10 93

10_92

10_94

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10 93 Quick Notes

10 93 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_16,6,17,5 X_20,8,1,7 X_18,13,19,14 X_14,9,15,10 X_10,3,11,4 X_4,11,5,12 X_12,17,13,18 X_8,20,9,19 X_2,16,3,15
Gauss code	1, -10, 6, -7, 2, -1, 3, -9, 5, -6, 7, -8, 4, -5, 10, -2, 8, -4, 9, -3
Dowker-Thistlethwaite code	6 10 16 20 14 4 18 2 12 8
Conway Notation	[.3.20.2]

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-8][-4]
Hyperbolic Volume	13.0165
A-Polynomial	See Data:10 93/A-polynomial

[edit Notes for 10 93's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	[math]\displaystyle{ 1 }[/math]
Topological 4 genus	[math]\displaystyle{ 1 }[/math]
Concordance genus	[math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant	2

[edit Notes for 10 93's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-8 t^2+15 t-17+15 t^{-1} -8 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+4 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 67, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+3 q^3-5 q^2+8 q-10+11 q^{-1} -10 q^{-2} +9 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^2 z^6+z^6-a^4 z^4+3 a^2 z^4-z^4 a^{-2} +3 z^4-2 a^4 z^2+3 a^2 z^2-2 z^2 a^{-2} +2 z^2-a^4+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+3 z^8 a^{-2} +9 z^8+9 a^3 z^7+5 a z^7-3 z^7 a^{-1} +z^7 a^{-3} +9 a^4 z^6-9 a^2 z^6-13 z^6 a^{-2} -31 z^6+6 a^5 z^5-17 a^3 z^5-29 a z^5-10 z^5 a^{-1} -4 z^5 a^{-3} +3 a^6 z^4-14 a^4 z^4-6 a^2 z^4+17 z^4 a^{-2} +28 z^4+a^7 z^3-4 a^5 z^3+7 a^3 z^3+25 a z^3+18 z^3 a^{-1} +5 z^3 a^{-3} +7 a^4 z^2+7 a^2 z^2-6 z^2 a^{-2} -6 z^2+a^5 z-a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} -a^4-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{18}+q^{16}-q^{14}-q^{12}+2 q^{10}-q^8+3 q^6+1-2 q^{-2} +2 q^{-4} + q^{-10} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+3 q^{92}-2 q^{90}-q^{88}+7 q^{86}-11 q^{84}+15 q^{82}-18 q^{80}+13 q^{78}-6 q^{76}-6 q^{74}+23 q^{72}-33 q^{70}+39 q^{68}-37 q^{66}+22 q^{64}-2 q^{62}-26 q^{60}+49 q^{58}-63 q^{56}+61 q^{54}-43 q^{52}+7 q^{50}+37 q^{48}-69 q^{46}+80 q^{44}-60 q^{42}+11 q^{40}+38 q^{38}-70 q^{36}+69 q^{34}-26 q^{32}-31 q^{30}+85 q^{28}-98 q^{26}+64 q^{24}+7 q^{22}-85 q^{20}+135 q^{18}-133 q^{16}+85 q^{14}-5 q^{12}-69 q^{10}+123 q^8-132 q^6+97 q^4-37 q^2-33+80 q^{-2} -93 q^{-4} +70 q^{-6} -15 q^{-8} -39 q^{-10} +75 q^{-12} -78 q^{-14} +39 q^{-16} +27 q^{-18} -87 q^{-20} +113 q^{-22} -92 q^{-24} +32 q^{-26} +43 q^{-28} -95 q^{-30} +111 q^{-32} -86 q^{-34} +36 q^{-36} +15 q^{-38} -54 q^{-40} +62 q^{-42} -47 q^{-44} +24 q^{-46} - q^{-48} -12 q^{-50} +13 q^{-52} -11 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_93.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+2 q^{11}-3 q^9+3 q^7-q^5+q^3+q-2 q^{-1} +3 q^{-3} -2 q^{-5} +2 q^{-7} - q^{-9} }[/math]
