3 q^{-5} - q^{-7} + q^{-11} }[/math]
5	[math]\displaystyle{ -q^{225}+q^{223}+q^{221}-q^{219}+q^{211}+2 q^{209}-2 q^{207}-5 q^{205}-2 q^{203}+q^{201}+6 q^{199}+10 q^{197}+6 q^{195}-10 q^{193}-20 q^{191}-14 q^{189}+q^{187}+24 q^{185}+33 q^{183}+18 q^{181}-18 q^{179}-47 q^{177}-45 q^{175}-12 q^{173}+41 q^{171}+77 q^{169}+61 q^{167}-7 q^{165}-82 q^{163}-110 q^{161}-63 q^{159}+45 q^{157}+139 q^{155}+144 q^{153}+33 q^{151}-125 q^{149}-205 q^{147}-136 q^{145}+56 q^{143}+229 q^{141}+231 q^{139}+40 q^{137}-199 q^{135}-287 q^{133}-141 q^{131}+129 q^{129}+300 q^{127}+217 q^{125}-49 q^{123}-268 q^{121}-250 q^{119}-22 q^{117}+209 q^{115}+246 q^{113}+68 q^{111}-149 q^{109}-210 q^{107}-84 q^{105}+93 q^{103}+162 q^{101}+81 q^{99}-56 q^{97}-118 q^{95}-61 q^{93}+32 q^{91}+81 q^{89}+54 q^{87}-14 q^{85}-71 q^{83}-55 q^{81}+3 q^{79}+54 q^{77}+78 q^{75}+28 q^{73}-62 q^{71}-112 q^{69}-65 q^{67}+55 q^{65}+155 q^{63}+129 q^{61}-27 q^{59}-192 q^{57}-201 q^{55}-24 q^{53}+205 q^{51}+268 q^{49}+97 q^{47}-181 q^{45}-316 q^{43}-182 q^{41}+112 q^{39}+311 q^{37}+246 q^{35}-22 q^{33}-259 q^{31}-272 q^{29}-75 q^{27}+167 q^{25}+257 q^{23}+139 q^{21}-60 q^{19}-179 q^{17}-160 q^{15}-24 q^{13}+106 q^{11}+134 q^9+68 q^7-23 q^5-83 q^3-73 q-17 q^{-1} +34 q^{-3} +46 q^{-5} +28 q^{-7} -2 q^{-9} -22 q^{-11} -23 q^{-13} -9 q^{-15} +5 q^{-17} +8 q^{-19} +7 q^{-21} +2 q^{-23} -2 q^{-25} -2 q^{-27} }[/math]
6	[math]\displaystyle{ q^{312}-q^{310}-q^{308}+q^{306}-2 q^{300}+2 q^{298}-2 q^{294}+4 q^{292}+3 q^{290}+q^{288}-6 q^{286}-2 q^{284}-6 q^{282}-8 q^{280}+6 q^{278}+14 q^{276}+19 q^{274}+3 q^{272}-22 q^{268}-40 q^{266}-22 q^{264}+6 q^{262}+44 q^{260}+47 q^{258}+54 q^{256}+10 q^{254}-57 q^{252}-94 q^{250}-87 q^{248}-25 q^{246}+37 q^{244}+136 q^{242}+155 q^{240}+90 q^{238}-35 q^{236}-160 q^{234}-220 q^{232}-207 q^{230}-24 q^{228}+183 q^{226}+330 q^{224}+318 q^{222}+144 q^{220}-155 q^{218}-457 q^{216}-501 q^{214}-287 q^{212}+138 q^{210}+538 q^{208}+719 q^{206}+481 q^{204}-108 q^{202}-668 q^{200}-921 q^{198}-639 q^{196}+44 q^{194}+810 q^{192}+1127 q^{190}+758 q^{188}-73 q^{186}-927 q^{184}-1253 q^{182}-834 q^{180}+168 q^{178}+1064 q^{176}+1300 q^{174}+741 q^{172}-292 q^{170}-1132 q^{168}-1261 q^{166}-539 q^{164}+471 q^{162}+1125 q^{160}+1038 q^{158}+294 q^{156}-587 q^{154}-1034 q^{152}-735 q^{150}-19 q^{148}+621 q^{146}+788 q^{144}+432 q^{142}-157 q^{140}-569 q^{138}-516 q^{136}-156 q^{134}+239 