10 66

10_65

10_67

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10 66 Quick Notes

10 66 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_3,10,4,11 X_5,14,6,15 X_7,16,8,17 X_15,6,16,7 X_17,20,18,1 X_11,18,12,19 X_19,12,20,13 X_13,8,14,9 X_9,2,10,3
Gauss code	-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6
Dowker-Thistlethwaite code	4 10 14 16 2 18 8 6 20 12
Conway Notation	[31,21,21]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	3
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-17][5]
Hyperbolic Volume	13.0293
A-Polynomial	See Data:10 66/A-polynomial

[edit Notes for 10 66's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	[math]\displaystyle{ 3 }[/math]
Topological 4 genus	[math]\displaystyle{ 3 }[/math]
Concordance genus	[math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant	-6

[edit Notes for 10 66's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^3-9 t^2+16 t-19+16 t^{-1} -9 t^{-2} +3 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^6+9 z^4+7 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 75, -6 }
Jones polynomial	[math]\displaystyle{ q^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -13 q^{-8} +12 q^{-9} -10 q^{-10} +7 q^{-11} -4 q^{-12} + q^{-13} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^{12}+a^{12}-3 z^4 a^{10}-8 z^2 a^{10}-4 a^{10}+2 z^6 a^8+8 z^4 a^8+9 z^2 a^8+2 a^8+z^6 a^6+4 z^4 a^6+5 z^2 a^6+2 a^6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{16}+4 z^5 a^{15}-3 z^3 a^{15}+7 z^6 a^{14}-8 z^4 a^{14}+2 z^2 a^{14}+7 z^7 a^{13}-7 z^5 a^{13}+z^3 a^{13}+4 z^8 a^{12}+3 z^6 a^{12}-13 z^4 a^{12}+5 z^2 a^{12}+a^{12}+z^9 a^{11}+11 z^7 a^{11}-28 z^5 a^{11}+22 z^3 a^{11}-6 z a^{11}+7 z^8 a^{10}-13 z^6 a^{10}+8 z^4 a^{10}-8 z^2 a^{10}+4 a^{10}+z^9 a^9+6 z^7 a^9-22 z^5 a^9+20 z^3 a^9-5 z a^9+3 z^8 a^8-8 z^6 a^8+8 z^4 a^8-6 z^2 a^8+2 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 a^6 }[/math]
The A2 invariant	[math]\displaystyle{ q^{40}-2 q^{36}+q^{34}-2 q^{32}-3 q^{26}+2 q^{24}-2 q^{22}+3 q^{20}+2 q^{18}+3 q^{14}-q^{12}+q^{10} }[/math]