2	[math]\displaystyle{ q^{36}-2 q^{34}+q^{32}+3 q^{30}-8 q^{28}+5 q^{26}+6 q^{24}-15 q^{22}+8 q^{20}+11 q^{18}-17 q^{16}+q^{14}+15 q^{12}-7 q^{10}-10 q^8+10 q^6+7 q^4-12 q^2-1+16 q^{-2} -8 q^{-4} -12 q^{-6} +17 q^{-8} -17 q^{-12} +11 q^{-14} +8 q^{-16} -12 q^{-18} + q^{-20} +7 q^{-22} -3 q^{-24} -2 q^{-26} + q^{-28} }[/math]
3	[math]\displaystyle{ -q^{69}+2 q^{67}-q^{65}-q^{63}+2 q^{61}+q^{59}-3 q^{57}-2 q^{55}+9 q^{53}-3 q^{51}-17 q^{49}+11 q^{47}+30 q^{45}-18 q^{43}-52 q^{41}+16 q^{39}+77 q^{37}-4 q^{35}-89 q^{33}-25 q^{31}+87 q^{29}+54 q^{27}-59 q^{25}-78 q^{23}+20 q^{21}+86 q^{19}+21 q^{17}-77 q^{15}-54 q^{13}+61 q^{11}+75 q^9-41 q^7-85 q^5+23 q^3+87 q-3 q^{-1} -89 q^{-3} -16 q^{-5} +87 q^{-7} +40 q^{-9} -76 q^{-11} -64 q^{-13} +58 q^{-15} +84 q^{-17} -27 q^{-19} -91 q^{-21} -10 q^{-23} +82 q^{-25} +39 q^{-27} -55 q^{-29} -57 q^{-31} +23 q^{-33} +54 q^{-35} +4 q^{-37} -37 q^{-39} -17 q^{-41} +17 q^{-43} +18 q^{-45} -4 q^{-47} -10 q^{-49} -2 q^{-51} +3 q^{-53} +2 q^{-55} - q^{-57} }[/math]
4	[math]\displaystyle{ q^{112}-2 q^{110}+q^{108}+q^{106}-4 q^{104}+5 q^{102}-3 q^{100}+2 q^{98}-q^{96}-8 q^{94}+21 q^{92}-8 q^{90}-15 q^{88}-13 q^{86}+7 q^{84}+83 q^{82}-4 q^{80}-100 q^{78}-109 q^{76}+39 q^{74}+277 q^{72}+111 q^{70}-233 q^{68}-387 q^{66}-74 q^{64}+513 q^{62}+458 q^{60}-145 q^{58}-678 q^{56}-474 q^{54}+410 q^{52}+768 q^{50}+309 q^{48}-509 q^{46}-793 q^{44}-128 q^{42}+550 q^{40}+668 q^{38}+91 q^{36}-575 q^{34}-553 q^{32}-46 q^{30}+541 q^{28}+531 q^{26}-77 q^{24}-546 q^{22}-431 q^{20}+216 q^{18}+576 q^{16}+220 q^{14}-390 q^{12}-503 q^{10}+49 q^8+527 q^6+321 q^4-335 q^2-542-56 q^{-2} +526 q^{-4} +485 q^{-6} -221 q^{-8} -627 q^{-10} -325 q^{-12} +383 q^{-14} +694 q^{-16} +135 q^{-18} -498 q^{-20} -634 q^{-22} -59 q^{-24} +613 q^{-26} +526 q^{-28} -18 q^{-30} -584 q^{-32} -504 q^{-34} +125 q^{-36} +498 q^{-38} +436 q^{-40} -112 q^{-42} -477 q^{-44} -304 q^{-46} +63 q^{-48} +395 q^{-50} +255 q^{-52} -81 q^{-54} -257 q^{-56} -221 q^{-58} +62 q^{-60} +186 q^{-62} +129 q^{-64} -7 q^{-66} -133 q^{-68} -75 q^{-70} +4 q^{-72} +60 q^{-74} +56 q^{-76} -9 q^{-78} -24 q^{-80} -23 q^{-82} -3 q^{-84} +13 q^{-86} +5 q^{-88} +2 q^{-90} -3 q^{-92} -2 q^{-94} + q^{-96} }[/math]
5	[math]\displaystyle{ -q^{165}+2 q^{163}-q^{161}-q^{159}+4 q^{157}-3 q^{155}-3 q^{153}+4 q^{151}+q^{149}-3 q^{147}-q^{145}-2 q^{143}+4 q^{141}+17 q^{139}+4 q^{137}-38 q^{135}-51 q^{133}+11 q^{131}+107 q^{129}+122 q^{127}-11 q^{125}-243 q^{123}-313 q^{121}-4 q^{119}+491 q^{117}+650 q^{115}+115 q^{113}-809 q^{111}-1238 q^{109}-452 q^{107}+1165 q^{105}+2102 q^{103}+1111 q^{101}-1341 q^{99}-3132 q^{97}-2242 q^{95}+1087 q^{93}+4093 q^{91}+3762 q^{89}-189 q^{87}-4564 q^{85}-5343 q^{83}-1421 q^{81}+4137 q^{79}+6533 q^{77}+3434 q^{75}-2725 q^{73}-6744 q^{71}-5260 q^{69}+473 q^{67}+5755 q^{65}+6348 q^{63}+1950 q^{61}-3729 q^{59}-6235 q^{57}-3913 q^{55}+1169 