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]

Minimum Braid Representative: Length is 8, width is 3. Braid index is 3.

A Morse Link Presentation:

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 $1$ 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 3} Rasmussen s-Invariant 2

[edit Notes for 8 16'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^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+2 z^4+z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 35, -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+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }
HOMFLY-PT polynomial (db, data sources)	$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-z^{4}-2a^{4}z^{2}+5a^{2}z^{2}-2z^{2}-a^{4}+2a^{2}$
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^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} }
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^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ 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^{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} }

Further Quantum Invariants

Further quantum knot invariants for 8_16.

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^{13}+2 q^{11}-2 q^9+q^7+2 q- q^{-1} +2 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^{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} }
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^{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} }
4	$q^{112}-2q^{110}+3q^{106}-3q^{104}-4q^{100}+8q^{98}+13q^{96}-16q^{94}-17q^{92}-17q^{90}+36q^{88}+65q^{86}-17q^{84}-75q^{82}-91q^{80}+44q^{78}+160q^{76}+60q^{74}-91q^{72}-198q^{70}-38q^{68}+185q^{66}+163q^{64}-9q^{62}-209q^{60}-134q^{58}+86q^{56}+165q^{54}+89q^{52}-103q^{50}-139q^{48}-29q^{46}+81q^{44}+115q^{42}+12q^{40}-83q^{38}-95q^{36}+3q^{34}+104q^{32}+79q^{30}-45q^{28}-130q^{26}-38q^{24}+99q^{22}+130q^{20}-20q^{18}-163q^{16}-79q^{14}+87q^{12}+180q^{10}+32q^{8}-165q^{6}-139q^{4}+17q^{2}+193+119q^{-2}-89q^{-4}-164q^{-6}-94q^{-8}+114q^{-10}+158q^{-12}+35q^{-14}-91q^{-16}-144q^{-18}-9q^{-20}+90q^{-22}+89q^{-24}+18q^{-26}-80q^{-28}-56q^{-30}-4q^{-32}+41q^{-34}+45q^{-36}-6q^{-38}-20q^{-40}-20q^{-42}-3q^{-44}+12q^{-46}+5q^{-48}+2q^{-50}-3q^{-52}-2q^{-54}+q^{-56}$
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^{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} }
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 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} }

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^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ 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^{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} }
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^{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} }

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^{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} }
1,0,0	$-q^{23}+q^{21}-2q^{19}+q^{17}-q^{15}+q^{13}+q^{9}+q^{7}+2q^{3}+2q^{-1}-q^{-3}+q^{-5}-q^{-7}$
1,0,1	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^{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} }

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^{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} }
1,0,0,0	$-q^{28}+q^{26}-2q^{24}-q^{18}+q^{16}+2q^{12}+2q^{8}+2q^{4}+1+q^{-2}-q^{-4}+q^{-6}-q^{-8}$

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

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

.

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. 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]=

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^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+2 z^4+z^2+1}

In[6]:=

Alexander[K, 2][t]

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

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 35, -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+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{2}z^{6}-a^{4}z^{4}+4a^{2}z^{4}-z^{4}-2a^{4}z^{2}+5a^{2}z^{2}-2z^{2}-a^{4}+2a^{2}$

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^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} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_156, K11n15, K11n56, K11n58, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_156, ...}

