8 16

8_15

8_17

Visit 8 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 16's page at Knotilus! Visit 8 16's page at the original Knot Atlas! 8 16 Quick Notes

Square depiction.

Knot presentations

Planar diagram presentation	X6271 X14,6,15,5 X16,11,1,12 X12,7,13,8 X8394 X4,9,5,10 X10,15,11,16 X2,14,3,13
Gauss code	1, -8, 5, -6, 2, -1, 4, -5, 6, -7, 3, -4, 8, -2, 7, -3
Dowker-Thistlethwaite code	6 8 14 12 4 16 2 10
Conway Notation	[.2.20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-2] Hyperbolic Volume 10.579 A-Polynomial See Data:8 16/A-polynomial

[edit Notes for 8 16'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 8 16's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 35, -2 }
Jones polynomial	[math]\displaystyle{ -q^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^2 z^6-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^2-2 z^2-a^4+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^3 a^7+3 z^4 a^6-z^2 a^6+5 z^5 a^5-5 z^3 a^5+2 z a^5+5 z^6 a^4-7 z^4 a^4+4 z^2 a^4-a^4+2 z^7 a^3+3 z^5 a^3-10 z^3 a^3+4 z a^3+8 z^6 a^2-18 z^4 a^2+10 z^2 a^2-2 a^2+2 z^7 a-z^5 a-6 z^3 a+3 z a+3 z^6-8 z^4+5 z^2+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} }[/math]
The A2 invariant	[math]\displaystyle{ -q^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ q^{-4} - q^{-6} }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+2 q^{92}-q^{90}-2 q^{88}+9 q^{86}-12 q^{84}+15 q^{82}-14 q^{80}+7 q^{78}+2 q^{76}-16 q^{74}+28 q^{72}-31 q^{70}+24 q^{68}-10 q^{66}-11 q^{64}+26 q^{62}-30 q^{60}+21 q^{58}-5 q^{56}-15 q^{54}+23 q^{52}-19 q^{50}+2 q^{48}+22 q^{46}-36 q^{44}+36 q^{42}-20 q^{40}-4 q^{38}+30 q^{36}-45 q^{34}+46 q^{32}-33 q^{30}+12 q^{28}+14 q^{26}-32 q^{24}+39 q^{22}-30 q^{20}+14 q^{18}+5 q^{16}-20 q^{14}+24 q^{12}-14 q^{10}-q^8+21 q^6-29 q^4+26 q^2-6-17 q^{-2} +35 q^{-4} -37 q^{-6} +28 q^{-8} -10 q^{-10} -12 q^{-12} +24 q^{-14} -25 q^{-16} +20 q^{-18} -9 q^{-20} -2 q^{-22} +6 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_16.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+2 q^{11}-2 q^9+q^7+2 q- q^{-1} +2 q^{-3} - q^{-5} }[/math]
2	[math]\displaystyle{ q^{36}-2 q^{34}+5 q^{30}-7 q^{28}-2 q^{26}+11 q^{24}-6 q^{22}-5 q^{20}+8 q^{18}-q^{16}-5 q^{14}+q^{12}+5 q^{10}-2 q^8-5 q^6+7 q^4+2 q^2-9+6 q^{-2} +6 q^{-4} -8 q^{-6} + q^{-8} +6 q^{-10} -3 q^{-12} -2 q^{-14} + q^{-16} }[/math]
3	[math]\displaystyle{ -q^{69}+2 q^{67}-3 q^{63}+q^{61}+6 q^{59}-16 q^{55}+26 q^{51}+6 q^{49}-33 q^{47}-19 q^{45}+35 q^{43}+28 q^{41}-30 q^{39}-32 q^{37}+19 q^{35}+34 q^{33}-5 q^{31}-28 q^{29}-8 q^{27}+20 q^{25}+15 q^{23}-12 q^{21}-24 q^{19}+5 q^{17}+29 q^{15}+2 q^{13}-32 q^{11}-6 q^9+34 q^7+16 q^5-32 q^3-25 q+28 q^{-1} +31 q^{-3} -16 q^{-5} -35 q^{-7} +5 q^{-9} +33 q^{-11} +7 q^{-13} -24 q^{-15} -13 q^{-17} +12 q^{-19} +15 q^{-21} -4 q^{-23} -9 q^{-25} -2 q^{-27} +3 q^{-29} +2 q^{-31} - q^{-33} }[/math]