q^{53}+5053 q^{51}+4933 q^{49}+1182 q^{47}-3207 q^{45}-4919 q^{43}-2825 q^{41}+1304 q^{39}+4200 q^{37}+3584 q^{35}+111 q^{33}-3238 q^{31}-3631 q^{29}-850 q^{27}+2482 q^{25}+3338 q^{23}+1013 q^{21}-2167 q^{19}-3097 q^{17}-875 q^{15}+2273 q^{13}+3155 q^{11}+796 q^9-2574 q^7-3589 q^5-1047 q^3+2801 q+4256 q^{-1} +1732 q^{-3} -2633 q^{-5} -4896 q^{-7} -2840 q^{-9} +1912 q^{-11} +5210 q^{-13} +4099 q^{-15} -572 q^{-17} -4877 q^{-19} -5195 q^{-21} -1220 q^{-23} +3785 q^{-25} +5716 q^{-27} +3068 q^{-29} -1949 q^{-31} -5348 q^{-33} -4535 q^{-35} -285 q^{-37} +4033 q^{-39} +5138 q^{-41} +2370 q^{-43} -1972 q^{-45} -4632 q^{-47} -3791 q^{-49} -286 q^{-51} +3173 q^{-53} +4111 q^{-55} +2099 q^{-57} -1171 q^{-59} -3339 q^{-61} -3014 q^{-63} -671 q^{-65} +1861 q^{-67} +2843 q^{-69} +1819 q^{-71} -260 q^{-73} -1892 q^{-75} -2060 q^{-77} -872 q^{-79} +691 q^{-81} +1552 q^{-83} +1277 q^{-85} +262 q^{-87} -733 q^{-89} -1082 q^{-91} -688 q^{-93} +51 q^{-95} +584 q^{-97} +638 q^{-99} +307 q^{-101} -134 q^{-103} -382 q^{-105} -329 q^{-107} -95 q^{-109} +117 q^{-111} +198 q^{-113} +145 q^{-115} +16 q^{-117} -75 q^{-119} -85 q^{-121} -43 q^{-123} +2 q^{-125} +30 q^{-127} +31 q^{-129} +8 q^{-131} -6 q^{-133} -8 q^{-135} -5 q^{-137} -2 q^{-139} +3 q^{-141} +2 q^{-143} - q^{-145} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{18}+q^{16}-q^{14}-q^{12}+2 q^{10}-q^8+3 q^6+1-2 q^{-2} +2 q^{-4} + q^{-10} - q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{52}-4 q^{50}+8 q^{48}-12 q^{46}+20 q^{44}-32 q^{42}+44 q^{40}-56 q^{38}+75 q^{36}-98 q^{34}+116 q^{32}-140 q^{30}+167 q^{28}-198 q^{26}+222 q^{24}-242 q^{22}+251 q^{20}-222 q^{18}+160 q^{16}-60 q^{14}-77 q^{12}+238 q^{10}-398 q^8+532 q^6-619 q^4+666 q^2-640+566 q^{-2} -444 q^{-4} +288 q^{-6} -118 q^{-8} -58 q^{-10} +201 q^{-12} -314 q^{-14} +382 q^{-16} -388 q^{-18} +352 q^{-20} -284 q^{-22} +212 q^{-24} -136 q^{-26} +74 q^{-28} -38 q^{-30} +14 q^{-32} -4 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{46}-q^{44}+2 q^{40}-2 q^{38}-2 q^{36}+3 q^{34}+q^{32}-5 q^{30}-4 q^{28}+7 q^{26}+2 q^{24}-11 q^{22}+3 q^{20}+9 q^{18}-5 q^{14}+3 q^{12}+4 q^{10}-3 q^8+4 q^4-q^2-3+7 q^{-2} -7 q^{-6} + q^{-8} +4 q^{-10} - q^{-12} -7 q^{-14} +2 q^{-16} +8 q^{-18} -4 q^{-22} + q^{-24} +3 q^{-26} -3 q^{-30} - q^{-32} + q^{-34} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{42}-2 q^{40}-q^{38}+6 q^{36}-3 q^{34}-7 q^{32}+10 q^{30}-q^{28}-12 q^{26}+11 q^{24}+2 q^{22}-13 q^{20}+8 q^{18}+4 q^{16}-9 q^{14}+3 q^{12}+4 q^{10}+2 q^8-5 q^6+2 q^4+11 q^2-8- q^{-2} +12 q^{-4} -8 q^{-6} -3 q^{-8} +10 q^{-10} -8 q^{-12} -2 q^{-14} +7 q^{-16} -6 q^{-18} + q^{-20} +2 q^{-22} -2 q^{-24} + q^{-26} }[/math]
1,0,0	[math]\displaystyle{ -q^{23}+q^{21}-2 q^{19}+q^{17}-2 q^{15}+2 q^{13}-q^{11}+3 q^9+q^7+q^5+q^3-q+ q^{-1} -2 q^{-3} +2 q^{-5} - q^{-7} +2 q^{-9} - q^{-11} + q^{-13} - q^{-15} }[/math]

