10 149

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-9 t+11-9 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 41, -4 }
Jones polynomial	[math]\displaystyle{ 2 q^{-2} -3 q^{-3} +6 q^{-4} -7 q^{-5} +7 q^{-6} -7 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8+2 z^2 a^8+a^8-z^6 a^6-4 z^4 a^6-6 z^2 a^6-4 a^6+2 z^4 a^4+6 z^2 a^4+4 a^4 }[/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}+z a^{11}+4 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-2 z^5 a^9-z^3 a^9+z a^9+z^8 a^8+3 z^6 a^8-4 z^4 a^8+a^8+4 z^7 a^7-6 z^5 a^7+5 z^3 a^7-3 z a^7+z^8 a^6-z^6 a^6+5 z^4 a^6-9 z^2 a^6+4 a^6+z^7 a^5-z^5 a^5+2 z^3 a^5-3 z a^5+3 z^4 a^4-7 z^2 a^4+4 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ q^{30}-q^{28}+q^{26}-2 q^{22}-3 q^{18}+q^{16}+q^{12}+3 q^{10}+2 q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+13 q^{148}-20 q^{146}+24 q^{144}-21 q^{142}+7 q^{140}+11 q^{138}-31 q^{136}+48 q^{134}-46 q^{132}+31 q^{130}-3 q^{128}-28 q^{126}+47 q^{124}-46 q^{122}+28 q^{120}+2 q^{118}-28 q^{116}+38 q^{114}-23 q^{112}-6 q^{110}+39 q^{108}-57 q^{106}+50 q^{104}-22 q^{102}-21 q^{100}+56 q^{98}-74 q^{96}+70 q^{94}-45 q^{92}+4 q^{90}+32 q^{88}-58 q^{86}+59 q^{84}-45 q^{82}+12 q^{80}+17 q^{78}-36 q^{76}+33 q^{74}-16 q^{72}-13 q^{70}+38 q^{68}-47 q^{66}+29 q^{64}+q^{62}-35 q^{60}+59 q^{58}-55 q^{56}+36 q^{54}-4 q^{52}-22 q^{50}+39 q^{48}-37 q^{46}+28 q^{44}-7 q^{42}-4 q^{40}+12 q^{38}-11 q^{36}+9 q^{34}-q^{32}+q^{30}+q^{28} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_149.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{21}-2 q^{19}+2 q^{17}-2 q^{15}-q^9+3 q^7-q^5+2 q^3 }[/math]
2	[math]\displaystyle{ q^{58}-2 q^{56}-q^{54}+6 q^{52}-5 q^{50}-6 q^{48}+13 q^{46}-2 q^{44}-12 q^{42}+11 q^{40}+3 q^{38}-9 q^{36}+2 q^{34}+5 q^{32}-2 q^{30}-8 q^{28}+5 q^{26}+6 q^{24}-13 q^{22}+2 q^{20}+12 q^{18}-10 q^{16}-2 q^{14}+10 q^{12}-2 q^{10}-3 q^8+3 q^6+q^4 }[/math]
3	[math]\displaystyle{ q^{111}-2 q^{109}-q^{107}+3 q^{105}+3 q^{103}-5 q^{101}-9 q^{99}+11 q^{97}+17 q^{95}-12 q^{93}-31 q^{91}+7 q^{89}+48 q^{87}+4 q^{85}-60 q^{83}-21 q^{81}+58 q^{79}+39 q^{77}-51 q^{75}-47 q^{73}+36 q^{71}+49 q^{69}-16 q^{67}-43 q^{65}-q^{63}+34 q^{61}+18 q^{59}-26 q^{57}-30 q^{55}+14 q^{53}+44 q^{51}-6 q^{49}-53 q^{47}-7 q^{45}+61 q^{43}+17 q^{41}-58 q^{39}-35 q^{37}+49 q^{35}+44 q^{33}-36 q^{31}-47 q^{29}+15 q^{27}+44 q^{25}+2 q^{23}-30 q^{21}-10 q^{19}+17 q^{17}+12 q^{15}-3 q^{13}-8 q^{11}+q^9+2 q^7+2 q^5 }[/math]