q^{132}+401 q^{130}+284 q^{128}-241 q^{124}-257 q^{122}-110 q^{120}+99 q^{118}+195 q^{116}+161 q^{114}+16 q^{112}-144 q^{110}-208 q^{108}-130 q^{106}+70 q^{104}+220 q^{102}+266 q^{100}+119 q^{98}-146 q^{96}-392 q^{94}-394 q^{92}-81 q^{90}+319 q^{88}+607 q^{86}+505 q^{84}+29 q^{82}-593 q^{80}-880 q^{78}-560 q^{76}+159 q^{74}+893 q^{72}+1090 q^{70}+575 q^{68}-422 q^{66}-1191 q^{64}-1195 q^{62}-443 q^{60}+652 q^{58}+1350 q^{56}+1212 q^{54}+254 q^{52}-833 q^{50}-1366 q^{48}-1076 q^{46}-128 q^{44}+854 q^{42}+1280 q^{40}+879 q^{38}+23 q^{36}-753 q^{34}-1050 q^{32}-717 q^{30}-11 q^{28}+622 q^{26}+795 q^{24}+536 q^{22}+48 q^{20}-409 q^{18}-574 q^{16}-399 q^{14}-46 q^{12}+238 q^{10}+356 q^8+290 q^6+79 q^4-125 q^2-210-171 q^{-2} -74 q^{-4} +32 q^{-6} +106 q^{-8} +103 q^{-10} +49 q^{-12} -6 q^{-14} -38 q^{-16} -48 q^{-18} -35 q^{-20} -5 q^{-22} +12 q^{-24} +16 q^{-26} +12 q^{-28} +7 q^{-30} + q^{-32} -5 q^{-34} -2 q^{-36} - q^{-38} - q^{-40} - q^{-42} + q^{-46} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{24}-q^{20}+q^{16}-q^{14}+q^{12}+2 q^8+2 q^6+q^2-1 }[/math]
1,1	[math]\displaystyle{ q^{68}-2 q^{66}+4 q^{64}-8 q^{62}+15 q^{60}-20 q^{58}+26 q^{56}-34 q^{54}+37 q^{52}-34 q^{50}+24 q^{48}-14 q^{46}-5 q^{44}+24 q^{42}-42 q^{40}+58 q^{38}-61 q^{36}+66 q^{34}-62 q^{32}+50 q^{30}-42 q^{28}+18 q^{26}-6 q^{24}-14 q^{22}+24 q^{20}-32 q^{18}+42 q^{16}-28 q^{14}+30 q^{12}-14 q^{10}+16 q^8-6 q^6+2 q^4-4 q^2-2 q^{-2} + q^{-4} }[/math]
2,0	[math]\displaystyle{ q^{62}+q^{56}+q^{54}-q^{52}-2 q^{50}-q^{48}-3 q^{44}-2 q^{42}+2 q^{40}+q^{38}+q^{36}+2 q^{34}+3 q^{32}-3 q^{28}-3 q^{26}-4 q^{24}-5 q^{22}+q^{20}+4 q^{18}+3 q^{16}+4 q^{14}+6 q^{12}+3 q^{10}-q^6+q^4-q^2-2 }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{54}-q^{52}+q^{48}-2 q^{46}+2 q^{44}+2 q^{42}-3 q^{40}+2 q^{38}-6 q^{34}-2 q^{32}-q^{30}-3 q^{28}+2 q^{24}+3 q^{22}+2 q^{20}+q^{18}+6 q^{16}+5 q^{10}-q^8-3 q^6+2 q^4-q^2-1+ q^{-2} }[/math]
1,0,0	[math]\displaystyle{ -q^{31}-2 q^{27}-q^{23}+q^{21}+q^{17}+q^{15}+q^{13}+2 q^{11}+q^9+2 q^7-q^5+q^3-q }[/math]