A4 Invariants.

Weight

Invariant

0,1,0,0

[math]\displaystyle{ q^{52}-q^{50}-2 q^{48}+3 q^{46}+3 q^{44}-4 q^{42}-3 q^{40}+4 q^{38}+4 q^{36}-8 q^{34}-6 q^{32}+9 q^{30}-11 q^{26}+4 q^{24}+11 q^{22}-6 q^{20}-3 q^{18}+8 q^{16}+q^{14}-11 q^{12}+q^{10}+11 q^8-9 q^6-4 q^4+19 q^2+6-10 q^{-2} +7 q^{-4} +11 q^{-6} -7 q^{-8} -8 q^{-10} +3 q^{-12} +2 q^{-14} -8 q^{-16} - q^{-18} +5 q^{-20} -2 q^{-22} -2 q^{-24} +3 q^{-26} - q^{-30} + q^{-32} }[/math]

1,0,0,0

[math]\displaystyle{ -q^{28}+q^{26}-2 q^{24}-2 q^{18}+2 q^{16}-q^{14}+3 q^{12}+q^{10}+2 q^8+q^6+q^4-1+ q^{-2} -2 q^{-4} +2 q^{-6} - q^{-8} + q^{-10} + q^{-12} - q^{-14} + q^{-16} - q^{-18} }[/math]

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ -q^{42}+2 q^{40}-3 q^{38}+6 q^{36}-9 q^{34}+11 q^{32}-16 q^{30}+17 q^{28}-18 q^{26}+17 q^{24}-14 q^{22}+9 q^{20}-8 q^{16}+19 q^{14}-25 q^{12}+34 q^{10}-36 q^8+37 q^6-34 q^4+27 q^2-20+9 q^{-2} -8 q^{-6} +15 q^{-8} -18 q^{-10} +20 q^{-12} -18 q^{-14} +17 q^{-16} -12 q^{-18} +9 q^{-20} -6 q^{-22} +2 q^{-24} - q^{-26} }[/math]