5	[math]\displaystyle{ q^{265}-2 q^{263}-q^{261}+3 q^{259}-3 q^{251}-3 q^{249}+8 q^{247}+9 q^{245}-5 q^{243}-16 q^{241}-19 q^{239}+35 q^{235}+61 q^{233}+19 q^{231}-78 q^{229}-130 q^{227}-77 q^{225}+90 q^{223}+254 q^{221}+237 q^{219}-50 q^{217}-407 q^{215}-487 q^{213}-132 q^{211}+489 q^{209}+839 q^{207}+497 q^{205}-424 q^{203}-1181 q^{201}-1008 q^{199}+121 q^{197}+1362 q^{195}+1573 q^{193}+408 q^{191}-1289 q^{189}-2030 q^{187}-1032 q^{185}+935 q^{183}+2205 q^{181}+1615 q^{179}-379 q^{177}-2080 q^{175}-1982 q^{173}-195 q^{171}+1672 q^{169}+2053 q^{167}+692 q^{165}-1136 q^{163}-1863 q^{161}-989 q^{159}+595 q^{157}+1498 q^{155}+1088 q^{153}-133 q^{151}-1092 q^{149}-1058 q^{147}-191 q^{145}+721 q^{143}+962 q^{141}+420 q^{139}-438 q^{137}-905 q^{135}-587 q^{133}+246 q^{131}+892 q^{129}+771 q^{127}-103 q^{125}-963 q^{123}-979 q^{121}-44 q^{119}+1061 q^{117}+1266 q^{115}+240 q^{113}-1147 q^{111}-1559 q^{109}-535 q^{107}+1113 q^{105}+1869 q^{103}+911 q^{101}-944 q^{99}-2033 q^{97}-1342 q^{95}+562 q^{93}+2024 q^{91}+1742 q^{89}-52 q^{87}-1759 q^{85}-1965 q^{83}-540 q^{81}+1255 q^{79}+1947 q^{77}+1046 q^{75}-620 q^{73}-1646 q^{71}-1313 q^{69}-29 q^{67}+1126 q^{65}+1306 q^{63}+502 q^{61}-548 q^{59}-1030 q^{57}-714 q^{55}+48 q^{53}+627 q^{51}+663 q^{49}+249 q^{47}-245 q^{45}-458 q^{43}-317 q^{41}-9 q^{39}+211 q^{37}+249 q^{35}+117 q^{33}-54 q^{31}-123 q^{29}-95 q^{27}-29 q^{25}+35 q^{23}+58 q^{21}+30 q^{19}+q^{17}-10 q^{15}-17 q^{13}-8 q^{11}+q^9+4 q^7+2 q^5+2 q^3 }[/math]
6	[math]\displaystyle{ q^{366}-2 q^{364}-q^{362}+3 q^{360}-3 q^{354}+5 q^{352}-2 q^{350}-7 q^{348}+11 q^{346}+2 q^{344}-6 q^{342}-18 q^{340}+3 q^{338}+3 q^{336}+6 q^{334}+52 q^{332}+29 q^{330}-27 q^{328}-106 q^{326}-77 q^{324}-49 q^{322}+52 q^{320}+260 q^{318}+273 q^{316}+92 q^{314}-295 q^{312}-505 q^{310}-564 q^{308}-210 q^{306}+614 q^{304}+1189 q^{302}+1124 q^{300}+131 q^{298}-1116 q^{296}-2211 q^{294}-2045 q^{292}-176 q^{290}+2251 q^{288}+3812 q^{286}+2998 q^{284}+186 q^{282}-3777 q^{280}-5987 q^{278}-4367 q^{276}+433 q^{274}+5897 q^{272}+8181 q^{270}+5703 q^{268}-1382 q^{266}-8446 q^{264}-10636 q^{262}-6118 q^{260}+2935 q^{258}+10738 q^{256}+12548 q^{254}+5828 q^{252}-4946 q^{250}-12943 q^{248}-12903 q^{246}-4535 q^{244}+6735 q^{242}+14172 q^{240}+12093 q^{238}+2490 q^{236}-8445 q^{234}-13670 q^{232}-10081 q^{230}-446 q^{228}+9229 q^{226}+12125 q^{224}+7406 q^{222}-1548 q^{220}-8668 q^{218}-9805 q^{216}-4859 q^{214}+2751 q^{212}+7553 q^{210}+7320 q^{208}+2586 