10 159: Difference between revisions

Revision as of 20:48, 27 August 2005

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10 159 Quick Notes

10 159 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₆₂₇ X₃₉₄₈ X_18,11,19,12 X_20,13,1,14 X_15,2,16,3 X_17,5,18,4 X_12,19,13,20 X_5,10,6,11 X_7,15,8,14 X_9,16,10,17
Gauss code	-1, 5, -2, 6, -8, 1, -9, 2, -10, 8, 3, -7, 4, 9, -5, 10, -6, -3, 7, -4
Dowker-Thistlethwaite code	6 8 10 14 16 -18 -20 2 4 -12
Conway Notation	[-30:2:20]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-10][0]
Hyperbolic Volume	11.7406
A-Polynomial	See Data:10 159/A-polynomial

[edit Notes for 10 159's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	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}
Topological 4 genus	$1$
Concordance genus	$3$
Rasmussen s-Invariant	-2

[edit Notes for 10 159's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+9t-11+9t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 39, -2 }
Jones polynomial	$-1+4q^{-1}-5q^{-2}+7q^{-3}-7q^{-4}+6q^{-5}-5q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-2z^{2}a^{6}-a^{6}+z^{6}a^{4}+4z^{4}a^{4}+5z^{2}a^{4}+a^{4}-z^{4}a^{2}-z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+3z^{2}a^{8}+3z^{7}a^{7}-5z^{5}a^{7}-z^{3}a^{7}+za^{7}+z^{8}a^{6}+3z^{6}a^{6}-8z^{4}a^{6}+z^{2}a^{6}+a^{6}+4z^{7}a^{5}-5z^{5}a^{5}+za^{5}+z^{8}a^{4}+3z^{4}a^{4}-4z^{2}a^{4}+a^{4}+z^{7}a^{3}+z^{5}a^{3}+za^{3}+4z^{4}a^{2}-2z^{2}a^{2}-a^{2}+z^{3}a$
The A2 invariant	$-q^{24}+q^{22}-q^{20}+q^{16}-2q^{14}+q^{12}-q^{10}+2q^{8}+2q^{6}+2q^{2}-1$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+7q^{120}-3q^{118}-6q^{116}+20q^{114}-28q^{112}+31q^{110}-22q^{108}-4q^{106}+28q^{104}-49q^{102}+51q^{100}-33q^{98}+2q^{96}+30q^{94}-46q^{92}+39q^{90}-14q^{88}-16q^{86}+37q^{84}-41q^{82}+19q^{80}+16q^{78}-43q^{76}+58q^{74}-48q^{72}+22q^{70}+12q^{68}-44q^{66}+59q^{64}-62q^{62}+43q^{60}-10q^{58}-25q^{56}+47q^{54}-51q^{52}+35q^{50}-6q^{48}-24q^{46}+37q^{44}-31q^{42}+8q^{40}+26q^{38}-46q^{36}+50q^{34}-24q^{32}-8q^{30}+37q^{28}-49q^{26}+44q^{24}-22q^{22}+2q^{20}+16q^{18}-24q^{16}+22q^{14}-12q^{12}+6q^{10}+2q^{8}-4q^{6}+q^{4}-2q^{2}+2-2q^{-2}+q^{-4}$

