10 30: Difference between revisions

Revision as of 20:08, 28 August 2005

10_29

10_31

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Visit 10 30's page at the original Knot Atlas!

10 30 Quick Notes

10 30 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_5,12,6,13 X_3,11,4,10 X_11,3,12,2 X_9,18,10,19 X_13,20,14,1 X_19,14,20,15 X_17,6,18,7 X_7,16,8,17 X_15,8,16,9
Gauss code	-1, 4, -3, 1, -2, 8, -9, 10, -5, 3, -4, 2, -6, 7, -10, 9, -8, 5, -7, 6
Dowker-Thistlethwaite code	4 10 12 16 18 2 20 8 6 14
Conway Notation	[312112]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	2
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-11][-1]
Hyperbolic Volume	11.8288
A-Polynomial	See Data:10 30/A-polynomial

[edit Notes for 10 30'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	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 2}
Rasmussen s-Invariant	-2

[edit Notes for 10 30's four dimensional invariants]

Polynomial invariants

Alexander polynomial	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 -4 t^2+17 t-25+17 t^{-1} -4 t^{-2} }
Conway polynomial	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 -4 z^4+z^2+1}
2nd Alexander ideal (db, data sources)	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\}}
Determinant and Signature	{ 67, -2 }
Jones polynomial	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-3+6 q^{-1} -8 q^{-2} +11 q^{-3} -11 q^{-4} +10 q^{-5} -8 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9} }
HOMFLY-PT polynomial (db, data sources)	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 z^2 a^8-z^4 a^6-2 z^4 a^4-2 z^2 a^4-a^4-z^4 a^2+z^2 a^2+2 a^2+z^2}
Kauffman polynomial (db, data sources)	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 z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+9 z^3 a^9-2 z a^9+3 z^8 a^8-7 z^6 a^8+2 z^4 a^8+z^2 a^8+z^9 a^7+5 z^7 a^7-20 z^5 a^7+18 z^3 a^7-6 z a^7+6 z^8 a^6-11 z^6 a^6+2 z^4 a^6+2 z^2 a^6+z^9 a^5+7 z^7 a^5-19 z^5 a^5+16 z^3 a^5-5 z a^5+3 z^8 a^4+2 z^6 a^4-11 z^4 a^4+9 z^2 a^4-a^4+5 z^7 a^3-6 z^5 a^3+4 z^3 a^3-z a^3+5 z^6 a^2-7 z^4 a^2+5 z^2 a^2-2 a^2+3 z^5 a-3 z^3 a+z^4-z^2}
The A2 invariant	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^{28}-q^{26}-q^{24}+2 q^{22}-2 q^{20}+q^{16}-2 q^{14}+q^{12}-q^{10}+2 q^8+2 q^6-q^4+3 q^2-1- q^{-2} + q^{-4} }
The G2 invariant	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^{142}-2 q^{140}+5 q^{138}-9 q^{136}+9 q^{134}-8 q^{132}-2 q^{130}+19 q^{128}-34 q^{126}+46 q^{124}-44 q^{122}+22 q^{120}+15 q^{118}-59 q^{116}+92 q^{114}-96 q^{112}+69 q^{110}-15 q^{108}-49 q^{106}+97 q^{104}-111 q^{102}+89 q^{100}-34 q^{98}-27 q^{96}+70 q^{94}-78 q^{92}+50 q^{90}-49 q^{86}+76 q^{84}-65 q^{82}+16 q^{80}+46 q^{78}-104 q^{76}+131 q^{74}-110 q^{72}+46 q^{70}+34 q^{68}-114 q^{66}+154 q^{64}-147 q^{62}+89 q^{60}-9 q^{58}-66 q^{56}+108 q^{54}-105 q^{52}+64 q^{50}-4 q^{48}-43 q^{46}+60 q^{44}-43 q^{42}+3 q^{40}+48 q^{38}-72 q^{36}+73 q^{34}-39 q^{32}-9 q^{30}+56 q^{28}-87 q^{26}+92 q^{24}-69 q^{22}+33 q^{20}+12 q^{18}-48 q^{16}+66 q^{14}-64 q^{12}+49 q^{10}-24 q^8-q^6+18 q^4-29 q^2+28-19 q^{-2} +12 q^{-4} -2 q^{-6} -3 q^{-8} +5 q^{-10} -6 q^{-12} +4 q^{-14} -2 q^{-16} + q^{-18} }

Further Quantum Invariants

Further quantum knot invariants for 10_30.