4	[math]\displaystyle{ q^{112}-2 q^{110}+3 q^{106}-3 q^{104}-4 q^{100}+8 q^{98}+13 q^{96}-16 q^{94}-17 q^{92}-17 q^{90}+36 q^{88}+65 q^{86}-17 q^{84}-75 q^{82}-91 q^{80}+44 q^{78}+160 q^{76}+60 q^{74}-91 q^{72}-198 q^{70}-38 q^{68}+185 q^{66}+163 q^{64}-9 q^{62}-209 q^{60}-134 q^{58}+86 q^{56}+165 q^{54}+89 q^{52}-103 q^{50}-139 q^{48}-29 q^{46}+81 q^{44}+115 q^{42}+12 q^{40}-83 q^{38}-95 q^{36}+3 q^{34}+104 q^{32}+79 q^{30}-45 q^{28}-130 q^{26}-38 q^{24}+99 q^{22}+130 q^{20}-20 q^{18}-163 q^{16}-79 q^{14}+87 q^{12}+180 q^{10}+32 q^8-165 q^6-139 q^4+17 q^2+193+119 q^{-2} -89 q^{-4} -164 q^{-6} -94 q^{-8} +114 q^{-10} +158 q^{-12} +35 q^{-14} -91 q^{-16} -144 q^{-18} -9 q^{-20} +90 q^{-22} +89 q^{-24} +18 q^{-26} -80 q^{-28} -56 q^{-30} -4 q^{-32} +41 q^{-34} +45 q^{-36} -6 q^{-38} -20 q^{-40} -20 q^{-42} -3 q^{-44} +12 q^{-46} +5 q^{-48} +2 q^{-50} -3 q^{-52} -2 q^{-54} + q^{-56} }[/math]
5	[math]\displaystyle{ -q^{165}+2 q^{163}-3 q^{159}+3 q^{157}+2 q^{155}-2 q^{153}-4 q^{151}-5 q^{149}+16 q^{145}+23 q^{143}-5 q^{141}-47 q^{139}-56 q^{137}+q^{135}+88 q^{133}+133 q^{131}+54 q^{129}-148 q^{127}-270 q^{125}-158 q^{123}+161 q^{121}+436 q^{119}+366 q^{117}-87 q^{115}-593 q^{113}-639 q^{111}-92 q^{109}+642 q^{107}+901 q^{105}+381 q^{103}-545 q^{101}-1081 q^{99}-686 q^{97}+319 q^{95}+1079 q^{93}+926 q^{91}-9 q^{89}-909 q^{87}-1027 q^{85}-283 q^{83}+632 q^{81}+953 q^{79}+492 q^{77}-306 q^{75}-772 q^{73}-588 q^{71}+26 q^{69}+532 q^{67}+573 q^{65}+188 q^{63}-301 q^{61}-515 q^{59}-314 q^{57}+121 q^{55}+446 q^{53}+394 q^{51}-6 q^{49}-406 q^{47}-447 q^{45}-64 q^{43}+409 q^{41}+512 q^{39}+102 q^{37}-444 q^{35}-591 q^{33}-159 q^{31}+485 q^{29}+704 q^{27}+242 q^{25}-502 q^{23}-819 q^{21}-375 q^{19}+450 q^{17}+918 q^{15}+557 q^{13}-325 q^{11}-945 q^9-746 q^7+108 q^5+868 q^3+905 q+178 q^{-1} -685 q^{-3} -956 q^{-5} -454 q^{-7} +383 q^{-9} +878 q^{-11} +668 q^{-13} -46 q^{-15} -663 q^{-17} -733 q^{-19} -257 q^{-21} +361 q^{-23} +642 q^{-25} +439 q^{-27} -53 q^{-29} -440 q^{-31} -465 q^{-33} -162 q^{-35} +194 q^{-37} +354 q^{-39} +259 q^{-41} +4 q^{-43} -201 q^{-45} -224 q^{-47} -99 q^{-49} +52 q^{-51} +134 q^{-53} +115 q^{-55} +21 q^{-57} -52 q^{-59} -68 q^{-61} -39 q^{-63} +26 q^{-67} +28 q^{-69} +8 q^{-71} -5 q^{-73} -8 q^{-75} -5 q^{-77} -2 q^{-79} +3 q^{-81} +2 q^{-83} - q^{-85} }[/math]
6	[math]\displaystyle{ q^{228}-2 q^{226}+3 q^{222}-3 q^{220}-2 q^{218}+10 q^{214}+q^{212}-8 q^{210}-19 q^{206}-10 q^{204}+21 q^{202}+57 q^{200}+33 q^{198}-31 q^{196}-73 q^{194}-135 q^{192}-75 q^{190}+106 q^{188}+304 q^{186}+281 q^{184}+25 q^{182}-308 q^{180}-659 q^{178}-562 q^{176}+42 q^{174}+883 q^{172}+1266 q^{170}+813 q^{168}-284 q^{166}-1689 q^{164}-2162 q^{162}-1163 q^{160}+975 q^{158}+2814 q^{156}+3056 q^{154}+1343 q^{152}-1837 q^{150}-4170 q^{148}-3998 q^{146}-992 q^{144}+2915 q^{142}+5291 q^{140}+4527 q^{138}+469 q^{136}-4039 q^{134}-6182 q^{132}-4287 q^{130}+289 q^{128}+4762 q^{126}+6374 q^{124}+3726 q^{122}-1108 q^{120}-5157 q^{118}-5719 q^{116}-2863 q^{114}+1612 q^{112}+4916 q^{110}+4828 q^{108}+1928 q^{106}-1918 q^{104}-4121 