Further Quantum Invariants

Further quantum knot invariants for 10_159.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-2q^{13}+q^{11}-q^{9}+2q^{5}-q^{3}+3q-q^{-1}$
2	$q^{48}-2q^{46}-2q^{44}+7q^{42}-q^{40}-10q^{38}+8q^{36}+6q^{34}-12q^{32}+2q^{30}+9q^{28}-7q^{26}-3q^{24}+6q^{22}-7q^{18}+10q^{14}-7q^{12}-6q^{10}+14q^{8}-2q^{6}-8q^{4}+8q^{2}+2-3q^{-2}$
3	$-q^{93}+2q^{91}+2q^{89}-3q^{87}-7q^{85}+q^{83}+17q^{81}+5q^{79}-23q^{77}-21q^{75}+21q^{73}+41q^{71}-10q^{69}-52q^{67}-11q^{65}+54q^{63}+33q^{61}-47q^{59}-48q^{57}+32q^{55}+53q^{53}-16q^{51}-52q^{49}+5q^{47}+47q^{45}+5q^{43}-36q^{41}-16q^{39}+28q^{37}+24q^{35}-17q^{33}-40q^{31}+4q^{29}+48q^{27}+14q^{25}-55q^{23}-33q^{21}+52q^{19}+48q^{17}-37q^{15}-52q^{13}+18q^{11}+49q^{9}+2q^{7}-35q^{5}-7q^{3}+15q+13q^{-1}-4q^{-3}-6q^{-5}-q^{-7}+q^{-11}$
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^{225}+2 q^{223}+2 q^{221}-3 q^{219}-3 q^{217}-3 q^{215}+8 q^{211}+17 q^{209}+5 q^{207}-19 q^{205}-34 q^{203}-31 q^{201}+8 q^{199}+67 q^{197}+100 q^{195}+37 q^{193}-85 q^{191}-182 q^{189}-170 q^{187}+6 q^{185}+254 q^{183}+379 q^{181}+205 q^{179}-191 q^{177}-553 q^{175}-570 q^{173}-108 q^{171}+576 q^{169}+954 q^{167}+613 q^{165}-288 q^{163}-1147 q^{161}-1235 q^{159}-315 q^{157}+1018 q^{155}+1719 q^{153}+1090 q^{151}-499 q^{149}-1867 q^{147}-1834 q^{145}-276 q^{143}+1631 q^{141}+2302 q^{139}+1087 q^{137}-1068 q^{135}-2400 q^{133}-1736 q^{131}+380 q^{129}+2158 q^{127}+2085 q^{125}+241 q^{123}-1709 q^{121}-2108 q^{119}-685 q^{117}+1213 q^{115}+1904 q^{113}+897 q^{111}-784 q^{109}-1587 q^{107}-915 q^{105}+468 q^{103}+1263 q^{101}+845 q^{99}-270 q^{97}-1016 q^{95}-760 q^{93}+140 q^{91}+838 q^{89}+760 q^{87}+2 q^{85}-764 q^{83}-857 q^{81}-200 q^{79}+680 q^{77}+1070 q^{75}+548 q^{73}-557 q^{71}-1327 q^{69}-1006 q^{67}+276 q^{65}+1528 q^{63}+1572 q^{61}+188 q^{59}-1555 q^{57}-2114 q^{55}-822 q^{53}+1313 q^{51}+2453 q^{49}+1516 q^{47}-779 q^{45}-2470 q^{43}-2099 q^{41}+52 q^{39}+2098 q^{37}+2352 q^{35}+694 q^{33}-1404 q^{31}-2210 q^{29}-1225 q^{27}+604 q^{25}+1720 q^{23}+1376 q^{21}+84 q^{19}-1029 q^{17}-1198 q^{15}-479 q^{13}+431 q^{11}+793 q^9+531 q^7+2 q^5-390 q^3-399 q-145 q^{-1} +112 q^{-3} +195 q^{-5} +135 q^{-7} +16 q^{-9} -61 q^{-11} -69 q^{-13} -31 q^{-15} +7 q^{-17} +15 q^{-19} +14 q^{-21} +5 q^{-23} -3 q^{-25} -3 q^{-27} }
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 q^{312}-2 q^{310}-2 q^{308}+3 q^{306}+3 q^{304}+3 q^{302}-4 q^{300}-8 q^{296}-17 q^{294}+4 q^{292}+19 q^{290}+34 q^{288}+18 q^{286}+9 q^{284}-43 q^{282}-100 q^{280}-80 q^{278}-7 q^{276}+118 q^{274}+185 q^{272}+239 q^{270}+75 q^{268}-222 q^{266}-453 q^{264}-498 q^{262}-214 q^{260}+241 q^{258}+868 q^{256}+1056 q^{254}+625 q^{252}-316 q^{250}-1349 q^{248}-1866 q^{246}-1489 q^{244}+144 q^{242}+1985 q^{240}+3123 q^{238}+2615 q^{236}+459 q^{234}-2521 q^{232}-4791 q^{230}-4348 q^{228}-1354 q^{226}+3101 q^{224}+6444 q^{222}+6618 q^{220}+2712 q^{218}-3620 q^{216}-8439 q^{214}-8989 q^{212}-4064 q^{210}+3838 q^{208}+10568 q^{206}+11424 q^{204}+5258 q^{202}-4465 q^{200}-12443 q^{198}-13299 q^{196}-6170 q^{194}+5480 q^{192}+14177 q^{190}+14421 q^{188}+5894 q^{186}-6669 q^{184}-15289 q^{182}-14629 q^{180}-4520 q^{178}+8197 q^{176}+15648 q^{174}+13209 q^{172}+2415 q^{170}-9513 q^{168}-15072 q^{166}-10522 q^{164}+283 q^{162}+10272 q^{160}+13020 q^{158}+7169 q^{156}-2786 q^{154}-10137 q^{152}-9989 q^{150}-3533 q^{148}+4582 q^{146}+8720 q^{144}+6635 q^{142}+416 q^{140}-5354 q^{138}-6624 q^{136}-3380 q^{134}+1833 q^{132}+5062 q^{130}+4453 q^{128}+764 q^{126}-3163 q^{124}-4462 q^{122}-2564 q^{120}+1217 q^{118}+3959 q^{116}+4007 q^{114}+1138 q^{112}-2712 q^{110}-4976 q^{108}-3971 q^{106}+56 q^{104}+4378 q^{102}+6410 q^{100}+4200 q^{98}-1200 q^{96}-6731 q^{94}-8368 q^{92}-4361 q^{90}+3030 q^{88}+9591 q^{86}+10372 q^{84}+4282 q^{82}-5692 q^{80}-12793 q^{78}-11929 q^{76}-3197 q^{74}+8601 q^{72}+15503 q^{70}+12786 q^{68}+1262 q^{66}-11398 q^{64}-17106 q^{62}-12176 q^{60}+719 q^{58}+13143 q^{56}+17409 q^{54}+10472 q^{52}-2507 q^{50}-13447 q^{48}-15869 q^{46}-8503 q^{44}+3412 q^{42}+12497 q^{40}+13179 q^{38}+6336 q^{36}-3402 q^{34}-10144 q^{32}-10240 q^{30}-4553 q^{28}+2904 q^{26}+7350 q^{24}+7190 q^{22}+3168 q^{20}-1851 q^{18}-4870 q^{16}-4657 q^{14}-2003 q^{12}+930 q^{10}+2755 q^8+2734 q^6+1306 q^4-392 q^2-1368-1357 q^{-2} -752 q^{-4} +46 q^{-6} +557 q^{-8} +618 q^{-10} +350 q^{-12} +38 q^{-14} -154 q^{-16} -218 q^{-18} -148 q^{-20} -35 q^{-22} +34 q^{-24} +49 q^{-26} +37 q^{-28} +19 q^{-30} -11 q^{-34} -5 q^{-36} - q^{-38} - q^{-40} - q^{-42} + q^{-46} }