A1 Invariants.

Weight	Invariant
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^{19}-2 q^{17}+2 q^{15}-3 q^{13}+2 q^{11}-q^9+3 q^5-2 q^3+3 q-2 q^{-1} + q^{-3} }
2	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}-2 q^{50}+7 q^{48}-2 q^{46}-10 q^{44}+12 q^{42}+4 q^{40}-18 q^{38}+11 q^{36}+11 q^{34}-20 q^{32}+3 q^{30}+14 q^{28}-12 q^{26}-5 q^{24}+10 q^{22}+3 q^{20}-11 q^{18}-2 q^{16}+18 q^{14}-10 q^{12}-11 q^{10}+21 q^8-5 q^6-12 q^4+14 q^2-2-7 q^{-2} +6 q^{-4} -2 q^{-8} + q^{-10} }
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 q^{105}-2 q^{103}-2 q^{101}+3 q^{99}+7 q^{97}-2 q^{95}-16 q^{93}-2 q^{91}+24 q^{89}+14 q^{87}-30 q^{85}-32 q^{83}+29 q^{81}+53 q^{79}-20 q^{77}-69 q^{75}-2 q^{73}+80 q^{71}+24 q^{69}-79 q^{67}-46 q^{65}+71 q^{63}+67 q^{61}-56 q^{59}-78 q^{57}+38 q^{55}+85 q^{53}-21 q^{51}-83 q^{49}-2 q^{47}+75 q^{45}+19 q^{43}-59 q^{41}-44 q^{39}+40 q^{37}+59 q^{35}-7 q^{33}-74 q^{31}-22 q^{29}+78 q^{27}+47 q^{25}-70 q^{23}-66 q^{21}+58 q^{19}+72 q^{17}-39 q^{15}-67 q^{13}+27 q^{11}+54 q^9-14 q^7-41 q^5+10 q^3+27 q-5 q^{-1} -18 q^{-3} +4 q^{-5} +12 q^{-7} -3 q^{-9} -6 q^{-11} + q^{-13} +3 q^{-15} -2 q^{-19} + q^{-21} }
4	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^{172}-2 q^{170}-2 q^{168}+3 q^{166}+3 q^{164}+7 q^{162}-9 q^{160}-16 q^{158}-q^{156}+10 q^{154}+42 q^{152}+q^{150}-47 q^{148}-45 q^{146}-16 q^{144}+102 q^{142}+77 q^{140}-29 q^{138}-121 q^{136}-144 q^{134}+95 q^{132}+196 q^{130}+121 q^{128}-103 q^{126}-321 q^{124}-79 q^{122}+191 q^{120}+333 q^{118}+111 q^{116}-357 q^{114}-321 q^{112}-28 q^{110}+396 q^{108}+392 q^{106}-167 q^{104}-425 q^{102}-318 q^{100}+254 q^{98}+536 q^{96}+95 q^{94}-346 q^{92}-478 q^{90}+51 q^{88}+503 q^{86}+268 q^{84}-204 q^{82}-489 q^{80}-96 q^{78}+381 q^{76}+340 q^{74}-67 q^{72}-415 q^{70}-216 q^{68}+203 q^{66}+374 q^{64}+107 q^{62}-262 q^{60}-339 q^{58}-70 q^{56}+333 q^{54}+326 q^{52}+17 q^{50}-387 q^{48}-389 q^{46}+151 q^{44}+440 q^{42}+332 q^{40}-252 q^{38}-557 q^{36}-108 