1,0,1	[math]\displaystyle{ q^{88}-2 q^{86}+3 q^{84}-3 q^{82}+q^{80}+6 q^{78}-11 q^{76}+13 q^{74}-10 q^{72}+13 q^{68}-21 q^{66}+26 q^{64}-25 q^{62}+8 q^{60}+4 q^{58}-25 q^{56}+31 q^{54}-23 q^{52}+23 q^{50}-4 q^{48}+9 q^{46}-7 q^{44}-5 q^{42}-4 q^{40}-25 q^{38}+20 q^{36}-38 q^{34}+34 q^{32}-20 q^{30}+4 q^{28}+22 q^{26}-24 q^{24}+32 q^{22}-9 q^{20}+12 q^{18}+11 q^{16}+q^{14}+7 q^{12}+q^{10}-6 q^8-q^6-2 q^4-4 q^2+3-2 q^{-2} + q^{-4} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{68}+q^{62}-q^{60}-q^{58}+3 q^{56}+2 q^{54}-q^{52}+2 q^{50}+2 q^{48}-4 q^{46}-9 q^{44}-6 q^{42}-6 q^{40}-10 q^{38}-3 q^{36}+5 q^{34}+3 q^{32}+5 q^{30}+12 q^{28}+7 q^{26}+3 q^{24}+5 q^{22}+4 q^{20}-q^{18}-3 q^{16}+2 q^{14}-3 q^{10}-q^8+2 q^6-q^4+1 }[/math]
1,0,0,0	[math]\displaystyle{ -q^{38}-2 q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+2 q^{22}+q^{20}+2 q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^8-q^6+q^4-q^2 }[/math]

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{74}-q^{72}+q^{70}-2 q^{68}+3 q^{66}-3 q^{64}+3 q^{62}-3 q^{60}+4 q^{58}-2 q^{56}+2 q^{54}-q^{52}-q^{50}-q^{48}-6 q^{46}-7 q^{42}+2 q^{40}-8 q^{38}+6 q^{36}-4 q^{34}+9 q^{32}-q^{30}+7 q^{28}+q^{26}+5 q^{24}+3 q^{22}+2 q^{18}-2 q^{16}+4 q^{14}-3 q^{12}+2 q^{10}-4 q^8+3 q^6-2 q^4+q^2-1+ q^{-2} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{128}-q^{126}+2 q^{124}-3 q^{122}+2 q^{120}-q^{118}-2 q^{116}+7 q^{114}-8 q^{112}+9 q^{110}-7 q^{108}+5 q^{104}-13 q^{102}+15 q^{100}-11 q^{98}+2 q^{96}+6 q^{94}-11 q^{92}+11 q^{90}-6 q^{88}-4 q^{86}+8 q^{84}-13 q^{82}+7 q^{80}+2 q^{78}-12 q^{76}+18 q^{74}-14 q^{72}+8 q^{70}+2 q^{68}-11 q^{66}+14 q^{64}-19 q^{62}+15 q^{60}-3 q^{58}-4 q^{56}+10 q^{54}-14 q^{52}+14 q^{50}-q^{48}-5 q^{46}+5 q^{44}-9 q^{42}+9 q^{40}+8 q^{38}-13 q^{36}+14 q^{34}-7 q^{32}+3 q^{30}+10 q^{28}-16 q^{26}+11 q^{24}-6 q^{22}+5 q^{20}+2 q^{18}-8 q^{16}+7 q^{14}-3 q^{12}+3 q^{10}-q^8-q^6-q^4-q^2+1- q^{-2} + q^{-4} }[/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["10 143"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^3-3 t^2+6 t-7+6 t^{-1} -3 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+3 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]=

{ 27, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -1+3 q^{-1} -3 q^{-2} +5 q^{-3} -5 q^{-4} +4 q^{-5} -3 q^{-6} +2 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^6-3 z^2 a^6-2 a^6+z^6 a^4+5 z^4 a^4+8 z^2 a^4+3 a^4-z^4 a^2-2 z^2 a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+3 z^2 a^8+2 z^7 a^7-6 z^5 a^7+5 z^3 a^7-2 z a^7+z^8 a^6-2 z^6 a^6+2 z^4 a^6-3 z^2 a^6+2 a^6+3 z^7 a^5-10 z^5 a^5+14 z^3 a^5-5 z a^5+z^8 a^4-4 z^6 a^4+11 z^4 a^4-10 z^2 a^4+3 a^4+z^7 a^3-3 z^5 a^3+7 z^3 a^3-3 z a^3+3 z^4 a^2-4 z^2 a^2+z^3 a-z a }[/math]