q^{102}-3794 q^{100}-1232 q^{98}+1821 q^{96}+3366 q^{94}+2838 q^{92}+684 q^{90}-1514 q^{88}-2714 q^{86}-2150 q^{84}-380 q^{82}+1434 q^{80}+2265 q^{78}+1615 q^{76}+57 q^{74}-1565 q^{72}-2025 q^{70}-1146 q^{68}+573 q^{66}+1878 q^{64}+1816 q^{62}+420 q^{60}-1428 q^{58}-2213 q^{56}-1420 q^{54}+671 q^{52}+2378 q^{50}+2383 q^{48}+523 q^{46}-1971 q^{44}-3147 q^{42}-2159 q^{40}+730 q^{38}+3282 q^{36}+3646 q^{34}+1333 q^{32}-2119 q^{30}-4283 q^{28}-3695 q^{26}-273 q^{24}+3444 q^{22}+5009 q^{20}+3166 q^{18}-818 q^{16}-4378 q^{14}-5287 q^{12}-2546 q^{10}+1786 q^8+5046 q^6+5012 q^4+1936 q^2-2346-5245 q^{-2} -4675 q^{-4} -1385 q^{-6} +2685 q^{-8} +4881 q^{-10} +4252 q^{-12} +1110 q^{-14} -2616 q^{-16} -4443 q^{-18} -3709 q^{-20} -876 q^{-22} +2111 q^{-24} +3844 q^{-26} +3229 q^{-28} +835 q^{-30} -1658 q^{-32} -3060 q^{-34} -2633 q^{-36} -978 q^{-38} +1160 q^{-40} +2332 q^{-42} +2106 q^{-44} +900 q^{-46} -623 q^{-48} -1572 q^{-50} -1682 q^{-52} -795 q^{-54} +263 q^{-56} +992 q^{-58} +1131 q^{-60} +696 q^{-62} +16 q^{-64} -602 q^{-66} -702 q^{-68} -501 q^{-70} -106 q^{-72} +244 q^{-74} +412 q^{-76} +347 q^{-78} +92 q^{-80} -78 q^{-82} -190 q^{-84} -179 q^{-86} -97 q^{-88} +17 q^{-90} +83 q^{-92} +70 q^{-94} +51 q^{-96} +7 q^{-98} -20 q^{-100} -34 q^{-102} -16 q^{-104} + q^{-108} +8 q^{-110} +5 q^{-112} +2 q^{-114} -3 q^{-116} -2 q^{-118} + q^{-120} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ q^{-4} - q^{-6} }[/math]
1,1	[math]\displaystyle{ q^{52}-4 q^{50}+8 q^{48}-12 q^{46}+22 q^{44}-36 q^{42}+46 q^{40}-62 q^{38}+82 q^{36}-92 q^{34}+96 q^{32}-90 q^{30}+67 q^{28}-34 q^{26}-18 q^{24}+70 q^{22}-120 q^{20}+166 q^{18}-188 q^{16}+200 q^{14}-190 q^{12}+164 q^{10}-124 q^8+70 q^6-20 q^4-28 q^2+76-100 q^{-2} +115 q^{-4} -106 q^{-6} +94 q^{-8} -70 q^{-10} +44 q^{-12} -28 q^{-14} +12 q^{-16} -4 q^{-18} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{46}-q^{44}+2 q^{40}-2 q^{38}-2 q^{36}+3 q^{32}-2 q^{30}-4 q^{28}+4 q^{26}+2 q^{24}-3 q^{22}+4 q^{18}-q^{16}-q^{14}+2 q^{12}-3 q^8-q^6+4 q^4-2 q^2+6 q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-10} -3 q^{-14} - q^{-16} + q^{-18} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{42}-2 q^{40}-q^{38}+5 q^{36}-3 q^{34}-3 q^{32}+8 q^{30}-4 q^{28}-5 q^{26}+6 q^{24}-4 q^{22}-4 q^{20}+2 q^{18}+q^{16}-q^{12}+5 q^{10}+4 q^8-5 q^6+4 q^4+6 q^2-6+2 q^{-2} +4 q^{-4} -6 q^{-6} +2 q^{-8} + q^{-10} -2 q^{-12} + q^{-14} }[/math]
1,0,0	[math]\displaystyle{ -q^{23}+q^{21}-2 q^{19}+q^{17}-q^{15}+q^{13}+q^9+q^7+2 q^3+2 q^{-1} - q^{-3} + q^{-5} - q^{-7} }[/math]