q^{34}+332 q^{32}+486 q^{30}-20 q^{28}-461 q^{26}-234 q^{24}+107 q^{22}+389 q^{20}+112 q^{18}-237 q^{16}-172 q^{14}-34 q^{12}+197 q^{10}+93 q^8-85 q^6-58 q^4-49 q^2+74+36 q^{-2} -33 q^{-4} -4 q^{-6} -23 q^{-8} +28 q^{-10} +7 q^{-12} -17 q^{-14} +5 q^{-16} -8 q^{-18} +11 q^{-20} +2 q^{-22} -7 q^{-24} +2 q^{-26} -2 q^{-28} +3 q^{-30} -2 q^{-34} + q^{-36} }
5	$q^{255}-2q^{253}-2q^{251}+3q^{249}+3q^{247}+3q^{245}-9q^{241}-16q^{239}-q^{237}+20q^{235}+28q^{233}+20q^{231}-17q^{229}-61q^{227}-65q^{225}+4q^{223}+92q^{221}+127q^{219}+66q^{217}-91q^{215}-226q^{213}-195q^{211}+34q^{209}+296q^{207}+375q^{205}+152q^{203}-285q^{201}-584q^{199}-444q^{197}+119q^{195}+701q^{193}+812q^{191}+255q^{189}-634q^{187}-1146q^{185}-792q^{183}+294q^{181}+1303q^{179}+1371q^{177}+324q^{175}-1140q^{173}-1857q^{171}-1119q^{169}+647q^{167}+2054q^{165}+1908q^{163}+172q^{161}-1894q^{159}-2546q^{157}-1107q^{155}+1378q^{153}+2857q^{151}+2003q^{149}-600q^{147}-2824q^{145}-2704q^{143}-239q^{141}+2484q^{139}+3086q^{137}+1007q^{135}-1951q^{133}-3184q^{131}-1577q^{129}+1389q^{127}+3023q^{125}+1907q^{123}-871q^{121}-2740q^{119}-2031q^{117}+479q^{115}+2403q^{113}+2034q^{111}-181q^{109}-2103q^{107}-1977q^{105}-66q^{103}+1800q^{101}+1983q^{99}+363q^{97}-1521q^{95}-1994q^{93}-757q^{91}+1098q^{89}+2049q^{87}+1302q^{85}-565q^{83}-2017q^{81}-1890q^{79}-217q^{77}+1807q^{75}+2485q^{73}+1112q^{71}-1351q^{69}-2878q^{67}-2046q^{65}+624q^{63}+2962q^{61}+2865q^{59}+251q^{57}-2687q^{55}-3356q^{53}-1131q^{51}+2064q^{49}+3459q^{47}+1838q^{45}-1278q^{43}-3138q^{41}-2230q^{39}+483q^{37}+2528q^{35}+2252q^{33}+155q^{31}-1779q^{29}-1989q^{27}-540q^{25}+1099q^{23}+1530q^{21}+666q^{19}-532q^{17}-1058q^{15}-623q^{13}+199q^{11}+641q^{9}+465q^{7}-9q^{5}-340q^{3}-311q-51q^{-1}+164q^{-3}+175q^{-5}+49q^{-7}-63q^{-9}-85q^{-11}-36q^{-13}+23q^{-15}+41q^{-17}+13q^{-19}-11q^{-21}-11q^{-23}-3q^{-25}+3q^{-27}+4q^{-29}+2q^{-31}-7q^{-33}-2q^{-35}+6q^{-37}+q^{-39}-2q^{-41}+q^{-43}-q^{-45}-2q^{-47}+3q^{-49}-2q^{-53}+q^{-55}$