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 $4$ $-8$ $8$ ${\frac {14}{3}}$ $-{\frac {38}{3}}$ $-32$ $-{\frac {80}{3}}$ ${\frac {64}{3}}$ $-8$ ${\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ $-{\frac {152}{3}}$ ${\frac {1951}{30}}$ $-{\frac {114}{5}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{298}{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{31}{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{449}{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 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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the 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 n+1} -dimensional representation of 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 sl(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 n}	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_n}
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^7-3 q^6-q^5+10 q^4-8 q^3-10 q^2+24 q-8-25 q^{-1} +35 q^{-2} -3 q^{-3} -37 q^{-4} +38 q^{-5} +4 q^{-6} -41 q^{-7} +32 q^{-8} +8 q^{-9} -32 q^{-10} +19 q^{-11} +7 q^{-12} -15 q^{-13} +6 q^{-14} +2 q^{-15} -3 q^{-16} + q^{-17} }
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^{15}+3 q^{14}+q^{13}-5 q^{12}-8 q^{11}+8 q^{10}+20 q^9-8 q^8-33 q^7-3 q^6+51 q^5+18 q^4-61 q^3-43 q^2+70 q+65-64 q^{-1} -96 q^{-2} +63 q^{-3} +113 q^{-4} -46 q^{-5} -136 q^{-6} +37 q^{-7} +147 q^{-8} -19 q^{-9} -160 q^{-10} +8 q^{-11} +159 q^{-12} +8 q^{-13} -155 q^{-14} -20 q^{-15} +139 q^{-16} +31 q^{-17} -116 q^{-18} -35 q^{-19} +88 q^{-20} +33 q^{-21} -58 q^{-22} -28 q^{-23} +34 q^{-24} +19 q^{-25} -19 q^{-26} -8 q^{-27} +8 q^{-28} +3 q^{-29} -3 q^{-30} -2 q^{-31} +3 q^{-32} - 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^{26}-3 q^{25}-q^{24}+5 q^{23}+3 q^{22}+8 q^{21}-18 q^{20}-18 q^{19}+5 q^{18}+17 q^{17}+59 q^{16}-22 q^{15}-63 q^{14}-47 q^{13}-7 q^{12}+157 q^{11}+49 q^{10}-62 q^9-146 q^8-142 q^7+210 q^6+175 q^5+61 q^4-190 q^3-350 q^2+140 q+250+269 q^{-1} -116 q^{-2} -526 q^{-3} -16 q^{-4} +224 q^{-5} +466 q^{-6} +32 q^{-7} -619 q^{-8} -182 q^{-9} +140 q^{-10} +609 q^{-11} +182 q^{-12} -650 q^{-13} -319 q^{-14} +48 q^{-15} +694 q^{-16} +306 q^{-17} -625 q^{-18} -420 q^{-19} -50 q^{-20} +706 q^{-21} +401 q^{-22} -522 q^{-23} -454 q^{-24} -160 q^{-25} +596 q^{-26} +437 q^{-27} -330 q^{-28} -378 q^{-29} -239 q^{-30} +376 q^{-31} +362 q^{-32} -130 q^{-33} -206 q^{-34} -217 q^{-35} +153 q^{-36} +202 q^{-37} -23 q^{-38} -55 q^{-39} -117 q^{-40} +37 q^{-41} +67 q^{-42} -7 q^{-43} +3 q^{-44} -35 q^{-45} +8 q^{-46} +14 q^{-47} -7 q^{-48} +4 q^{-49} -6 q^{-50} +3 q^{-51} +2 q^{-52} -3 q^{-53} + q^{-54} }