1,0

[math]\displaystyle{ q^{68}-2 q^{64}-2 q^{62}+q^{60}+6 q^{58}+3 q^{56}-6 q^{54}-9 q^{52}-q^{50}+13 q^{48}+8 q^{46}-9 q^{44}-15 q^{42}+17 q^{38}+8 q^{36}-13 q^{34}-15 q^{32}+7 q^{30}+17 q^{28}-17 q^{24}-5 q^{22}+14 q^{20}+10 q^{18}-10 q^{16}-10 q^{14}+7 q^{12}+11 q^{10}-5 q^8-11 q^6+5 q^4+15 q^2+1-17 q^{-2} -7 q^{-4} +16 q^{-6} +16 q^{-8} -8 q^{-10} -21 q^{-12} - q^{-14} +19 q^{-16} +10 q^{-18} -13 q^{-20} -15 q^{-22} +5 q^{-24} +14 q^{-26} + q^{-28} -9 q^{-30} -5 q^{-32} +5 q^{-34} +4 q^{-36} -2 q^{-38} -2 q^{-40} + q^{-44} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{58}-2 q^{56}+q^{54}-2 q^{52}+6 q^{50}-6 q^{48}+5 q^{46}-9 q^{44}+13 q^{42}-12 q^{40}+11 q^{38}-15 q^{36}+14 q^{34}-12 q^{32}+10 q^{30}-9 q^{28}+3 q^{26}+q^{24}-6 q^{22}+12 q^{20}-17 q^{18}+22 q^{16}-24 q^{14}+29 q^{12}-28 q^{10}+30 q^8-25 q^6+25 q^4-19 q^2+17-7 q^{-2} +5 q^{-4} -6 q^{-8} +11 q^{-10} -13 q^{-12} +13 q^{-14} -17 q^{-16} +16 q^{-18} -15 q^{-20} +12 q^{-22} -11 q^{-24} +9 q^{-26} -6 q^{-28} +4 q^{-30} -2 q^{-32} + q^{-34} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+3 q^{92}-2 q^{90}-q^{88}+7 q^{86}-11 q^{84}+15 q^{82}-18 q^{80}+13 q^{78}-6 q^{76}-6 q^{74}+23 q^{72}-33 q^{70}+39 q^{68}-37 q^{66}+22 q^{64}-2 q^{62}-26 q^{60}+49 q^{58}-63 q^{56}+61 q^{54}-43 q^{52}+7 q^{50}+37 q^{48}-69 q^{46}+80 q^{44}-60 q^{42}+11 q^{40}+38 q^{38}-70 q^{36}+69 q^{34}-26 q^{32}-31 q^{30}+85 q^{28}-98 q^{26}+64 q^{24}+7 q^{22}-85 q^{20}+135 q^{18}-133 q^{16}+85 q^{14}-5 q^{12}-69 q^{10}+123 q^8-132 q^6+97 q^4-37 q^2-33+80 q^{-2} -93 q^{-4} +70 q^{-6} -15 q^{-8} -39 q^{-10} +75 q^{-12} -78 q^{-14} +39 q^{-16} +27 q^{-18} -87 q^{-20} +113 q^{-22} -92 q^{-24} +32 q^{-26} +43 q^{-28} -95 q^{-30} +111 q^{-32} -86 q^{-34} +36 q^{-36} +15 q^{-38} -54 q^{-40} +62 q^{-42} -47 q^{-44} +24 q^{-46} - q^{-48} -12 q^{-50} +13 q^{-52} -11 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60} }[/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 93"];

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

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

Out[4]= [math]\displaystyle{ 2 t^3-8 t^2+15 t-17+15 t^{-1} -8 t^{-2} +2 t^{-3} }[/math]

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

Out[5]= [math]\displaystyle{ 2 z^6+4 z^4+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]= { 67, -2 }

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

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

Out[8]= [math]\displaystyle{ -q^4+3 q^3-5 q^2+8 q-10+11 q^{-1} -10 q^{-2} +9 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6} }[/math]

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

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

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

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

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

Out[10]= [math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+3 z^8 a^{-2} +9 z^8+9 a^3 z^7+5 a z^7-3 z^7 a^{-1} +z^7 a^{-3} +9 a^4 z^6-9 a^2 z^6-13 z^6 a^{-2} -31 z^6+6 a^5 z^5-17 a^3 z^5-29 a z^5-10 z^5 a^{-1} -4 z^5 a^{-3} +3 a^6 z^4-14 a^4 z^4-6 a^2 z^4+17 z^4 a^{-2} +28 z^4+a^7 z^3-4 a^5 z^3+7 a^3 z^3+25 a z^3+18 z^3 a^{-1} +5 z^3 a^{-3} +7 a^4 z^2+7 a^2 z^2-6 z^2 a^{-2} -6 z^2+a^5 z-a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} -a^4-2 a^2 }[/math]

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

[math]\displaystyle{ 4 }[/math]