The G2 invariant	[math]\displaystyle{ q^{210}-3 q^{208}+6 q^{206}-10 q^{204}+9 q^{202}-6 q^{200}-2 q^{198}+19 q^{196}-34 q^{194}+50 q^{192}-53 q^{190}+33 q^{188}-2 q^{186}-44 q^{184}+90 q^{182}-117 q^{180}+119 q^{178}-79 q^{176}+9 q^{174}+73 q^{172}-134 q^{170}+158 q^{168}-136 q^{166}+64 q^{164}+21 q^{162}-95 q^{160}+126 q^{158}-92 q^{156}+21 q^{154}+64 q^{152}-114 q^{150}+100 q^{148}-35 q^{146}-74 q^{144}+168 q^{142}-211 q^{140}+179 q^{138}-68 q^{136}-76 q^{134}+198 q^{132}-256 q^{130}+228 q^{128}-135 q^{126}-5 q^{124}+115 q^{122}-177 q^{120}+178 q^{118}-106 q^{116}+q^{114}+82 q^{112}-117 q^{110}+87 q^{108}-17 q^{106}-76 q^{104}+136 q^{102}-139 q^{100}+90 q^{98}+6 q^{96}-107 q^{94}+178 q^{92}-175 q^{90}+118 q^{88}-30 q^{86}-60 q^{84}+118 q^{82}-125 q^{80}+102 q^{78}-47 q^{76}-2 q^{74}+38 q^{72}-49 q^{70}+42 q^{68}-25 q^{66}+11 q^{64}+2 q^{62}-6 q^{60}+7 q^{58}-5 q^{56}+4 q^{54}-q^{52}+q^{50} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_66.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{27}-3 q^{25}+3 q^{23}-3 q^{21}+2 q^{19}-q^{17}-2 q^{15}+3 q^{13}-2 q^{11}+4 q^9-q^7+q^5 }[/math]
2	[math]\displaystyle{ q^{74}-3 q^{72}+9 q^{68}-11 q^{66}-4 q^{64}+22 q^{62}-16 q^{60}-11 q^{58}+30 q^{56}-12 q^{54}-19 q^{52}+20 q^{50}+2 q^{48}-16 q^{46}-q^{44}+16 q^{42}-2 q^{40}-20 q^{38}+18 q^{36}+10 q^{34}-30 q^{32}+10 q^{30}+18 q^{28}-22 q^{26}+q^{24}+17 q^{22}-8 q^{20}-3 q^{18}+7 q^{16}-q^{12}+q^{10} }[/math]
3	[math]\displaystyle{ q^{141}-3 q^{139}+6 q^{135}+q^{133}-11 q^{131}-7 q^{129}+26 q^{127}+10 q^{125}-39 q^{123}-22 q^{121}+57 q^{119}+39 q^{117}-79 q^{115}-64 q^{113}+95 q^{111}+96 q^{109}-99 q^{107}-127 q^{105}+86 q^{103}+151 q^{101}-55 q^{99}-158 q^{97}+13 q^{95}+145 q^{93}+33 q^{91}-112 q^{89}-72 q^{87}+69 q^{85}+105 q^{83}-23 q^{81}-123 q^{79}-21 q^{77}+125 q^{75}+64 q^{73}-130 q^{71}-97 q^{69}+110 q^{67}+131 q^{65}-89 q^{63}-147 q^{61}+53 q^{59}+157 q^{57}-14 q^{55}-149 q^{53}-24 q^{51}+122 q^{49}+50 q^{47}-88 q^{45}-63 q^{43}+47 q^{41}+62 q^{39}-20 q^{37}-42 q^{35}-q^{33}+28 q^{31}+8 q^{29}-10 q^{27}-5 q^{25}+5 q^{23}+3 q^{21}-q^{17}+q^{15} }[/math]
4	[math]\displaystyle{ q^{228}-3 q^{226}+6 q^{222}-2 q^{220}+q^{218}-14 q^{216}+3 q^{214}+26 q^{212}-8 q^{210}-3 q^{208}-50 q^{206}+10 q^{204}+91 q^{202}-2 q^{200}-41 q^{198}-150 q^{196}+25 q^{194}+248 q^{192}+71 q^{190}-129 q^{188}-395 q^{186}-34 q^{184}+511 q^{182}+344 q^{180}-157 q^{178}-786 q^{176}-321 q^{174}+682 q^{172}+793 q^{170}+94 q^{168}-1036 q^{166}-799 q^{164}+458 q^{162}+1063 q^{160}+586 q^{158}-789 q^{156}-1070 q^{154}-114 q^{152}+821 q^{150}+919 q^{148}-153 q^{146}-859 q^{144}-609 q^{142}+227 