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^{40}+3 q^{39}+q^{38}-5 q^{37}-3 q^{36}-3 q^{35}+2 q^{34}+16 q^{33}+21 q^{32}-7 q^{31}-29 q^{30}-42 q^{29}-27 q^{28}+32 q^{27}+94 q^{26}+87 q^{25}-10 q^{24}-124 q^{23}-178 q^{22}-93 q^{21}+117 q^{20}+292 q^{19}+245 q^{18}-29 q^{17}-338 q^{16}-449 q^{15}-186 q^{14}+317 q^{13}+632 q^{12}+463 q^{11}-135 q^{10}-730 q^9-804 q^8-159 q^7+702 q^6+1080 q^5+579 q^4-520 q^3-1299 q^2-996 q+200+1351 q^{-1} +1442 q^{-2} +207 q^{-3} -1336 q^{-4} -1756 q^{-5} -654 q^{-6} +1152 q^{-7} +2062 q^{-8} +1089 q^{-9} -975 q^{-10} -2224 q^{-11} -1479 q^{-12} +708 q^{-13} +2379 q^{-14} +1833 q^{-15} -513 q^{-16} -2443 q^{-17} -2123 q^{-18} +276 q^{-19} +2526 q^{-20} +2379 q^{-21} -103 q^{-22} -2546 q^{-23} -2596 q^{-24} -107 q^{-25} +2567 q^{-26} +2779 q^{-27} +297 q^{-28} -2494 q^{-29} -2921 q^{-30} -542 q^{-31} +2366 q^{-32} +2993 q^{-33} +786 q^{-34} -2109 q^{-35} -2962 q^{-36} -1048 q^{-37} +1752 q^{-38} +2809 q^{-39} +1252 q^{-40} -1311 q^{-41} -2501 q^{-42} -1369 q^{-43} +837 q^{-44} +2065 q^{-45} +1370 q^{-46} -411 q^{-47} -1566 q^{-48} -1216 q^{-49} +77 q^{-50} +1060 q^{-51} +975 q^{-52} +125 q^{-53} -640 q^{-54} -696 q^{-55} -182 q^{-56} +326 q^{-57} +428 q^{-58} +171 q^{-59} -134 q^{-60} -243 q^{-61} -112 q^{-62} +51 q^{-63} +109 q^{-64} +59 q^{-65} -12 q^{-66} -41 q^{-67} -33 q^{-68} +6 q^{-69} +22 q^{-70} +2 q^{-71} -3 q^{-72} + q^{-73} -5 q^{-74} - q^{-75} +6 q^{-76} -3 q^{-77} -2 q^{-78} +3 q^{-79} - q^{-80} }
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 q^{57}-3 q^{56}-q^{55}+5 q^{54}+3 q^{53}+3 q^{52}-7 q^{51}-19 q^{49}-19 q^{48}+19 q^{47}+30 q^{46}+47 q^{45}+12 q^{44}+13 q^{43}-85 q^{42}-133 q^{41}-63 q^{40}+19 q^{39}+159 q^{38}+182 q^{37}+268 q^{36}-20 q^{35}-301 q^{34}-413 q^{33}-376 q^{32}-42 q^{31}+282 q^{30}+886 q^{29}+660 q^{28}+134 q^{27}-552 q^{26}-1105 q^{25}-1100 q^{24}-605 q^{23}+996 q^{22}+1609 q^{21}+1657 q^{20}+654 q^{19}-879 q^{18}-2272 q^{17}-2743 q^{16}-659 q^{15}+1182 q^{14}+3059 q^{13}+3147 q^{12}+1515 q^{11}-1657 q^{10}-4476 q^9-3646 q^8-1651 q^7+2325 q^6+4974 q^5+5241 q^4+1485 q^3-3847 q^2-5842 q-5721-965 q^{-1} +4404 q^{-2} +8140 q^{-3} +5767 q^{-4} -771 q^{-5} -5808 q^{-6} -8981 q^{-7} -5297 q^{-8} +1663 q^{-9} +9049 q^{-10} +9327 q^{-11} +3213 q^{-12} -3965 q^{-13} -10546 q^{-14} -9014 q^{-15} -1759 q^{-16} +8461 q^{-17} +11491 q^{-18} +6665 q^{-19} -1652 q^{-20} -10910 q^{-21} -11566 q^{-22} -4648 q^{-23} +7473 q^{-24} +12667 q^{-25} +9159 q^{-26} +208 q^{-27} -10915 q^{-28} -13273 q^{-29} -6739 q^{-30} +6680 q^{-31} +13452 q^{-32} +11007 q^{-33} +1604 q^{-34} -10853 q^{-35} -14578 q^{-36} -8458 q^{-37} +5801 q^{-38} +13912 q^{-39} +12629 q^{-40} +3162 q^{-41} -10203 q^{-42} -15409 q^{-43} -10272 q^{-44} +4031 q^{-45} +13348 q^{-46} +13829 q^{-47} +5360 q^{-48} -8049 q^{-49} -14881 q^{-50} -11817 q^{-51} +978 q^{-52} +10786 q^{-53} +13502 q^{-54} +7563 q^{-55} -4203 q^{-56} -11981 q^{-57} -11729 q^{-58} -2326 q^{-59} +6311 q^{-60} +10646 q^{-61} +8125 q^{-62} -154 q^{-63} -7147 q^{-64} -9081 q^{-65} -3938 q^{-66} +1838 q^{-67} +6070 q^{-68} +6230 q^{-69} +1989 q^{-70} -2639 q^{-71} -5023 q^{-72} -3174 q^{-73} -538 q^{-74} +2163 q^{-75} +3224 q^{-76} +1817 q^{-77} -306 q^{-78} -1843 q^{-79} -1461 q^{-80} -780 q^{-81} +324 q^{-82} +1086 q^{-83} +818 q^{-84} +167 q^{-85} -438 q^{-86} -364 q^{-87} -327 q^{-88} -59 q^{-89} +245 q^{-90} +214 q^{-91} +70 q^{-92} -87 q^{-93} -31 q^{-94} -71 q^{-95} -36 q^{-96} +47 q^{-97} +33 q^{-98} +10 q^{-99} -25 q^{-100} +11 q^{-101} -7 q^{-102} -12 q^{-103} +11 