q^{206}-3049 q^{204}-6218 q^{202}-5220 q^{200}-1048 q^{198}+3168 q^{196}+5101 q^{194}+3559 q^{192}-3430 q^{188}-4384 q^{186}-2400 q^{184}+1196 q^{182}+4070 q^{180}+4003 q^{178}+1157 q^{176}-2854 q^{174}-4952 q^{172}-3683 q^{170}+646 q^{168}+5007 q^{166}+5952 q^{164}+2647 q^{162}-3199 q^{160}-7211 q^{158}-6588 q^{156}-735 q^{154}+6290 q^{152}+9322 q^{150}+6044 q^{148}-1983 q^{146}-9142 q^{144}-10766 q^{142}-4600 q^{140}+5071 q^{138}+11688 q^{136}+10782 q^{134}+2503 q^{132}-7581 q^{130}-13317 q^{128}-9992 q^{126}-241 q^{124}+9619 q^{122}+13372 q^{120}+8614 q^{118}-1344 q^{116}-10625 q^{114}-12652 q^{112}-7014 q^{110}+2585 q^{108}+10140 q^{106}+11276 q^{104}+5779 q^{102}-3062 q^{100}-9058 q^{98}-9496 q^{96}-4483 q^{94}+2537 q^{92}+7492 q^{90}+7793 q^{88}+3493 q^{86}-1894 q^{84}-5681 q^{82}-5884 q^{80}-2989 q^{78}+1166 q^{76}+4066 q^{74}+4208 q^{72}+2353 q^{70}-475 q^{68}-2517 q^{66}-2981 q^{64}-1757 q^{62}+78 q^{60}+1384 q^{58}+1842 q^{56}+1258 q^{54}+220 q^{52}-730 q^{50}-1022 q^{48}-772 q^{46}-272 q^{44}+260 q^{42}+509 q^{40}+462 q^{38}+180 q^{36}-51 q^{34}-191 q^{32}-217 q^{30}-125 q^{28}-11 q^{26}+71 q^{24}+74 q^{22}+55 q^{20}+23 q^{18}-9 q^{16}-24 q^{14}-19 q^{12}-7 q^{10}-q^8+3 q^6+3 q^4+3 q^2+1 }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{30}-q^{28}+q^{26}-2 q^{22}-3 q^{18}+q^{16}+q^{12}+3 q^{10}+2 q^6 }[/math]
1,1	[math]\displaystyle{ q^{84}-4 q^{82}+10 q^{80}-20 q^{78}+36 q^{76}-60 q^{74}+88 q^{72}-120 q^{70}+152 q^{68}-174 q^{66}+178 q^{64}-158 q^{62}+114 q^{60}-40 q^{58}-52 q^{56}+148 q^{54}-240 q^{52}+308 q^{50}-348 q^{48}+356 q^{46}-324 q^{44}+270 q^{42}-186 q^{40}+92 q^{38}-q^{36}-88 q^{34}+146 q^{32}-192 q^{30}+193 q^{28}-180 q^{26}+146 q^{24}-104 q^{22}+69 q^{20}-32 q^{18}+20 q^{16}+2 q^{12}+2 q^{10} }[/math]
2,0	[math]\displaystyle{ q^{76}-q^{74}-q^{72}+2 q^{70}-q^{68}-2 q^{66}-q^{64}+4 q^{62}+q^{60}-6 q^{58}+2 q^{56}+5 q^{54}-q^{52}-q^{50}+5 q^{48}+3 q^{46}-4 q^{44}-2 q^{42}-5 q^{38}-6 q^{36}+3 q^{34}-q^{32}-6 q^{30}+3 q^{28}+4 q^{26}-q^{24}-2 q^{22}+5 q^{20}+5 q^{18}-q^{16}+4 q^{12}+q^{10} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{68}-2 q^{66}+4 q^{62}-6 q^{60}+8 q^{56}-9 q^{54}+10 q^{50}-7 q^{48}+6 q^{44}-q^{40}+3 q^{36}-4 q^{34}-11 q^{32}+2 q^{30}-2 q^{28}-12 q^{26}+9 q^{24}+5 q^{22}-3 q^{20}+9 q^{18}+4 q^{16}-q^{14}+3 q^{12} }[/math]
1,0,0	[math]\displaystyle{ q^{39}-q^{37}+2 q^{35}-q^{33}+q^{31}-2 q^{29}-q^{27}-3 q^{25}-2 q^{23}+3 q^{17}+q^{15}+4 q^{13}+2 q^9 }[/math]