q^{140}+840 q^{138}+449 q^{136}-360 q^{134}-811 q^{132}-319 q^{130}+547 q^{128}+814 q^{126}+94 q^{124}-820 q^{122}-673 q^{120}+235 q^{118}+1007 q^{116}+467 q^{114}-726 q^{112}-929 q^{110}-129 q^{108}+1031 q^{106}+823 q^{104}-424 q^{102}-1031 q^{100}-588 q^{98}+754 q^{96}+1044 q^{94}+104 q^{92}-793 q^{90}-931 q^{88}+188 q^{86}+882 q^{84}+557 q^{82}-245 q^{80}-856 q^{78}-307 q^{76}+376 q^{74}+579 q^{72}+223 q^{70}-416 q^{68}-390 q^{66}-62 q^{64}+267 q^{62}+299 q^{60}-42 q^{58}-175 q^{56}-151 q^{54}+15 q^{52}+134 q^{50}+52 q^{48}-10 q^{46}-60 q^{44}-30 q^{42}+24 q^{40}+17 q^{38}+13 q^{36}-7 q^{34}-8 q^{32}+3 q^{30}+q^{28}+3 q^{26}-q^{22}+q^{20} }[/math]
5	[math]\displaystyle{ q^{335}-3 q^{333}+6 q^{329}-2 q^{327}-2 q^{325}-2 q^{323}-4 q^{321}+3 q^{319}+14 q^{317}+2 q^{315}-21 q^{313}-17 q^{311}+8 q^{309}+37 q^{307}+32 q^{305}-9 q^{303}-87 q^{301}-97 q^{299}+54 q^{297}+199 q^{295}+169 q^{293}-70 q^{291}-379 q^{289}-405 q^{287}+82 q^{285}+715 q^{283}+779 q^{281}+q^{279}-1114 q^{277}-1451 q^{275}-317 q^{273}+1579 q^{271}+2425 q^{269}+966 q^{267}-1920 q^{265}-3624 q^{263}-2100 q^{261}+1908 q^{259}+4915 q^{257}+3688 q^{255}-1357 q^{253}-5925 q^{251}-5542 q^{249}+107 q^{247}+6325 q^{245}+7336 q^{243}+1717 q^{241}-5860 q^{239}-8593 q^{237}-3810 q^{235}+4458 q^{233}+8949 q^{231}+5739 q^{229}-2360 q^{227}-8267 q^{225}-7033 q^{223}-12 q^{221}+6636 q^{219}+7459 q^{217}+2214 q^{215}-4427 q^{213}-7041 q^{211}-3883 q^{209}+2090 q^{207}+5971 q^{205}+4893 q^{203}+39 q^{201}-4630 q^{199}-5384 q^{197}-1700 q^{195}+3328 q^{193}+5523 q^{191}+2937 q^{189}-2252 q^{187}-5555 q^{185}-3895 q^{183}+1452 q^{181}+5661 q^{179}+4693 q^{177}-776 q^{175}-5782 q^{173}-5573 q^{171}+22 q^{169}+5936 q^{167}+6471 q^{165}+923 q^{163}-5746 q^{161}-7404 q^{159}-2238 q^{157}+5203 q^{155}+8090 q^{153}+3751 q^{151}-3987 q^{149}-8307 q^{147}-5358 q^{145}+2246 q^{143}+7823 q^{141}+6653 q^{139}-86 q^{137}-6545 q^{135}-7336 q^{133}-2094 q^{131}+4562 q^{129}+7163 q^{127}+3876 q^{125}-2222 q^{123}-6102 q^{121}-4890 q^{119}-44 q^{117}+4350 q^{115}+4975 q^{113}+1791 q^{111}-2353 q^{109}-4220 q^{107}-2702 q^{105}+546 q^{103}+2900 q^{101}+2826 q^{99}+715 q^{97}-1541 q^{95}-2271 q^{93}-1272 q^{91}+387 q^{89}+1461 q^{87}+1294 q^{85}+247 q^{83}-693 q^{81}-936 q^{79}-497 q^{77}+164 q^{75}+536 q^{73}+433 q^{71}+81 q^{69}-214 q^{67}-278 q^{65}-130 q^{63}+47 q^{61}+125 q^{59}+102 q^{57}+14 q^{55}-47 q^{53}-46 q^{51}-13 q^{49}+6 q^{47}+21 q^{45}+14 q^{43}-3 q^{41}-5 q^{39}-q^{35}+q^{33}+3 q^{31}-q^{27}+q^{25} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{40}-2 q^{36}+q^{34}-2 q^{32}-3 q^{26}+2 q^{24}-2 q^{22}+3 q^{20}+2 q^{18}+3 q^{14}-q^{12}+q^{10} }[/math]
