9 40: Difference between revisions

Revision as of 20:50, 27 August 2005

Visit 9 40's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 40's page at the original Knot Atlas!

In three-fold symmetrical form		Symmetrical triangular form		(less open)		(alternate)		Variant
Obtained by an epitrochoid.		Cylindrical depiction.		9.40 as a geodesic line of the oblate spheroid		Photo of an alsatian chair, musée de l'oeuvre Notre Dame, Strasbourg, France.

Knot presentations

Planar diagram presentation	X₁₆₂₇ X_7,12,8,13 X_5,15,6,14 X_11,3,12,2 X_15,10,16,11 X_3,16,4,17 X_9,4,10,5 X_17,9,18,8 X_13,18,14,1
Gauss code	-1, 4, -6, 7, -3, 1, -2, 8, -7, 5, -4, 2, -9, 3, -5, 6, -8, 9
Dowker-Thistlethwaite code	6 16 14 12 4 2 18 10 8
Conway Notation	[9*]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	4
Nakanishi index	2
Maximal Thurston-Bennequin number	[-9][-2]
Hyperbolic Volume	15.0183
A-Polynomial	See Data:9 40/A-polynomial

[edit Notes for 9 40'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	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}
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 [1,3]}
Rasmussen s-Invariant	-2

[edit Notes for 9 40'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 t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3} }
Conway polynomial	$z^{6}-z^{4}-z^{2}+1$
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 \left\{t^2-3 t+1\right\}}
Determinant and Signature	{ 75, -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^2+5 q-8+11 q^{-1} -13 q^{-2} +13 q^{-3} -11 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} }
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^6-2 z^4 a^4-2 z^2 a^4+a^4+z^6 a^2+2 z^4 a^2-2 a^2-z^4+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^4 a^8+4 z^5 a^7-2 z^3 a^7+8 z^6 a^6-9 z^4 a^6+4 z^2 a^6+9 z^7 a^5-12 z^5 a^5+6 z^3 a^5-z a^5+4 z^8 a^4+7 z^6 a^4-20 z^4 a^4+7 z^2 a^4+a^4+17 z^7 a^3-32 z^5 a^3+14 z^3 a^3-z a^3+4 z^8 a^2+4 z^6 a^2-17 z^4 a^2+3 z^2 a^2+2 a^2+8 z^7 a-15 z^5 a+6 z^3 a+5 z^6-7 z^4+2+z^5 a^{-1} }
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^{22}-q^{20}-2 q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^{10}-3 q^8+q^6-4 q^4+3 q^2+1+3 q^{-4} - q^{-6} }
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^{114}-3 q^{112}+6 q^{110}-10 q^{108}+10 q^{106}-8 q^{104}+q^{102}+17 q^{100}-35 q^{98}+57 q^{96}-69 q^{94}+58 q^{92}-26 q^{90}-41 q^{88}+121 q^{86}-182 q^{84}+197 q^{82}-139 q^{80}+14 q^{78}+135 q^{76}-248 q^{74}+274 q^{72}-196 q^{70}+38 q^{68}+122 q^{66}-223 q^{64}+212 q^{62}-79 q^{60}-87 q^{58}+218 q^{56}-237 q^{54}+135 q^{52}+47 q^{50}-232 q^{48}+337 q^{46}-328 q^{44}+209 q^{42}-7 q^{40}-197 q^{38}+334 q^{36}-361 q^{34}+269 q^{32}-104 q^{30}-93 q^{28}+225 q^{26}-263 q^{24}+194 q^{22}-39 q^{20}-123 q^{18}+217 q^{16}-194 q^{14}+58 q^{12}+116 q^{10}-252 q^8+282 q^6-192 q^4+31 q^2+141-245 q^{-2} +261 q^{-4} -179 q^{-6} +57 q^{-8} +53 q^{-10} -121 q^{-12} +126 q^{-14} -87 q^{-16} +44 q^{-18} -2 q^{-20} -19 q^{-22} +24 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }

Further Quantum Invariants

Further quantum knot invariants for 9_40.