1,0,1	[math]\displaystyle{ q^{68}-4 q^{66}+6 q^{64}-2 q^{62}-7 q^{60}+19 q^{58}-24 q^{56}+9 q^{54}+19 q^{52}-46 q^{50}+54 q^{48}-25 q^{46}-26 q^{44}+76 q^{42}-97 q^{40}+71 q^{38}-10 q^{36}-66 q^{34}+109 q^{32}-114 q^{30}+64 q^{28}-2 q^{26}-41 q^{24}+53 q^{22}-21 q^{20}+3 q^{18}+3 q^{16}+30 q^{14}-75 q^{12}+94 q^{10}-80 q^8+18 q^6+56 q^4-104 q^2+124-82 q^{-2} +36 q^{-4} +28 q^{-6} -59 q^{-8} +60 q^{-10} -43 q^{-12} +10 q^{-14} +7 q^{-16} -14 q^{-18} +10 q^{-20} -4 q^{-22} + q^{-24} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{52}-q^{50}-2 q^{48}+3 q^{46}+q^{44}-5 q^{42}+2 q^{40}+6 q^{38}-q^{36}-4 q^{34}+3 q^{32}-9 q^{28}-5 q^{26}+3 q^{24}-4 q^{22}-5 q^{20}+10 q^{18}+4 q^{16}-2 q^{14}+7 q^{12}+9 q^{10}-2 q^8-q^6+5 q^4+2 q^2-5- q^{-2} +3 q^{-4} -2 q^{-6} -2 q^{-8} +2 q^{-10} - q^{-14} + q^{-16} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{28}+q^{26}-2 q^{24}-q^{18}+q^{16}+2 q^{12}+2 q^8+2 q^4+1+ q^{-2} - q^{-4} + q^{-6} - q^{-8} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{42}+2 q^{40}-3 q^{38}+5 q^{36}-7 q^{34}+7 q^{32}-8 q^{30}+6 q^{28}-3 q^{26}+4 q^{22}-6 q^{20}+10 q^{18}-13 q^{16}+14 q^{14}-13 q^{12}+11 q^{10}-8 q^8+5 q^6-2 q^2+6-6 q^{-2} +8 q^{-4} -6 q^{-6} +6 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-14} }[/math]
1,0	[math]\displaystyle{ q^{68}-2 q^{64}-2 q^{62}+q^{60}+5 q^{58}+2 q^{56}-5 q^{54}-5 q^{52}+2 q^{50}+8 q^{48}+q^{46}-8 q^{44}-4 q^{42}+5 q^{40}+5 q^{38}-4 q^{36}-6 q^{34}+q^{32}+6 q^{30}-6 q^{26}-q^{24}+5 q^{22}+3 q^{20}-3 q^{18}-3 q^{16}+4 q^{14}+5 q^{12}-2 q^{10}-6 q^8+2 q^6+8 q^4+4 q^2-6-6 q^{-2} +4 q^{-4} +8 q^{-6} - q^{-8} -6 q^{-10} -3 q^{-12} +4 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{58}-2 q^{56}+q^{54}-2 q^{52}+5 q^{50}-5 q^{48}+4 q^{46}-5 q^{44}+8 q^{42}-5 q^{40}+3 q^{38}-4 q^{36}+q^{32}-5 q^{30}+2 q^{28}-9 q^{26}+9 q^{24}-9 q^{22}+11 q^{20}-9 q^{18}+12 q^{16}-5 q^{14}+10 q^{12}-4 q^{10}+4 q^8+q^6+2 q^2-4+6 q^{-2} -6 q^{-4} +5 q^{-6} -6 q^{-8} +5 q^{-10} -4 q^{-12} +3 q^{-14} -2 q^{-16} + q^{-18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-4 q^{94}+2 q^{92}-q^{90}-2 q^{88}+9 q^{86}-12 q^{84}+15 q^{82}-14 q^{80}+7 q^{78}+2 q^{76}-16 q^{74}+28 q^{72}-31 q^{70}+24 q^{68}-10 q^{66}-11 q^{64}+26 q^{62}-30 q^{60}+21 q^{58}-5 q^{56}-15 q^{54}+23 q^{52}-19 q^{50}+2 q^{48}+22 q^{46}-36 q^{44}+36 q^{42}-20 q^{40}-4 q^{38}+30 q^{36}-45 q^{34}+46 q^{32}-33 q^{30}+12 q^{28}+14 q^{26}-32 q^{24}+39 q^{22}-30 q^{20}+14 q^{18}+5 q^{16}-20 q^{14}+24 q^{12}-14 q^{10}-q^8+21 q^6-29 q^4+26 q^2-6-17 q^{-2} +35 q^{-4} -37 q^{-6} +28 q^{-8} -10 q^{-10} -12 q^{-12} +24 q^{-14} -25 q^{-16} +20 q^{-18} -9 q^{-20} -2 q^{-22} +6 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32} }[/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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 16"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+2 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]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 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-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^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{ z^3 a^7+3 z^4 a^6-z^2 a^6+5 z^5 a^5-5 z^3 a^5+2 z a^5+5 z^6 a^4-7 z^4 a^4+4 z^2 a^4-a^4+2 z^7 a^3+3 z^5 a^3-10 z^3 a^3+4 z a^3+8 z^6 a^2-18 z^4 a^2+10 z^2 a^2-2 a^2+2 z^7 a-z^5 a-6 z^3 a+3 z a+3 z^6-8 z^4+5 z^2+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} }[/math]