1,0,1	[math]\displaystyle{ q^{110}-4 q^{108}+8 q^{106}-8 q^{104}-2 q^{102}+23 q^{100}-44 q^{98}+45 q^{96}-13 q^{94}-50 q^{92}+111 q^{90}-130 q^{88}+81 q^{86}+27 q^{84}-142 q^{82}+211 q^{80}-188 q^{78}+83 q^{76}+54 q^{74}-158 q^{72}+179 q^{70}-123 q^{68}+30 q^{66}+22 q^{64}-16 q^{62}-40 q^{60}+85 q^{58}-54 q^{56}-24 q^{54}+156 q^{52}-216 q^{50}+211 q^{48}-112 q^{46}-39 q^{44}+133 q^{42}-206 q^{40}+141 q^{38}-82 q^{36}-20 q^{34}+64 q^{32}-61 q^{30}+52 q^{28}+2 q^{26}+4 q^{24}+18 q^{22}+q^{20}+3 q^{18}+2 q^{16} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{86}-q^{84}-q^{82}+4 q^{80}-q^{78}-7 q^{76}+2 q^{74}+4 q^{72}-6 q^{70}-4 q^{68}+7 q^{66}+3 q^{64}-5 q^{62}+4 q^{60}+12 q^{58}-q^{56}-q^{54}+13 q^{52}-q^{50}-11 q^{48}-q^{46}-4 q^{44}-19 q^{42}-12 q^{40}-q^{38}-3 q^{36}-6 q^{34}+6 q^{32}+13 q^{30}+4 q^{28}+6 q^{26}+9 q^{24}+4 q^{22}+3 q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ q^{48}-q^{46}+2 q^{44}+q^{38}-2 q^{36}-q^{34}-4 q^{32}-2 q^{30}-3 q^{28}+3 q^{22}+3 q^{20}+2 q^{18}+4 q^{16}+2 q^{12} }[/math]

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{94}-2 q^{92}+2 q^{90}-3 q^{88}+5 q^{86}-7 q^{84}+6 q^{82}-7 q^{80}+9 q^{78}-8 q^{76}+5 q^{74}-4 q^{72}+4 q^{70}+2 q^{68}-5 q^{66}+6 q^{64}-7 q^{62}+14 q^{60}-12 q^{58}+14 q^{56}-13 q^{54}+15 q^{52}-12 q^{50}+7 q^{48}-14 q^{46}-6 q^{42}-6 q^{40}-2 q^{38}-7 q^{36}+9 q^{34}-4 q^{32}+12 q^{30}-3 q^{28}+12 q^{26}-q^{24}+7 q^{22}-2 q^{20}+3 q^{18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+13 q^{148}-20 q^{146}+24 q^{144}-21 q^{142}+7 q^{140}+11 q^{138}-31 q^{136}+48 q^{134}-46 q^{132}+31 q^{130}-3 q^{128}-28 q^{126}+47 q^{124}-46 q^{122}+28 q^{120}+2 q^{118}-28 q^{116}+38 q^{114}-23 q^{112}-6 q^{110}+39 q^{108}-57 q^{106}+50 q^{104}-22 q^{102}-21 q^{100}+56 q^{98}-74 q^{96}+70 q^{94}-45 q^{92}+4 q^{90}+32 q^{88}-58 q^{86}+59 q^{84}-45 q^{82}+12 q^{80}+17 q^{78}-36 q^{76}+33 q^{74}-16 q^{72}-13 q^{70}+38 q^{68}-47 q^{66}+29 q^{64}+q^{62}-35 q^{60}+59 q^{58}-55 q^{56}+36 q^{54}-4 q^{52}-22 q^{50}+39 q^{48}-37 q^{46}+28 q^{44}-7 q^{42}-4 q^{40}+12 q^{38}-11 q^{36}+9 q^{34}-q^{32}+q^{30}+q^{28} }[/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["10 149"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-9 t+11-9 t^{-1} +5 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-z^4+2 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]=

{ 41, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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