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^{28}-q^{26}-q^{24}+2 q^{22}-2 q^{20}+q^{16}-2 q^{14}+q^{12}-q^{10}+2 q^8+2 q^6-q^4+3 q^2-1- q^{-2} + q^{-4} }
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^{76}-4 q^{74}+12 q^{72}-30 q^{70}+60 q^{68}-106 q^{66}+168 q^{64}-244 q^{62}+324 q^{60}-390 q^{58}+438 q^{56}-448 q^{54}+407 q^{52}-316 q^{50}+178 q^{48}-4 q^{46}-196 q^{44}+398 q^{42}-578 q^{40}+720 q^{38}-811 q^{36}+834 q^{34}-792 q^{32}+686 q^{30}-534 q^{28}+354 q^{26}-166 q^{24}-8 q^{22}+155 q^{20}-268 q^{18}+336 q^{16}-362 q^{14}+366 q^{12}-338 q^{10}+298 q^8-242 q^6+191 q^4-142 q^2+98-62 q^{-2} +38 q^{-4} -20 q^{-6} +10 q^{-8} -4 q^{-10} + q^{-12} }
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^{72}-q^{70}-2 q^{68}+4 q^{64}+2 q^{62}-7 q^{60}-2 q^{58}+8 q^{56}+5 q^{54}-9 q^{52}-6 q^{50}+11 q^{48}+7 q^{46}-12 q^{44}-8 q^{42}+9 q^{40}+4 q^{38}-8 q^{36}-3 q^{34}+7 q^{32}+q^{30}-3 q^{28}+3 q^{26}-4 q^{24}-7 q^{22}+9 q^{20}+6 q^{18}-10 q^{16}-4 q^{14}+15 q^{12}+7 q^{10}-13 q^8-4 q^6+13 q^4+2 q^2-9- q^{-2} +5 q^{-4} +2 q^{-6} -2 q^{-8} - q^{-10} + q^{-12} }

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^{60}-2 q^{58}+q^{56}+2 q^{54}-7 q^{52}+5 q^{50}+3 q^{48}-11 q^{46}+10 q^{44}+6 q^{42}-14 q^{40}+11 q^{38}+8 q^{36}-16 q^{34}+2 q^{32}+5 q^{30}-9 q^{28}-5 q^{26}+3 q^{24}+7 q^{22}-7 q^{20}-2 q^{18}+17 q^{16}-9 q^{14}-7 q^{12}+18 q^{10}-5 q^8-9 q^6+13 q^4-q^2-6+5 q^{-2} -2 q^{-6} + q^{-8} }

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^{37}-q^{35}-q^{31}+2 q^{29}-2 q^{27}+q^{25}-q^{23}+q^{21}-2 q^{19}-q^{13}+2 q^{11}+q^9+3 q^7-q^5+3 q^3-q- q^{-3} + q^{-5} }

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^{60}-2 q^{58}+5 q^{56}-8 q^{54}+11 q^{52}-15 q^{50}+17 q^{48}-19 q^{46}+18 q^{44}-16 q^{42}+10 q^{40}-3 q^{38}-6 q^{36}+16 q^{34}-24 q^{32}+31 q^{30}-35 q^{28}+37 q^{26}-35 q^{24}+29 q^{22}-21 q^{20}+12 q^{18}-3 q^{16}-5 q^{14}+13 q^{12}-16 q^{10}+19 q^8-17 q^6+17 q^4-13 q^2+10-7 q^{-2} +4 q^{-4} -2 q^{-6} + q^{-8} }

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^{98}-2 q^{94}-2 q^{92}+3 q^{90}+5 q^{88}-3 q^{86}-9 q^{84}-2 q^{82}+12 q^{80}+9 q^{78}-10 q^{76}-16 q^{74}+3 q^{72}+20 q^{70}+9 q^{68}-16 q^{66}-16 q^{64}+8 q^{62}+20 q^{60}+2 q^{58}-17 q^{56}-8 q^{54}+10 q^{52}+8 q^{50}-9 q^{48}-11 q^{46}+5 q^{44}+10 q^{42}-5 q^{40}-13 q^{38}+2 q^{36}+15 q^{34}+3 q^{32}-14 q^{30}-7 q^{28}+15 q^{26}+14 q^{24}-8 q^{22}-18 q^{20}+q^{18}+19 q^{16}+10 q^{14}-12 q^{12}-15 q^{10}+2 q^8+15 q^6+6 q^4-7 q^2-8+ q^{-2} +6 q^{-4} +2 q^{-6} -2 q^{-8} -2 q^{-10} + q^{-14} }