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^{24}+q^{22}-q^{20}+q^{16}-2 q^{14}+q^{12}-q^{10}+2 q^8+2 q^6+2 q^2-1}
1,1	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^{68}-4 q^{66}+12 q^{64}-28 q^{62}+50 q^{60}-80 q^{58}+116 q^{56}-144 q^{54}+158 q^{52}-154 q^{50}+122 q^{48}-62 q^{46}-11 q^{44}+92 q^{42}-172 q^{40}+234 q^{38}-273 q^{36}+284 q^{34}-272 q^{32}+234 q^{30}-172 q^{28}+94 q^{26}-18 q^{24}-60 q^{22}+114 q^{20}-150 q^{18}+160 q^{16}-140 q^{14}+119 q^{12}-76 q^{10}+52 q^8-28 q^6+15 q^4-4 q^2+2-4 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^{62}-q^{60}-2 q^{58}+2 q^{56}+2 q^{54}-2 q^{52}-2 q^{50}+2 q^{48}+5 q^{46}-4 q^{44}-3 q^{42}+4 q^{40}-q^{38}-3 q^{36}-q^{34}+3 q^{32}-q^{30}-2 q^{28}+q^{26}-2 q^{24}-5 q^{22}+2 q^{20}+5 q^{18}-3 q^{16}+q^{14}+7 q^{12}+3 q^{10}-q^8-q^6+5 q^4-3}