[math]\displaystyle{ -8 }[/math]

[math]\displaystyle{ 8 }[/math]

[math]\displaystyle{ -\frac{34}{3} }[/math]

[math]\displaystyle{ -\frac{86}{3} }[/math]

[math]\displaystyle{ -32 }[/math]

[math]\displaystyle{ -\frac{176}{3} }[/math]

[math]\displaystyle{ \frac{160}{3} }[/math]

[math]\displaystyle{ -72 }[/math]

[math]\displaystyle{ \frac{32}{3} }[/math]

[math]\displaystyle{ 32 }[/math]

[math]\displaystyle{ -\frac{136}{3} }[/math]

[math]\displaystyle{ -\frac{344}{3} }[/math]

[math]\displaystyle{ \frac{751}{30} }[/math]

[math]\displaystyle{ \frac{698}{15} }[/math]

[math]\displaystyle{ -\frac{3658}{45} }[/math]

[math]\displaystyle{ -\frac{655}{18} }[/math]

[math]\displaystyle{ -\frac{2129}{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]-2 is the signature of 10 93. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

9

1

-1

7

2

5

3

1

-2

3

5

2

3

1

5

3

-2

-1

6

5

1

-3

5

6

1

-5

4

5

-1

-7

2

5

3

-9

1

4

-3

-11

2

-13

1

-1

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, 93]]
Out[2]=	10
In[3]:=	PD[Knot[10, 93]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 6, 17, 5], X[20, 8, 1, 7], X[18, 13, 19, 14], X[14, 9, 15, 10], X[10, 3, 11, 4], X[4, 11, 5, 12], X[12, 17, 13, 18], X[8, 20, 9, 19], X[2, 16, 3, 15]]
In[4]:=	GaussCode[Knot[10, 93]]
Out[4]=	GaussCode[1, -10, 6, -7, 2, -1, 3, -9, 5, -6, 7, -8, 4, -5, 10, -2, 8, -4, 9, -3]
In[5]:=	BR[Knot[10, 93]]
Out[5]=	BR[4, {-1, -1, 2, -1, -1, 2, -1, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[10, 93]][t]
Out[6]=	2 8 15 2 3 -17 + -- - -- + -- + 15 t - 8 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 93]][z]
Out[7]=	2 4 6 1 + z + 4 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 93]}
In[9]:=	{KnotDet[Knot[10, 93]], KnotSignature[Knot[10, 93]]}
Out[9]=	{67, -2}
In[10]:=	J=Jones[Knot[10, 93]][q]
Out[10]=	-6 3 6 9 10 11 2 3 4 -10 - q + -- - -- + -- - -- + -- + 8 q - 5 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 93]}
In[12]:=	A2Invariant[Knot[10, 93]][q]
Out[12]=	-18 -16 -14 -12 2 -8 3 2 4 10 12 1 - q + q - q - q + --- - q + -- - 2 q + 2 q + q - q 10 6 q q
In[13]:=	Kauffman[Knot[10, 93]][a, z]
Out[13]=	2 2 4 2 z 6 z 3 5 2 6 z 2 2 -2 a - a - --- - --- - 6 a z - a z + a z - 6 z - ---- + 7 a z + 3 a 2 a a 3 3 4 2 5 z 18 z 3 3 3 5 3 7 3 7 a z + ---- + ----- + 25 a z + 7 a z - 4 a z + a z + 3 a a 4 5 5 4 17 z 2 4 4 4 6 4 4 z 10 z 28 z + ----- - 6 a z - 14 a z + 3 a z - ---- - ----- - 2 3 a a a 6 5 3 5 5 5 6 13 z 2 6 4 6 29 a z - 17 a z + 6 a z - 31 z - ----- - 9 a z + 9 a z + 2 a 7 7 8 9 z 3 z 7 3 7 8 3 z 2 8 2 z 9 -- - ---- + 5 a z + 9 a z + 9 z + ---- + 6 a z + ---- + 2 a z 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[10, 93]], Vassiliev[3][Knot[10, 93]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 93]][q, t]
Out[15]=	6 6 1 2 1 4 2 5 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t 5 5 5 t 2 3 2 3 3 5 3 ---- + ---- + --- + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + 5 3 q q t q t 5 4 7 4 9 5 q t + 2 q t + q t

10 93

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