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^{15}-3 q^{13}+4 q^{11}-3 q^9+2 q^7-2 q^3+3 q-3 q^{-1} +4 q^{-3} - q^{-5} }
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^{42}-3 q^{40}+q^{38}+9 q^{36}-16 q^{34}-3 q^{32}+33 q^{30}-22 q^{28}-22 q^{26}+40 q^{24}-8 q^{22}-30 q^{20}+20 q^{18}+12 q^{16}-17 q^{14}-8 q^{12}+23 q^{10}+2 q^8-31 q^6+22 q^4+22 q^2-39+6 q^{-2} +30 q^{-4} -23 q^{-6} -9 q^{-8} +17 q^{-10} - q^{-12} -4 q^{-14} + q^{-16} }
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^{81}-3 q^{79}+q^{77}+6 q^{75}-4 q^{73}-15 q^{71}+6 q^{69}+47 q^{67}-7 q^{65}-96 q^{63}-23 q^{61}+150 q^{59}+98 q^{57}-188 q^{55}-190 q^{53}+175 q^{51}+278 q^{49}-113 q^{47}-333 q^{45}+25 q^{43}+330 q^{41}+63 q^{39}-274 q^{37}-133 q^{35}+194 q^{33}+176 q^{31}-106 q^{29}-195 q^{27}+19 q^{25}+205 q^{23}+57 q^{21}-209 q^{19}-133 q^{17}+203 q^{15}+208 q^{13}-176 q^{11}-271 q^9+115 q^7+321 q^5-34 q^3-324 q-59 q^{-1} +279 q^{-3} +139 q^{-5} -192 q^{-7} -173 q^{-9} +96 q^{-11} +153 q^{-13} -16 q^{-15} -102 q^{-17} -28 q^{-19} +52 q^{-21} +24 q^{-23} -11 q^{-25} -13 q^{-27} + q^{-29} +4 q^{-31} - q^{-33} }
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^{132}-3 q^{130}+q^{128}+6 q^{126}-7 q^{124}-3 q^{122}-6 q^{120}+26 q^{118}+38 q^{116}-57 q^{114}-83 q^{112}-54 q^{110}+169 q^{108}+307 q^{106}-58 q^{104}-449 q^{102}-561 q^{100}+209 q^{98}+1103 q^{96}+687 q^{94}-602 q^{92}-1796 q^{90}-834 q^{88}+1520 q^{86}+2270 q^{84}+609 q^{82}-2404 q^{80}-2683 q^{78}+308 q^{76}+2944 q^{74}+2520 q^{72}-1207 q^{70}-3340 q^{68}-1547 q^{66}+1770 q^{64}+3142 q^{62}+602 q^{60}-2211 q^{58}-2299 q^{56}+53 q^{54}+2248 q^{52}+1548 q^{50}-641 q^{48}-1962 q^{46}-1023 q^{44}+1075 q^{42}+1812 q^{40}+483 q^{38}-1542 q^{36}-1702 q^{34}+205 q^{32}+2090 q^{30}+1450 q^{28}-1214 q^{26}-2450 q^{24}-829 q^{22}+2208 q^{20}+2605 q^{18}-303 q^{16}-2836 q^{14}-2279 q^{12}+1320 q^{10}+3252 q^8+1350 q^6-1886 q^4-3163 q^2-544+2305 q^{-2} +2419 q^{-4} +110 q^{-6} -2297 q^{-8} -1724 q^{-10} +331 q^{-12} +1714 q^{-14} +1253 q^{-16} -522 q^{-18} -1171 q^{-20} -683 q^{-22} +324 q^{-24} +800 q^{-26} +287 q^{-28} -187 q^{-30} -393 q^{-32} -164 q^{-34} +136 q^{-36} +134 q^{-38} +65 q^{-40} -44 q^{-42} -53 q^{-44} -4 q^{-46} +7 q^{-48} +13 q^{-50} - q^{-52} -4 q^{-54} + q^{-56} }
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^{195}-3 q^{193}+q^{191}+6 q^{189}-7 q^{187}-6 q^{185}+6 q^{183}+14 q^{181}+17 q^{179}-6 q^{177}-68 q^{175}-90 q^{173}+31 q^{171}+223 q^{169}+281 q^{167}+9 q^{165}-504 q^{163}-829 q^{161}-385 q^{159}+910 q^{157}+1976 q^{155}+1479 q^{153}-919 q^{151}-3669 q^{149}-3993 q^{147}-320 q^{145}+5365 q^{143}+7993 q^{141}+3795 q^{139}-5578 q^{137}-12701 q^{135}-10047 q^{133}+2784 q^{131}+16391 q^{129}+18101 q^{127}+3797 q^{125}-16705 q^{123}-25866 q^{121}-13539 q^{119}+12444 q^{117}+30646 q^{115}+24001 q^{113}-3906 q^{111}-30412 q^{109}-32341 q^{107}-6868 q^{105}+25059 q^{103}+36221 q^{101}+16940 q^{99}-16089 q^{97}-34824 q^{95}-23986 q^{93}+5996 q^{91}+29197 q^{89}+26869 q^{87}+2813 q^{85}-21320 q^{83}-25896 q^{81}-9071 q^{79}+13300 q^{77}+22554 q^{75}+12559 q^{73}-6644 q^{71}-18493 q^{69}-13997 q^{67}+1817 q^{65}+15043 q^{63}+14672 q^{61}+1434 q^{59}-12928 q^{57}-15598 q^{55}-3914 q^{53}+11985 q^{51}+17517 q^{49}+6674 q^{47}-11643 q^{45}-20517 q^{43}-10461 q^{41}+10796 q^{39}+23989 q^{37}+15718 q^{35}-8296 q^{33}-26901 q^{31}-22122 q^{29}+3419 q^{27}+27710 q^{25}+28570 q^{23}+4041 q^{21}-25170 q^{19}-33454 q^{17}-13019 q^{15}+18728 q^{13}+34795 q^{11}+21734 q^9-9014 q^7-31506 q^5-27744 q^3-2006 q+23734 q^{-1} +29144 q^{-3} +11595 q^{-5} -13164 q^{-7} -25450 q^{-9} -17354 q^{-11} +2643 q^{-13} +17992 q^{-15} +18081 q^{-17} +5151 q^{-19} -9254 q^{-21} -14599 q^{-23} -8790 q^{-25} +1983 q^{-27} +9097 q^{-29} +8440 q^{-31} +2256 q^{-33} -3848 q^{-35} -5856 q^{-37} -3509 q^{-39} +517 q^{-41} +2960 q^{-43} +2723 q^{-45} +860 q^{-47} -888 q^{-49} -1496 q^{-51} -918 q^{-53} +30 q^{-55} +527 q^{-57} +501 q^{-59} +179 q^{-61} -102 q^{-63} -195 q^{-65} -107 q^{-67} +7 q^{-69} +45 q^{-71} +33 q^{-73} +8 q^{-75} -7 q^{-77} -13 q^{-79} + q^{-81} +4 q^{-83} - q^{-85} }