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^{54}-2 q^{52}+q^{50}+3 q^{48}-6 q^{46}+4 q^{44}+3 q^{42}-9 q^{40}+6 q^{38}+2 q^{36}-8 q^{34}+2 q^{32}+3 q^{30}-3 q^{28}-q^{26}+q^{24}+2 q^{22}-3 q^{20}-4 q^{18}+9 q^{16}-3 q^{14}-2 q^{12}+12 q^{10}-2 q^8-3 q^6+6 q^4-2 q^2-2+ q^{-2} }
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^{31}+q^{29}-2 q^{27}+q^{25}-q^{23}+q^{21}-q^{19}+2 q^{11}+q^9+3 q^7-q^5+2 q^3-q}
1,0,1	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^{88}-4 q^{86}+10 q^{84}-14 q^{82}+7 q^{80}+15 q^{78}-48 q^{76}+71 q^{74}-64 q^{72}+19 q^{70}+54 q^{68}-120 q^{66}+153 q^{64}-133 q^{62}+60 q^{60}+37 q^{58}-117 q^{56}+150 q^{54}-124 q^{52}+68 q^{50}-22 q^{48}+9 q^{46}-27 q^{44}+36 q^{42}-15 q^{40}-50 q^{38}+121 q^{36}-173 q^{34}+156 q^{32}-91 q^{30}-14 q^{28}+99 q^{26}-138 q^{24}+123 q^{22}-57 q^{20}+q^{18}+51 q^{16}-41 q^{14}+30 q^{12}+6 q^{10}-15 q^8+11 q^6-2 q^4-7 q^2+6-4 q^{-2} + q^{-4} }

A4 Invariants.

Weight

Invariant

0,1,0,0

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^{68}-q^{66}+3 q^{62}-3 q^{60}-2 q^{58}+6 q^{56}+q^{54}-7 q^{52}+q^{50}+5 q^{48}-5 q^{46}-8 q^{44}+4 q^{42}+4 q^{40}-9 q^{38}+q^{36}+9 q^{34}-6 q^{32}-5 q^{30}+8 q^{28}-2 q^{26}-7 q^{24}+4 q^{22}+8 q^{20}-q^{18}-q^{16}+10 q^{14}+5 q^{12}-4 q^{10}+4 q^6-3 q^4-q^2+1}

1,0,0,0

-q^{38}+q^{36}-2q^{34}-q^{28}+q^{26}-q^{24}+q^{22}-q^{20}+q^{18}+2q^{14}+q^{12}+2q^{10}+2q^{8}-q^{6}+2q^{4}-q^{2}

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^{54}+2 q^{52}-5 q^{50}+7 q^{48}-8 q^{46}+10 q^{44}-9 q^{42}+7 q^{40}-4 q^{38}+4 q^{34}-10 q^{32}+13 q^{30}-17 q^{28}+17 q^{26}-17 q^{24}+14 q^{22}-9 q^{20}+6 q^{18}+q^{16}-3 q^{14}+8 q^{12}-8 q^{10}+10 q^8-9 q^6+8 q^4-4 q^2+2- q^{-2} }

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^{88}-2 q^{84}-2 q^{82}+3 q^{80}+5 q^{78}-2 q^{76}-7 q^{74}-2 q^{72}+9 q^{70}+6 q^{68}-8 q^{66}-9 q^{64}+3 q^{62}+10 q^{60}+q^{58}-9 q^{56}-4 q^{54}+6 q^{52}+4 q^{50}-4 q^{48}-5 q^{46}+3 q^{44}+6 q^{42}-q^{40}-8 q^{38}-q^{36}+7 q^{34}+2 q^{32}-7 q^{30}-6 q^{28}+7 q^{26}+8 q^{24}-3 q^{22}-9 q^{20}+3 q^{18}+11 q^{16}+5 q^{14}-6 q^{12}-6 q^{10}+3 q^8+7 q^6-3 q^2-2+ q^{-4} }

D4 Invariants.