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^{142}-2 q^{140}+5 q^{138}-9 q^{136}+9 q^{134}-8 q^{132}-2 q^{130}+19 q^{128}-34 q^{126}+46 q^{124}-44 q^{122}+22 q^{120}+15 q^{118}-59 q^{116}+92 q^{114}-96 q^{112}+69 q^{110}-15 q^{108}-49 q^{106}+97 q^{104}-111 q^{102}+89 q^{100}-34 q^{98}-27 q^{96}+70 q^{94}-78 q^{92}+50 q^{90}-49 q^{86}+76 q^{84}-65 q^{82}+16 q^{80}+46 q^{78}-104 q^{76}+131 q^{74}-110 q^{72}+46 q^{70}+34 q^{68}-114 q^{66}+154 q^{64}-147 q^{62}+89 q^{60}-9 q^{58}-66 q^{56}+108 q^{54}-105 q^{52}+64 q^{50}-4 q^{48}-43 q^{46}+60 q^{44}-43 q^{42}+3 q^{40}+48 q^{38}-72 q^{36}+73 q^{34}-39 q^{32}-9 q^{30}+56 q^{28}-87 q^{26}+92 q^{24}-69 q^{22}+33 q^{20}+12 q^{18}-48 q^{16}+66 q^{14}-64 q^{12}+49 q^{10}-24 q^8-q^6+18 q^4-29 q^2+28-19 q^{-2} +12 q^{-4} -2 q^{-6} -3 q^{-8} +5 q^{-10} -6 q^{-12} +4 q^{-14} -2 q^{-16} + q^{-18} }

.

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 30"];

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

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

Out[4]= 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 -4 t^2+17 t-25+17 t^{-1} -4 t^{-2} }

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

Out[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 -4 z^4+z^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]= 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\}}

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

Out[7]= { 67, -2 }

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

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

Out[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 q-3+6 q^{-1} -8 q^{-2} +11 q^{-3} -11 q^{-4} +10 q^{-5} -8 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9} }

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

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

Out[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 z^2 a^8-z^4 a^6-2 z^4 a^4-2 z^2 a^4-a^4-z^4 a^2+z^2 a^2+2 a^2+z^2}

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

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

Out[10]= 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 z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+9 z^3 a^9-2 z a^9+3 z^8 a^8-7 z^6 a^8+2 z^4 a^8+z^2 a^8+z^9 a^7+5 z^7 a^7-20 z^5 a^7+18 z^3 a^7-6 z a^7+6 z^8 a^6-11 z^6 a^6+2 z^4 a^6+2 z^2 a^6+z^9 a^5+7 z^7 a^5-19 z^5 a^5+16 z^3 a^5-5 z a^5+3 z^8 a^4+2 z^6 a^4-11 z^4 a^4+9 z^2 a^4-a^4+5 z^7 a^3-6 z^5 a^3+4 z^3 a^3-z a^3+5 z^6 a^2-7 z^4 a^2+5 z^2 a^2-2 a^2+3 z^5 a-3 z^3 a+z^4-z^2}

Vassiliev invariants

V₂ and V₃:

(1, -1)

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

4

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 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{158}{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{106}{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 -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{752}{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{224}{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 -136}

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 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{632}{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{424}{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{26671}{30}}

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{1954}{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 \frac{50102}{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{1265}{18}}

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{5551}{30}}

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 30. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