1,1	[math]\displaystyle{ q^{108}-6 q^{106}+18 q^{104}-38 q^{102}+71 q^{100}-128 q^{98}+206 q^{96}-296 q^{94}+403 q^{92}-522 q^{90}+628 q^{88}-696 q^{86}+717 q^{84}-676 q^{82}+550 q^{80}-328 q^{78}+36 q^{76}+310 q^{74}-676 q^{72}+1016 q^{70}-1301 q^{68}+1474 q^{66}-1528 q^{64}+1452 q^{62}-1242 q^{60}+964 q^{58}-608 q^{56}+228 q^{54}+119 q^{52}-424 q^{50}+620 q^{48}-762 q^{46}+786 q^{44}-738 q^{42}+632 q^{40}-496 q^{38}+368 q^{36}-236 q^{34}+152 q^{32}-76 q^{30}+43 q^{28}-16 q^{26}+8 q^{24}-2 q^{22}+q^{20} }[/math]
2,0	[math]\displaystyle{ q^{100}-2 q^{96}-q^{94}+q^{92}+q^{90}-5 q^{88}+q^{86}+7 q^{84}-3 q^{80}+6 q^{78}+10 q^{76}-7 q^{74}-10 q^{72}+6 q^{70}+2 q^{68}-12 q^{66}-q^{64}+10 q^{62}-4 q^{60}-6 q^{58}+6 q^{56}+q^{54}-13 q^{52}-2 q^{50}+8 q^{48}-9 q^{46}-8 q^{44}+11 q^{42}+7 q^{40}-7 q^{38}+2 q^{36}+11 q^{34}+3 q^{32}-5 q^{30}+3 q^{28}+5 q^{26}-q^{24}-q^{22}+q^{20} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{88}-3 q^{86}+8 q^{82}-9 q^{80}-2 q^{78}+17 q^{76}-15 q^{74}-8 q^{72}+23 q^{70}-13 q^{68}-10 q^{66}+22 q^{64}-3 q^{62}-8 q^{60}+7 q^{58}+3 q^{56}-6 q^{54}-15 q^{52}+8 q^{50}+2 q^{48}-24 q^{46}+10 q^{44}+11 q^{42}-20 q^{40}+10 q^{38}+14 q^{36}-12 q^{34}+9 q^{32}+7 q^{30}-4 q^{28}+4 q^{26}+2 q^{24}-q^{22}+q^{20} }[/math]
1,0,0	[math]\displaystyle{ q^{53}+q^{49}-2 q^{47}+q^{45}-4 q^{43}+q^{41}-3 q^{39}-2 q^{35}+q^{31}+4 q^{27}+4 q^{23}-q^{21}+3 q^{19}-q^{17}+q^{15} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{114}-3 q^{110}-2 q^{108}+4 q^{106}+4 q^{104}-6 q^{102}-2 q^{100}+11 q^{98}-q^{96}-16 q^{94}+3 q^{92}+12 q^{90}-8 q^{88}-3 q^{86}+20 q^{84}+11 q^{82}-9 q^{80}+7 q^{78}+12 q^{76}-19 q^{74}-18 q^{72}+7 q^{70}-13 q^{68}-30 q^{66}+3 q^{64}+9 q^{62}-13 q^{60}-6 q^{58}+17 q^{56}+7 q^{54}-6 q^{52}+7 q^{50}+13 q^{48}+10 q^{42}+2 q^{40}+3 q^{36}+2 q^{34}-q^{32}+q^{30} }[/math]
1,0,0,0	[math]\displaystyle{ q^{66}+q^{62}+q^{60}-2 q^{58}+q^{56}-4 q^{54}-q^{52}-2 q^{50}-3 q^{48}-2 q^{44}+q^{42}-q^{40}+3 q^{38}+4 q^{34}+q^{32}+2 q^{30}+3 q^{28}-q^{26}+3 q^{24}-q^{22}+q^{20} }[/math]