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^{22}-q^{20}-2 q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^{10}-3 q^8+q^6-4 q^4+3 q^2+1+3 q^{-4} - q^{-6} }
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^{60}-6 q^{58}+18 q^{56}-38 q^{54}+75 q^{52}-144 q^{50}+244 q^{48}-382 q^{46}+576 q^{44}-798 q^{42}+1004 q^{40}-1168 q^{38}+1223 q^{36}-1110 q^{34}+828 q^{32}-360 q^{30}-236 q^{28}+880 q^{26}-1502 q^{24}+1988 q^{22}-2316 q^{20}+2426 q^{18}-2296 q^{16}+1966 q^{14}-1452 q^{12}+850 q^{10}-198 q^8-408 q^6+885 q^4-1194 q^2+1304-1250 q^{-2} +1061 q^{-4} -814 q^{-6} +568 q^{-8} -338 q^{-10} +184 q^{-12} -88 q^{-14} +32 q^{-16} -8 q^{-18} + q^{-20} }
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^{56}-q^{54}-3 q^{52}+2 q^{50}+6 q^{48}-q^{46}-11 q^{44}-q^{42}+16 q^{40}-3 q^{38}-13 q^{36}+6 q^{34}+16 q^{32}-3 q^{30}-19 q^{28}+7 q^{26}+4 q^{24}-12 q^{22}-q^{20}+8 q^{18}-2 q^{16}+16 q^{12}-12 q^8+2 q^6+14 q^4-10 q^2-18+12 q^{-2} +10 q^{-4} -8 q^{-6} -6 q^{-8} +6 q^{-10} +9 q^{-12} -3 q^{-14} -3 q^{-16} + q^{-18} }