2

-2

-1

4

1

3

-3

5

3

-2

-5

6

3

-7

5

0

-9

5

6

-1

-11

3

5

2

-13

2

5

-3

-15

1

3

2

-17

2

-2

-19

1

Integral Khovanov Homology

(db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	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 i=-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 i=-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 r=-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 {\mathbb Z}}
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=-7}	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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2}	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 {\mathbb Z}}
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=-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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	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 {\mathbb Z}^{2}}
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=-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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-4}	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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-2}	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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{4}}
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=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 {\mathbb Z}\oplus{\mathbb Z}_2^{2}}	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 {\mathbb Z}^{2}}
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=2}	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 {\mathbb Z}_2}	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 {\mathbb Z}}

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, 30]]
Out[2]=	10
In[3]:=	PD[Knot[10, 30]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[9, 18, 10, 19], X[13, 20, 14, 1], X[19, 14, 20, 15], X[17, 6, 18, 7], X[7, 16, 8, 17], X[15, 8, 16, 9]]
In[4]:=	GaussCode[Knot[10, 30]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 8, -9, 10, -5, 3, -4, 2, -6, 7, -10, 9, -8, 5, -7, 6]
In[5]:=	BR[Knot[10, 30]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -2, -3, 2, -3, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 30]][t]
Out[6]=	4 17 2 -25 - -- + -- + 17 t - 4 t 2 t t
In[7]:=	Conway[Knot[10, 30]][z]
Out[7]=	2 4 1 + z - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 30], Knot[11, Alternating, 154]}
In[9]:=	{KnotDet[Knot[10, 30]], KnotSignature[Knot[10, 30]]}
Out[9]=	{67, -2}
In[10]:=	J=Jones[Knot[10, 30]][q]
Out[10]=	-9 3 5 8 10 11 11 8 6 -3 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 30]}
In[12]:=	A2Invariant[Knot[10, 30]][q]
Out[12]=	-28 -26 -24 2 2 -16 2 -12 -10 2 -1 + q - q - q + --- - --- + q - --- + q - q + -- + 22 20 14 8 q q q q 2 -4 3 2 4 -- - q + -- - q + q 6 2 q q
In[13]:=	Kauffman[Knot[10, 30]][a, z]
Out[13]=	2 4 3 5 7 9 2 2 2 4 2 -2 a - a - a z - 5 a z - 6 a z - 2 a z - z + 5 a z + 9 a z + 6 2 8 2 10 2 3 3 3 5 3 7 3 2 a z + a z + 2 a z - 3 a z + 4 a z + 16 a z + 18 a z + 9 3 4 2 4 4 4 6 4 8 4 10 4 9 a z + z - 7 a z - 11 a z + 2 a z + 2 a z - 3 a z + 5 3 5 5 5 7 5 9 5 2 6 3 a z - 6 a z - 19 a z - 20 a z - 10 a z + 5 a z + 4 6 6 6 8 6 10 6 3 7 5 7 7 7 2 a z - 11 a z - 7 a z + a z + 5 a z + 7 a z + 5 a z + 9 7 4 8 6 8 8 8 5 9 7 9 3 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 30]], Vassiliev[3][Knot[10, 30]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 30]][q, t]
Out[15]=	3 4 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 5 5 6 5 5 6 3 5 t ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 2 q t + q t

@@ Line 1: / Line 1: @@
 <!--  -->
+<!-- -->
+<!--  -->
+<!-- -->
 <!-- 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}}
+{{Rolfsen Knot Page Header|n=10|k=30|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,8,-9,10,-5,3,-4,2,-6,7,-10,9,-8,5,-7,6/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=10|k=30|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,8,-9,10,-5,3,-4,2,-6,7,-10,9,-8,5,-7,6/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-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=13.3333%><table cellpadding=0 cellspacing=0>
@@ Line 48: / Line 41: @@
 <tr align=center><td>-17</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>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
 <tr align=center><td>-19</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>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 136: / Line 128: @@
 q t + q  t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 30: Difference between revisions

Revision as of 20:08, 28 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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