Weight

Invariant

1,0,0,0

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^{74}-2 q^{72}+3 q^{70}-4 q^{68}+6 q^{66}-7 q^{64}+7 q^{62}-8 q^{60}+8 q^{58}-6 q^{56}+3 q^{54}-2 q^{52}+2 q^{48}-8 q^{46}+8 q^{44}-10 q^{42}+11 q^{40}-14 q^{38}+13 q^{36}-12 q^{34}+13 q^{32}-10 q^{30}+6 q^{28}-5 q^{26}+4 q^{24}+2 q^{22}-3 q^{20}+6 q^{18}-4 q^{16}+11 q^{14}-5 q^{12}+7 q^{10}-7 q^8+7 q^6-3 q^4+q^2-2+ q^{-2} }

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^{128}-2 q^{126}+5 q^{124}-8 q^{122}+7 q^{120}-3 q^{118}-6 q^{116}+20 q^{114}-28 q^{112}+31 q^{110}-22 q^{108}-4 q^{106}+28 q^{104}-49 q^{102}+51 q^{100}-33 q^{98}+2 q^{96}+30 q^{94}-46 q^{92}+39 q^{90}-14 q^{88}-16 q^{86}+37 q^{84}-41 q^{82}+19 q^{80}+16 q^{78}-43 q^{76}+58 q^{74}-48 q^{72}+22 q^{70}+12 q^{68}-44 q^{66}+59 q^{64}-62 q^{62}+43 q^{60}-10 q^{58}-25 q^{56}+47 q^{54}-51 q^{52}+35 q^{50}-6 q^{48}-24 q^{46}+37 q^{44}-31 q^{42}+8 q^{40}+26 q^{38}-46 q^{36}+50 q^{34}-24 q^{32}-8 q^{30}+37 q^{28}-49 q^{26}+44 q^{24}-22 q^{22}+2 q^{20}+16 q^{18}-24 q^{16}+22 q^{14}-12 q^{12}+6 q^{10}+2 q^8-4 q^6+q^4-2 q^2+2-2 q^{-2} + q^{-4} }

.

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.

In[3]:= K = Knot["10 159"];

In[4]:= Alexander[K][t]

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

Out[4]= $t^{3}-4t^{2}+9t-11+9t^{-1}-4t^{-2}+t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $z^{6}+2z^{4}+2z^{2}+1$

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]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 39, -2 }

In[8]:= Jones[K][q]

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

Out[8]= $-1+4q^{-1}-5q^{-2}+7q^{-3}-7q^{-4}+6q^{-5}-5q^{-6}+3q^{-7}-q^{-8}$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $-z^{4}a^{6}-2z^{2}a^{6}-a^{6}+z^{6}a^{4}+4z^{4}a^{4}+5z^{2}a^{4}+a^{4}-z^{4}a^{2}-z^{2}a^{2}+a^{2}$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+3z^{2}a^{8}+3z^{7}a^{7}-5z^{5}a^{7}-z^{3}a^{7}+za^{7}+z^{8}a^{6}+3z^{6}a^{6}-8z^{4}a^{6}+z^{2}a^{6}+a^{6}+4z^{7}a^{5}-5z^{5}a^{5}+za^{5}+z^{8}a^{4}+3z^{4}a^{4}-4z^{2}a^{4}+a^{4}+z^{7}a^{3}+z^{5}a^{3}+za^{3}+4z^{4}a^{2}-2z^{2}a^{2}-a^{2}+z^{3}a$

Vassiliev invariants

V₂ and V₃:

(2, -3)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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 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 -24}

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 32}

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 \frac{220}{3}}

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 -\frac{4}{3}}

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 -192}

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 -272}

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 -32}

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 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 \frac{256}{3}}