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^{48}-3 q^{46}+9 q^{42}-10 q^{40}-4 q^{38}+22 q^{36}-19 q^{34}-10 q^{32}+29 q^{30}-20 q^{28}-8 q^{26}+24 q^{24}-7 q^{22}-6 q^{20}+5 q^{18}+7 q^{16}-4 q^{14}-15 q^{12}+13 q^{10}+7 q^8-28 q^6+15 q^4+15 q^2-26+16 q^{-2} +10 q^{-4} -15 q^{-6} +9 q^{-8} +2 q^{-10} -4 q^{-12} + q^{-14} }

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^{29}-q^{27}-2 q^{23}+3 q^{21}-2 q^{19}+4 q^{17}+q^{13}-2 q^{11}-2 q^9-q^7-3 q^5+3 q^3+4 q^{-1} - q^{-3} +3 q^{-5} - q^{-7} }

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^{62}-q^{60}-3 q^{58}+q^{56}+6 q^{54}+q^{52}-10 q^{50}+2 q^{48}+15 q^{46}-9 q^{44}-19 q^{42}+16 q^{40}+13 q^{38}-21 q^{36}-6 q^{34}+23 q^{32}-21 q^{28}+13 q^{26}+18 q^{24}-20 q^{22}-4 q^{20}+29 q^{18}-12 q^{16}-23 q^{14}+20 q^{12}+8 q^{10}-28 q^8-8 q^6+21 q^4-15+11 q^{-2} +17 q^{-4} -6 q^{-6} -4 q^{-8} +8 q^{-10} - q^{-12} -3 q^{-14} + q^{-16} }

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^{36}-q^{34}-2 q^{28}+3 q^{26}-2 q^{24}+3 q^{22}+2 q^{20}+q^{16}-2 q^{14}-q^{12}-4 q^{10}-3 q^6+3 q^4+3+3 q^{-2} - q^{-4} +3 q^{-6} - q^{-8} }

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^{48}-3 q^{46}+6 q^{44}-11 q^{42}+18 q^{40}-24 q^{38}+30 q^{36}-33 q^{34}+28 q^{32}-21 q^{30}+10 q^{28}+4 q^{26}-18 q^{24}+37 q^{22}-48 q^{20}+57 q^{18}-59 q^{16}+54 q^{14}-47 q^{12}+31 q^{10}-17 q^8-2 q^6+15 q^4-23 q^2+30-30 q^{-2} +32 q^{-4} -23 q^{-6} +17 q^{-8} -10 q^{-10} +4 q^{-12} - q^{-14} }