Vassiliev invariants

V2 and V3:

(1, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 4 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -\frac{80}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{56}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ \frac{1951}{30} }[/math] [math]\displaystyle{ -\frac{114}{5} }[/math] [math]\displaystyle{ -\frac{298}{45} }[/math] [math]\displaystyle{ -\frac{31}{18} }[/math] [math]\displaystyle{ -\frac{449}{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 8 16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 χ 5                 1 -1 3               2   2 1             2 1   -1 -1           4 2     2 -3         3 3       0 -5       3 3         0 -7     2 3           1 -9   1 3             -2 -11   2               2 -13 1                 -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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[8, 16]]
Out[2]=	8
In[3]:=	PD[Knot[8, 16]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 6, 15, 5], X[16, 11, 1, 12], X[12, 7, 13, 8], X[8, 3, 9, 4], X[4, 9, 5, 10], X[10, 15, 11, 16], X[2, 14, 3, 13]]
In[4]:=	GaussCode[Knot[8, 16]]
Out[4]=	GaussCode[1, -8, 5, -6, 2, -1, 4, -5, 6, -7, 3, -4, 8, -2, 7, -3]
In[5]:=	BR[Knot[8, 16]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[8, 16]][t]
Out[6]=	-3 4 8 2 3 -9 + t - -- + - + 8 t - 4 t + t 2 t t
In[7]:=	Conway[Knot[8, 16]][z]
Out[7]=	2 4 6 1 + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 16], Knot[10, 156], Knot[11, NonAlternating, 15], Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[9]:=	{KnotDet[Knot[8, 16]], KnotSignature[Knot[8, 16]]}
Out[9]=	{35, -2}
In[10]:=	J=Jones[Knot[8, 16]][q]
Out[10]=	-6 3 5 6 6 6 2 -4 - 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[8, 16], Knot[10, 156]}
In[12]:=	A2Invariant[Knot[8, 16]][q]
Out[12]=	-18 -16 -14 -10 -8 2 -4 2 4 6 1 - q + q - q + q - q + -- - q + -- + q - q 6 2 q q
In[13]:=	Kauffman[Knot[8, 16]][a, z]
Out[13]=	2 4 z 3 5 2 2 2 4 2 -2 a - a + - + 3 a z + 4 a z + 2 a z + 5 z + 10 a z + 4 a z - a 3 6 2 2 z 3 3 3 5 3 7 3 4 a z - ---- - 6 a z - 10 a z - 5 a z + a z - 8 z - a 5 2 4 4 4 6 4 z 5 3 5 5 5 6 18 a z - 7 a z + 3 a z + -- - a z + 3 a z + 5 a z + 3 z + a 2 6 4 6 7 3 7 8 a z + 5 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[8, 16]], Vassiliev[3][Knot[8, 16]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[8, 16]][q, t]
Out[15]=	3 4 1 2 1 3 2 3 3 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 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 3 3 2 t 2 3 2 5 3 ---- + ---- + --- + 2 q t + q t + 2 q t + q t 5 3 q q t q t