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ q^{88}-3 q^{86}+6 q^{84}-10 q^{82}+15 q^{80}-20 q^{78}+23 q^{76}-25 q^{74}+24 q^{72}-21 q^{70}+13 q^{68}-2 q^{66}-10 q^{64}+23 q^{62}-34 q^{60}+43 q^{58}-49 q^{56}+48 q^{54}-45 q^{52}+34 q^{50}-26 q^{48}+12 q^{46}-11 q^{42}+20 q^{40}-22 q^{38}+26 q^{36}-22 q^{34}+21 q^{32}-15 q^{30}+12 q^{28}-6 q^{26}+4 q^{24}-q^{22}+q^{20} }[/math]

1,0

[math]\displaystyle{ q^{142}-3 q^{138}-3 q^{136}+3 q^{134}+9 q^{132}+2 q^{130}-12 q^{128}-11 q^{126}+9 q^{124}+20 q^{122}+2 q^{120}-23 q^{118}-16 q^{116}+15 q^{114}+25 q^{112}-3 q^{110}-25 q^{108}-7 q^{106}+21 q^{104}+16 q^{102}-13 q^{100}-17 q^{98}+8 q^{96}+18 q^{94}-3 q^{92}-19 q^{90}-2 q^{88}+15 q^{86}+3 q^{84}-18 q^{82}-8 q^{80}+15 q^{78}+12 q^{76}-16 q^{74}-23 q^{72}+5 q^{70}+26 q^{68}+6 q^{66}-25 q^{64}-18 q^{62}+15 q^{60}+26 q^{58}-18 q^{54}-8 q^{52}+14 q^{50}+13 q^{48}-2 q^{46}-8 q^{44}+5 q^{40}+3 q^{38}-q^{36}-q^{34}+q^{30} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{122}-3 q^{120}+3 q^{118}-4 q^{116}+9 q^{114}-12 q^{112}+12 q^{110}-14 q^{108}+20 q^{106}-20 q^{104}+16 q^{102}-19 q^{100}+19 q^{98}-12 q^{96}+5 q^{94}-3 q^{92}-2 q^{90}+17 q^{88}-17 q^{86}+26 q^{84}-30 q^{82}+38 q^{80}-37 q^{78}+33 q^{76}-43 q^{74}+27 q^{72}-31 q^{70}+16 q^{68}-21 q^{66}+5 q^{64}-4 q^{60}+11 q^{58}-14 q^{56}+22 q^{54}-16 q^{52}+21 q^{50}-16 q^{48}+20 q^{46}-11 q^{44}+14 q^{42}-7 q^{40}+8 q^{38}-2 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{210}-3 q^{208}+6 q^{206}-10 q^{204}+9 q^{202}-6 q^{200}-2 q^{198}+19 q^{196}-34 q^{194}+50 q^{192}-53 q^{190}+33 q^{188}-2 q^{186}-44 q^{184}+90 q^{182}-117 q^{180}+119 q^{178}-79 q^{176}+9 q^{174}+73 q^{172}-134 q^{170}+158 q^{168}-136 q^{166}+64 q^{164}+21 q^{162}-95 q^{160}+126 q^{158}-92 q^{156}+21 q^{154}+64 q^{152}-114 q^{150}+100 q^{148}-35 q^{146}-74 q^{144}+168 q^{142}-211 q^{140}+179 q^{138}-68 q^{136}-76 q^{134}+198 q^{132}-256 q^{130}+228 q^{128}-135 q^{126}-5 q^{124}+115 q^{122}-177 q^{120}+178 q^{118}-106 q^{116}+q^{114}+82 q^{112}-117 q^{110}+87 q^{108}-17 q^{106}-76 q^{104}+136 q^{102}-139 q^{100}+90 q^{98}+6 q^{96}-107 q^{94}+178 q^{92}-175 q^{90}+118 q^{88}-30 q^{86}-60 q^{84}+118 q^{82}-125 q^{80}+102 q^{78}-47 q^{76}-2 q^{74}+38 q^{72}-49 q^{70}+42 q^{68}-25 q^{66}+11 q^{64}+2 q^{62}-6 q^{60}+7 q^{58}-5 q^{56}+4 q^{54}-q^{52}+q^{50} }[/math]

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 66"];

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

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

Out[4]= [math]\displaystyle{ 3 t^3-9 t^2+16 t-19+16 t^{-1} -9 t^{-2} +3 t^{-3} }[/math]

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

Out[5]= [math]\displaystyle{ 3 z^6+9 z^4+7 z^2+1 }[/math]

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= [math]\displaystyle{ \{1\} }[/math]

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

Out[7]= { 75, -6 }

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

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

Out[8]= [math]\displaystyle{ q^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -13 q^{-8} +12 q^{-9} -10 q^{-10} +7 q^{-11} -4 q^{-12} + q^{-13} }[/math]

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

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

Out[9]= [math]\displaystyle{ z^2 a^{12}+a^{12}-3 z^4 a^{10}-8 z^2 a^{10}-4 a^{10}+2 z^6 a^8+8 z^4 a^8+9 z^2 a^8+2 a^8+z^6 a^6+4 z^4 a^6+5 z^2 a^6+2 a^6 }[/math]

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

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

Out[10]= [math]\displaystyle{ z^4 a^{16}+4 z^5 a^{15}-3 z^3 a^{15}+7 z^6 a^{14}-8 z^4 a^{14}+2 z^2 a^{14}+7 z^7 a^{13}-7 z^5 a^{13}+z^3 a^{13}+4 z^8 a^{12}+3 z^6 a^{12}-13 z^4 a^{12}+5 z^2 a^{12}+a^{12}+z^9 a^{11}+11 z^7 a^{11}-28 z^5 a^{11}+22 z^3 a^{11}-6 z a^{11}+7 z^8 a^{10}-13 z^6 a^{10}+8 z^4 a^{10}-8 z^2 a^{10}+4 a^{10}+z^9 a^9+6 z^7 a^9-22 z^5 a^9+20 z^3 a^9-5 z a^9+3 z^8 a^8-8 z^6 a^8+8 z^4 a^8-6 z^2 a^8+2 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 a^6 }[/math]

Vassiliev invariants

V₂ and V₃:

(7, -17)

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

[math]\displaystyle{ 28 }[/math]

[math]\displaystyle{ -136 }[/math]

[math]\displaystyle{ 392 }[/math]

[math]\displaystyle{ \frac{2594}{3} }[/math]

[math]\displaystyle{ \frac{358}{3} }[/math]