q^{-104} +2 q^{-105} + q^{-106} -6 q^{-107} +3 q^{-108} +2 q^{-109} -3 q^{-110} + q^{-111} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{77}+3 q^{76}+q^{75}-5 q^{74}-3 q^{73}-3 q^{72}+7 q^{71}+5 q^{70}+3 q^{69}+17 q^{68}+7 q^{67}-20 q^{66}-35 q^{65}-50 q^{64}-13 q^{63}+24 q^{62}+39 q^{61}+119 q^{60}+117 q^{59}+50 q^{58}-63 q^{57}-236 q^{56}-265 q^{55}-206 q^{54}-102 q^{53}+231 q^{52}+496 q^{51}+605 q^{50}+507 q^{49}-59 q^{48}-577 q^{47}-990 q^{46}-1207 q^{45}-684 q^{44}+181 q^{43}+1236 q^{42}+2115 q^{41}+1858 q^{40}+879 q^{39}-694 q^{38}-2595 q^{37}-3326 q^{36}-2899 q^{35}-954 q^{34}+2197 q^{33}+4399 q^{32}+5232 q^{31}+3816 q^{30}-72 q^{29}-4116 q^{28}-7274 q^{27}-7578 q^{26}-3711 q^{25}+1847 q^{24}+7752 q^{23}+10998 q^{22}+8840 q^{21}+2904 q^{20}-5758 q^{19}-13028 q^{18}-14181 q^{17}-9460 q^{16}+851 q^{15}+12286 q^{14}+18243 q^{13}+16969 q^{12}+6719 q^{11}-8340 q^{10}-19818 q^9-23784 q^8-15878 q^7+1107 q^6+18085 q^5+28705 q^4+25303 q^3+8356 q^2-12909 q-30600-33687 q^{-1} -19117 q^{-2} +4988 q^{-3} +29521 q^{-4} +39865 q^{-5} +29561 q^{-6} +4823 q^{-7} -25363 q^{-8} -43600 q^{-9} -39112 q^{-10} -15197 q^{-11} +19379 q^{-12} +44807 q^{-13} +46617 q^{-14} +25329 q^{-15} -11993 q^{-16} -44047 q^{-17} -52503 q^{-18} -34421 q^{-19} +4638 q^{-20} +41971 q^{-21} +56398 q^{-22} +42097 q^{-23} +2550 q^{-24} -39239 q^{-25} -59199 q^{-26} -48383 q^{-27} -8627 q^{-28} +36527 q^{-29} +60871 q^{-30} +53343 q^{-31} +13846 q^{-32} -34098 q^{-33} -62288 q^{-34} -57325 q^{-35} -17878 q^{-36} +32218 q^{-37} +63392 q^{-38} +60622 q^{-39} +21301 q^{-40} -30840 q^{-41} -64672 q^{-42} -63577 q^{-43} -24173 q^{-44} +29684 q^{-45} +65879 q^{-46} +66514 q^{-47} +27181 q^{-48} -28413 q^{-49} -67044 q^{-50} -69481 q^{-51} -30530 q^{-52} +26397 q^{-53} +67545 q^{-54} +72462 q^{-55} +34715 q^{-56} -23142 q^{-57} -66976 q^{-58} -75019 q^{-59} -39541 q^{-60} +18181 q^{-61} +64426 q^{-62} +76462 q^{-63} +44850 q^{-64} -11370 q^{-65} -59471 q^{-66} -75993 q^{-67} -49674 q^{-68} +3119 q^{-69} +51715 q^{-70} +72699 q^{-71} +53060 q^{-72} +5813 q^{-73} -41513 q^{-74} -66212 q^{-75} -53989 q^{-76} -14044 q^{-77} +29826 q^{-78} +56667 q^{-79} +51610 q^{-80} +20314 q^{-81} -17896 q^{-82} -44973 q^{-83} -46094 q^{-84} -23700 q^{-85} +7529 q^{-86} +32625 q^{-87} +37962 q^{-88} +23751 q^{-89} +298 q^{-90} -21075 q^{-91} -28709 q^{-92} -21093 q^{-93} -4955 q^{-94} +11823 q^{-95} +19723 q^{-96} +16633 q^{-97} +6638 q^{-98} -5311 q^{-99} -12093 q^{-100} -11767 q^{-101} -6321 q^{-102} +1507 q^{-103} +6674 q^{-104} +7423 q^{-105} +4806 q^{-106} +201 q^{-107} -3153 q^{-108} -4130 q^{-109} -3212 q^{-110} -698 q^{-111} +1305 q^{-112} +2093 q^{-113} +1843 q^{-114} +559 q^{-115} -456 q^{-116} -903 q^{-117} -921 q^{-118} -363 q^{-119} +107 q^{-120} +370 q^{-121} +465 q^{-122} +154 q^{-123} -63 q^{-124} -123 q^{-125} -162 q^{-126} -53 q^{-127} -15 q^{-128} +35 q^{-129} +99 q^{-130} +17 q^{-131} -25 q^{-132} -15 q^{-133} -16 q^{-134} +9 q^{-135} -8 q^{-136} -6 q^{-137} +22 q^{-138} -8 q^{-140} -2 q^{-141} - q^{-142} +6 q^{-143} -3 q^{-144} -2 q^{-145} +3 q^{-146} - q^{-147} }