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^{78}-3 q^{74}-3 q^{72}+3 q^{70}+10 q^{68}+3 q^{66}-14 q^{64}-14 q^{62}+9 q^{60}+26 q^{58}+4 q^{56}-31 q^{54}-19 q^{52}+22 q^{50}+29 q^{48}-9 q^{46}-32 q^{44}-2 q^{42}+29 q^{40}+12 q^{38}-23 q^{36}-14 q^{34}+17 q^{32}+18 q^{30}-11 q^{28}-21 q^{26}+6 q^{24}+21 q^{22}-q^{20}-23 q^{18}-5 q^{16}+24 q^{14}+15 q^{12}-24 q^{10}-27 q^8+15 q^6+34 q^4-32-15 q^{-2} +23 q^{-4} +25 q^{-6} -10 q^{-8} -19 q^{-10} - q^{-12} +13 q^{-14} +6 q^{-16} -4 q^{-18} -4 q^{-20} + q^{-24} }

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^{66}-3 q^{64}+3 q^{62}-4 q^{60}+10 q^{58}-14 q^{56}+14 q^{54}-17 q^{52}+26 q^{50}-27 q^{48}+21 q^{46}-23 q^{44}+20 q^{42}-11 q^{40}+2 q^{38}+q^{36}-10 q^{34}+28 q^{32}-29 q^{30}+36 q^{28}-41 q^{26}+48 q^{24}-43 q^{22}+40 q^{20}-43 q^{18}+30 q^{16}-24 q^{14}+14 q^{12}-12 q^{10}-4 q^8+14 q^6-15 q^4+21 q^2-22+29 q^{-2} -22 q^{-4} +24 q^{-6} -19 q^{-8} +14 q^{-10} -8 q^{-12} +6 q^{-14} -4 q^{-16} + q^{-18} }

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^{114}-3 q^{112}+6 q^{110}-10 q^{108}+10 q^{106}-8 q^{104}+q^{102}+17 q^{100}-35 q^{98}+57 q^{96}-69 q^{94}+58 q^{92}-26 q^{90}-41 q^{88}+121 q^{86}-182 q^{84}+197 q^{82}-139 q^{80}+14 q^{78}+135 q^{76}-248 q^{74}+274 q^{72}-196 q^{70}+38 q^{68}+122 q^{66}-223 q^{64}+212 q^{62}-79 q^{60}-87 q^{58}+218 q^{56}-237 q^{54}+135 q^{52}+47 q^{50}-232 q^{48}+337 q^{46}-328 q^{44}+209 q^{42}-7 q^{40}-197 q^{38}+334 q^{36}-361 q^{34}+269 q^{32}-104 q^{30}-93 q^{28}+225 q^{26}-263 q^{24}+194 q^{22}-39 q^{20}-123 q^{18}+217 q^{16}-194 q^{14}+58 q^{12}+116 q^{10}-252 q^8+282 q^6-192 q^4+31 q^2+141-245 q^{-2} +261 q^{-4} -179 q^{-6} +57 q^{-8} +53 q^{-10} -121 q^{-12} +126 q^{-14} -87 q^{-16} +44 q^{-18} -2 q^{-20} -19 q^{-22} +24 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }

.

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["9 40"];

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 t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3} }

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

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

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 \left\{t^2-3 t+1\right\}}

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

Out[7]= { 75, -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^2+5 q-8+11 q^{-1} -13 q^{-2} +13 q^{-3} -11 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} }

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^6-2 z^4 a^4-2 z^2 a^4+a^4+z^6 a^2+2 z^4 a^2-2 a^2-z^4+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^4 a^8+4 z^5 a^7-2 z^3 a^7+8 z^6 a^6-9 z^4 a^6+4 z^2 a^6+9 z^7 a^5-12 z^5 a^5+6 z^3 a^5-z a^5+4 z^8 a^4+7 z^6 a^4-20 z^4 a^4+7 z^2 a^4+a^4+17 z^7 a^3-32 z^5 a^3+14 z^3 a^3-z a^3+4 z^8 a^2+4 z^6 a^2-17 z^4 a^2+3 z^2 a^2+2 a^2+8 z^7 a-15 z^5 a+6 z^3 a+5 z^6-7 z^4+2+z^5 a^{-1} }