[math]\displaystyle{ -3808 }[/math]

[math]\displaystyle{ -\frac{18448}{3} }[/math]

[math]\displaystyle{ -\frac{3232}{3} }[/math]

[math]\displaystyle{ -680 }[/math]

[math]\displaystyle{ \frac{10976}{3} }[/math]

[math]\displaystyle{ 9248 }[/math]

[math]\displaystyle{ \frac{72632}{3} }[/math]

[math]\displaystyle{ \frac{10024}{3} }[/math]

[math]\displaystyle{ \frac{1349977}{30} }[/math]

[math]\displaystyle{ \frac{41206}{15} }[/math]

[math]\displaystyle{ \frac{677354}{45} }[/math]

[math]\displaystyle{ \frac{3431}{18} }[/math]

[math]\displaystyle{ \frac{55417}{30} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-6 is the signature of 10 66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

2

1

-1

-9

4

-11

4

2

-2

-13

7

4

3

-15

6

4

-2

-17

6

7

-1

-19

4

6

2

-21

3

6

-3

-23

1

4

3

-25

3

-3

-27

1

Integral Khovanov Homology

(db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]	[math]\displaystyle{ i=-7 }[/math]	[math]\displaystyle{ i=-5 }[/math]
[math]\displaystyle{ r=-10 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-9 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-8 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-7 }[/math]	[math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-6 }[/math]	[math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math]	[math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-5 }[/math]	[math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math]	[math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-4 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math]	[math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-3 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 66]]
Out[2]=	10
In[3]:=	PD[Knot[10, 66]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 14, 6, 15], X[7, 16, 8, 17], X[15, 6, 16, 7], X[17, 20, 18, 1], X[11, 18, 12, 19], X[19, 12, 20, 13], X[13, 8, 14, 9], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[10, 66]]
Out[4]=	GaussCode[-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6]
In[5]:=	BR[Knot[10, 66]]
Out[5]=	BR[4, {-1, -1, -1, 2, -1, -3, -2, -2, -2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 66]][t]
Out[6]=	3 9 16 2 3 -19 + -- - -- + -- + 16 t - 9 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[10, 66]][z]
Out[7]=	2 4 6 1 + 7 z + 9 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 66], Knot[11, Alternating, 245]}
In[9]:=	{KnotDet[Knot[10, 66]], KnotSignature[Knot[10, 66]]}
Out[9]=	{75, -6}
In[10]:=	J=Jones[Knot[10, 66]][q]
Out[10]=	-13 4 7 10 12 13 11 8 6 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q 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, 66]}
In[12]:=	A2Invariant[Knot[10, 66]][q]
Out[12]=	-40 2 -34 2 3 2 2 3 2 3 -12 q - --- + q - --- - --- + --- - --- + --- + --- + --- - q + 36 32 26 24 22 20 18 14 q q q q q q q q -10 q
In[13]:=	Kauffman[Knot[10, 66]][a, z]
Out[13]=	6 8 10 12 7 9 11 6 2 -2 a + 2 a + 4 a + a + a z - 5 a z - 6 a z + 5 a z - 8 2 10 2 12 2 14 2 7 3 9 3 6 a z - 8 a z + 5 a z + 2 a z + 2 a z + 20 a z + 11 3 13 3 15 3 6 4 8 4 10 4 22 a z + a z - 3 a z - 4 a z + 8 a z + 8 a z - 12 4 14 4 16 4 7 5 9 5 11 5 13 a z - 8 a z + a z - 5 a z - 22 a z - 28 a z - 13 5 15 5 6 6 8 6 10 6 12 6 7 a z + 4 a z + a z - 8 a z - 13 a z + 3 a z + 14 6 7 7 9 7 11 7 13 7 8 8 7 a z + 2 a z + 6 a z + 11 a z + 7 a z + 3 a z + 10 8 12 8 9 9 11 9 7 a z + 4 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 66]], Vassiliev[3][Knot[10, 66]]}
Out[14]=	{0, -17}
In[15]:=	Kh[Knot[10, 66]][q, t]
Out[15]=	-7 -5 1 3 1 4 3 6 q + q + ------- + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 q t q t q t q t q t q t 4 6 6 7 6 4 7 ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 7 19 6 17 6 17 5 15 5 15 4 13 4 q t q t q t q t q t q t q t 4 4 2 4 2 ------ + ------ + ------ + ----- + ---- 13 3 11 3 11 2 9 2 7 q t q t q t q t q t

10 66

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

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