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 288}

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 \frac{1760}{3}}

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 -\frac{32}{3}}

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 \frac{16591}{15}}

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 \frac{3836}{15}}

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 \frac{484}{45}}

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 -\frac{175}{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 -\frac{449}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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 j} , alternation over 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 r} ). The squares with yellow highlighting are those on the "critical diagonals", where 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 j-2r=s+1} or 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 j-2r=s+1} , where 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 s=} -2 is the signature of 10 159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

χ

1

-1

3

-3

3

2

-1

-5

4

2

-7

3

0

-9

3

4

-1

-11

2

3

1

-13

1

3

-2

-15

2

-17

1

-1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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 \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 159]]
Out[2]=	10
In[3]:=	PD[Knot[10, 159]]
Out[3]=	PD[X[1, 6, 2, 7], X[3, 9, 4, 8], X[18, 11, 19, 12], X[20, 13, 1, 14], X[15, 2, 16, 3], X[17, 5, 18, 4], X[12, 19, 13, 20], X[5, 10, 6, 11], X[7, 15, 8, 14], X[9, 16, 10, 17]]
In[4]:=	GaussCode[Knot[10, 159]]
Out[4]=	GaussCode[-1, 5, -2, 6, -8, 1, -9, 2, -10, 8, 3, -7, 4, 9, -5, 10, -6, -3, 7, -4]
In[5]:=	BR[Knot[10, 159]]
Out[5]=	BR[3, {-1, -1, -1, -2, 1, -2, 1, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 159]][t]
Out[6]=	-3 4 9 2 3 -11 + t - -- + - + 9 t - 4 t + t 2 t t
In[7]:=	Conway[Knot[10, 159]][z]
Out[7]=	2 4 6 1 + 2 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 159]}
In[9]:=	{KnotDet[Knot[10, 159]], KnotSignature[Knot[10, 159]]}
Out[9]=	{39, -2}
In[10]:=	J=Jones[Knot[10, 159]][q]
Out[10]=	-8 3 5 6 7 7 5 4 -1 - q + -- - -- + -- - -- + -- - -- + - 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 159]}
In[12]:=	A2Invariant[Knot[10, 159]][q]
Out[12]=	-24 -22 -20 -16 2 -12 -10 2 2 2 -1 - q + q - q + q - --- + q - q + -- + -- + -- 14 8 6 2 q q q q
In[13]:=	Kauffman[Knot[10, 159]][a, z]
Out[13]=	2 4 6 3 5 7 9 2 2 4 2 6 2 -a + a + a + a z + a z + a z + a z - 2 a z - 4 a z + a z + 8 2 3 7 3 9 3 2 4 4 4 6 4 3 a z + a z - a z - 2 a z + 4 a z + 3 a z - 8 a z - 8 4 3 5 5 5 7 5 9 5 6 6 8 6 7 a z + a z - 5 a z - 5 a z + a z + 3 a z + 3 a z + 3 7 5 7 7 7 4 8 6 8 a z + 4 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 159]], Vassiliev[3][Knot[10, 159]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[10, 159]][q, t]
Out[15]=	2 3 1 2 1 3 2 3 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 4 3 3 4 2 3 ----- + ----- + ----- + ----- + ---- + ---- + q t 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t