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

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}

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{34}{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{38}{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{208}{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{160}{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 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{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{136}{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{152}{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{2129}{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{2102}{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{4742}{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{113}{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{751}{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 9 40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

4

1

4

1

-3

-1

7

4

3

-3

7

5

-2

-5

6

0

-7

5

7

2

-9

3

6

-3

-11

1

5

4

-13

3

-3

-15

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[9, 40]]
Out[2]=	9
In[3]:=	PD[Knot[9, 40]]
Out[3]=	PD[X[1, 6, 2, 7], X[7, 12, 8, 13], X[5, 15, 6, 14], X[11, 3, 12, 2], X[15, 10, 16, 11], X[3, 16, 4, 17], X[9, 4, 10, 5], X[17, 9, 18, 8], X[13, 18, 14, 1]]
In[4]:=	GaussCode[Knot[9, 40]]
Out[4]=	GaussCode[-1, 4, -6, 7, -3, 1, -2, 8, -7, 5, -4, 2, -9, 3, -5, 6, -8, 9]
In[5]:=	BR[Knot[9, 40]]
Out[5]=	BR[4, {-1, 2, -1, -3, 2, -1, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 40]][t]
Out[6]=	-3 7 18 2 3 -23 + t - -- + -- + 18 t - 7 t + t 2 t t
In[7]:=	Conway[Knot[9, 40]][z]
Out[7]=	2 4 6 1 - z - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 40], Knot[10, 59], Knot[11, NonAlternating, 66]}
In[9]:=	{KnotDet[Knot[9, 40]], KnotSignature[Knot[9, 40]]}
Out[9]=	{75, -2}
In[10]:=	J=Jones[Knot[9, 40]][q]
Out[10]=	-7 4 8 11 13 13 11 2 -8 + q - -- + -- - -- + -- - -- + -- + 5 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 40]}
In[12]:=	A2Invariant[Knot[9, 40]][q]
Out[12]=	-22 -20 2 3 -14 2 -10 3 -6 4 3 1 + q - q - --- + --- - q + --- + q - -- + q - -- + -- + 18 16 12 8 4 2 q q q q q q 4 6 3 q - q
In[13]:=	Kauffman[Knot[9, 40]][a, z]
Out[13]=	2 4 3 5 2 2 4 2 6 2 3 2 + 2 a + a - a z - a z + 3 a z + 7 a z + 4 a z + 6 a z + 3 3 5 3 7 3 4 2 4 4 4 6 4 14 a z + 6 a z - 2 a z - 7 z - 17 a z - 20 a z - 9 a z + 5 8 4 z 5 3 5 5 5 7 5 6 a z + -- - 15 a z - 32 a z - 12 a z + 4 a z + 5 z + a 2 6 4 6 6 6 7 3 7 5 7 2 8 4 a z + 7 a z + 8 a z + 8 a z + 17 a z + 9 a z + 4 a z + 4 8 4 a z
In[14]:=	{Vassiliev[2][Knot[9, 40]], Vassiliev[3][Knot[9, 40]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[9, 40]][q, t]
Out[15]=	5 7 1 3 1 5 3 6 5 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 7 6 6 7 4 t 2 3 2 5 3 ----- + ----- + ---- + ---- + --- + 4 q t + q t + 4 q t + q t 7 2 5 2 5 3 q q t q t q t q t