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[8, 16]]
Out[2]=	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[3]:=	GaussCode[Knot[8, 16]]
Out[3]=	GaussCode[1, -8, 5, -6, 2, -1, 4, -5, 6, -7, 3, -4, 8, -2, 7, -3]
In[4]:=	DTCode[Knot[8, 16]]
Out[4]=	DTCode[6, 8, 14, 12, 4, 16, 2, 10]
In[5]:=	br = BR[Knot[8, 16]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 16]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 16]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 16]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, 4, 1}
In[10]:=	alex = Alexander[Knot[8, 16]][t]
Out[10]=	-3 4 8 2 3 -9 + t - -- + - + 8 t - 4 t + t 2 t t
In[11]:=	Conway[Knot[8, 16]][z]
Out[11]=	2 4 6 1 + z + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 16], Knot[10, 156], Knot[11, NonAlternating, 15], Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[13]:=	{KnotDet[Knot[8, 16]], KnotSignature[Knot[8, 16]]}
Out[13]=	{35, -2}
In[14]:=	Jones[Knot[8, 16]][q]
Out[14]=	-6 3 5 6 6 6 2 -4 - q + -- - -- + -- - -- + - + 3 q - q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 16], Knot[10, 156]}
In[16]:=	A2Invariant[Knot[8, 16]][q]
Out[16]=	-18 -16 -14 -10 -8 2 -4 2 4 6 1 - q + q - q + q - q + -- - q + -- + q - q 6 2 q q
In[17]:=	HOMFLYPT[Knot[8, 16]][a, z]
Out[17]=	2 4 2 2 2 4 2 4 2 4 4 4 2 6 2 a - a - 2 z + 5 a z - 2 a z - z + 4 a z - a z + a z
In[18]:=	Kauffman[Knot[8, 16]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[8, 16]], Vassiliev[3][Knot[8, 16]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[8, 16]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[8, 16], 2][q]
Out[21]=	-17 3 2 6 15 7 19 32 8 32 41 -8 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + 16 15 14 13 12 11 10 9 8 7 q q q q q q q q q q 4 38 37 3 35 25 2 3 4 5 -- + -- - -- - -- + -- - -- + 24 q - 10 q - 8 q + 10 q - q - 6 5 4 3 2 q q q q q q 6 7 3 q + q

See/edit the Rolfsen_Splice_Template.