@@ Line 1: / Line 1: @@
+<!--  -->
-{{Template:Basic Knot Invariants|name=10_159}}
+<!-- provide an anchor so we can return to the top of the page -->
+<span id="top"></span>
+<!-- this relies on transclusion for next and previous links -->
+{{Knot Navigation Links|ext=gif}}
+{| align=left
+|- valign=top
+|[[Image:{{PAGENAME}}.gif]]
+|{{Rolfsen Knot Site Links|n=10|k=159|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,5,-2,6,-8,1,-9,2,-10,8,3,-7,4,9,-5,10,-6,-3,7,-4/goTop.html}}
+|{{:{{PAGENAME}} Quick Notes}}
+|}
+<br style="clear:both" />
+{{:{{PAGENAME}} Further Notes and Views}}
+{{Knot Presentations}}
+{{3D Invariants}}
+{{4D Invariants}}
+{{Polynomial Invariants}}
+{{Vassiliev Invariants}}
+===[[Khovanov Homology]]===
+The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
+<center><table border=1>
+<tr align=center>
+<td width=15.3846%><table cellpadding=0 cellspacing=0>
+ <tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
+</table></td>
+ <td width=7.69231%>-7</td  ><td width=7.69231%>-6</td  ><td width=7.69231%>-5</td  ><td width=7.69231%>-4</td  ><td width=7.69231%>-3</td  ><td width=7.69231%>-2</td  ><td width=7.69231%>-1</td  ><td width=7.69231%>0</td  ><td width=7.69231%>1</td  ><td width=15.3846%>&chi;</td></tr>
+<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>-1</td></tr>
+<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>&nbsp;</td><td>3</td></tr>
+<tr align=center><td>-3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>-1</td></tr>
+<tr align=center><td>-5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
+<tr align=center><td>-7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
+<tr align=center><td>-9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
+<tr align=center><td>-11</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
+<tr align=center><td>-13</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
+<tr align=center><td>-15</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
+<tr align=center><td>-17</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
+</table></center>
+{{Computer Talk Header}}
+<table>
+<tr valign=top>
+<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
+<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
+</tr>
+<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 17, 2005, 14:44:34)...</pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Knot[10, 159]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>10</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 159]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 6, 2, 7], X[3, 9, 4, 8], X[18, 11, 19, 12], X[20, 13, 1, 14],
+  X[15, 2, 16, 3], X[17, 5, 18, 4], X[12, 19, 13, 20], X[5, 10, 6, 11],
+  X[7, 15, 8, 14], X[9, 16, 10, 17]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 159]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 5, -2, 6, -8, 1, -9, 2, -10, 8, 3, -7, 4, 9, -5, 10, -6,
+  -3, 7, -4]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[Knot[10, 159]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[3, {-1, -1, -1, -2, 1, -2, 1, 1, -2, -2}]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 159]][t]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       -3   4    9            2    3
+-11 + t   - -- + - + 9 t - 4 t  + t
+t
+            t</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[10, 159]][z]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       2      4    6
++ 2 z  + 2 z  + z</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 159]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[Knot[10, 159]], KnotSignature[Knot[10, 159]]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{39, -2}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[Knot[10, 159]][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>      -8   3    5    6    7    7    5    4
+-1 - q   + -- - -- + -- - -- + -- - -- + -
+6    5    4    3    2   q
+           q    q    q    q    q    q</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 159]}</nowiki></pre></td></tr>
+<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 159]][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>      -24    -22    -20    -16    2     -12    -10   2    2    2
+-1 - q    + q    - q    + q    - --- + q    - q    + -- + -- + --
+8    6    2
+                                 q                   q    q    q</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 159]][a, z]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>  2    4    6    3      5      7      9        2  2      4  2    6  2
+-a  + a  + a  + a  z + a  z + a  z + a  z - 2 a  z  - 4 a  z  + a  z  +
+2      3    7  3      9  3      2  4      4  4      6  4
+a  z  + a z  - a  z  - 2 a  z  + 4 a  z  + 3 a  z  - 8 a  z  -
+4    3  5      5  5      7  5    9  5      6  6      8  6
+a  z  + a  z  - 5 a  z  - 5 a  z  + a  z  + 3 a  z  + 3 a  z  +
+7      5  7      7  7    4  8    6  8
+  a  z  + 4 a  z  + 3 a  z  + a  z  + a  z</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[10, 159]], Vassiliev[3][Knot[10, 159]]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, -3}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[10, 159]][q, t]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>2    3     1        2        1        3        2        3        3
+-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- +
+q    17  7    15  6    13  6    13  5    11  5    11  4    9  4
+q        q   t    q   t    q   t    q   t    q   t    q   t    q  t
+3       3       4      2      3
+  ----- + ----- + ----- + ----- + ---- + ---- + q t
+3    7  3    7  2    5  2    5      3
+  q  t    q  t    q  t    q  t    q  t   q  t</nowiki></pre></td></tr>
+</table>

10 159: Difference between revisions

Revision as of 20:48, 27 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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