@@ Line 1: / Line 1: @@
+<!--  -->
-{{Template:Basic Knot Invariants|name=9_40}}
+<!-- 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=9|k=40|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-6,7,-3,1,-2,8,-7,5,-4,2,-9,3,-5,6,-8,9/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=14.2857%><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=7.14286%>-6</td  ><td width=7.14286%>-5</td  ><td width=7.14286%>-4</td  ><td width=7.14286%>-3</td  ><td width=7.14286%>-2</td  ><td width=7.14286%>-1</td  ><td width=7.14286%>0</td  ><td width=7.14286%>1</td  ><td width=7.14286%>2</td  ><td width=7.14286%>3</td  ><td width=14.2857%>&chi;</td></tr>
+<tr align=center><td>5</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 bgcolor=yellow>1</td><td>-1</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>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>&nbsp;</td><td>4</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>4</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>-3</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 bgcolor=yellow>7</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&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 bgcolor=yellow>7</td><td bgcolor=yellow>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
+<tr align=center><td>-5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>6</td><td bgcolor=yellow>6</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
+<tr align=center><td>-7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>5</td><td bgcolor=yellow>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
+<tr align=center><td>-9</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>6</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-3</td></tr>
+<tr align=center><td>-11</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>4</td></tr>
+<tr align=center><td>-13</td><td bgcolor=yellow>&nbsp;</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>&nbsp;</td><td>&nbsp;</td><td>-3</td></tr>
+<tr align=center><td>-15</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>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[9, 40]]</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>9</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[9, 40]]</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[7, 12, 8, 13], X[5, 15, 6, 14], X[11, 3, 12, 2],
+  X[15, 10, 16, 11], X[3, 16, 4, 17], X[9, 4, 10, 5], X[17, 9, 18, 8],
+  X[13, 18, 14, 1]]</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[9, 40]]</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, 4, -6, 7, -3, 1, -2, 8, -7, 5, -4, 2, -9, 3, -5, 6, -8, 9]</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[9, 40]]</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[4, {-1, 2, -1, -3, 2, -1, -3, 2, -3}]</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[9, 40]][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   7    18             2    3
+-23 + t   - -- + -- + 18 t - 7 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[9, 40]][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
+- z  - 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[9, 40], Knot[10, 59], Knot[11, NonAlternating, 66]}</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[9, 40]], KnotSignature[Knot[9, 40]]}</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>{75, -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[9, 40]][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>      -7   4    8    11   13   13   11          2
+-8 + q   - -- + -- - -- + -- - -- + -- + 5 q - q
+5    4    3    2   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[9, 40]}</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[9, 40]][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>     -22    -20    2     3     -14    2     -10   3     -6   4    3
++ q    - q    - --- + --- - q    + --- + q    - -- + q   - -- + -- +
+16           12           8          4    2
+                  q     q            q            q          q    q
+6
+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[9, 40]][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    3      5        2  2      4  2      6  2        3
++ 2 a  + a  - a  z - a  z + 3 a  z  + 7 a  z  + 4 a  z  + 6 a z  +
+3      5  3      7  3      4       2  4       4  4      6  4
+a  z  + 6 a  z  - 2 a  z  - 7 z  - 17 a  z  - 20 a  z  - 9 a  z  +
+4   z          5       3  5       5  5      7  5      6
+  a  z  + -- - 15 a z  - 32 a  z  - 12 a  z  + 4 a  z  + 5 z  +
+          a
+6      4  6      6  6        7       3  7      5  7      2  8
+a  z  + 7 a  z  + 8 a  z  + 8 a z  + 17 a  z  + 9 a  z  + 4 a  z  +
+8
+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[9, 40]], Vassiliev[3][Knot[9, 40]]}</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, 1}</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[9, 40]][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>5    7     1        3        1        5        3       6       5
+-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- +
+q    15  6    13  5    11  5    11  4    9  4    9  3    7  3
+q        q   t    q   t    q   t    q   t    q  t    q  t    q  t
+6      6      7     4 t              2      3  2    5  3
+  ----- + ----- + ---- + ---- + --- + 4 q t + q t  + 4 q  t  + q  t
+2    5  2    5      3      q
+  q  t    q  t    q  t   q  t</nowiki></pre></td></tr>
+</table>

9 40: Difference between revisions

Revision as of 20:50, 27 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

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