10 116

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Visit 10 116's page at the original Knot Atlas!

Square depiction.		Triquetra-heart decorative depiction.		Medieval manuscript.
As decorative double heart knot.		Mongolian ornament.

Knot presentations

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

Minimum Braid Representative:

Length is 10, width is 3.

Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-9][-3]
Hyperbolic Volume	15.4239
A-Polynomial	See Data:10 116/A-polynomial

[edit Notes for 10 116's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Topological 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Concordance genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4}
Rasmussen s-Invariant	2

[edit Notes for 10 116'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^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }
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^8-3 z^6-2 z^4+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	{ 95, -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^3-4 q^2+8 q-11+15 q^{-1} -16 q^{-2} +15 q^{-3} -12 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-8 a^2 z^4+3 z^4+2 a^4 z^2-4 a^2 z^2+2 z^2+1}
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 3 a^3 z^9+3 a z^9+8 a^4 z^8+14 a^2 z^8+6 z^8+10 a^5 z^7+9 a^3 z^7+3 a z^7+4 z^7 a^{-1} +8 a^6 z^6-8 a^4 z^6-32 a^2 z^6+z^6 a^{-2} -15 z^6+4 a^7 z^5-13 a^5 z^5-29 a^3 z^5-22 a z^5-10 z^5 a^{-1} +a^8 z^4-8 a^6 z^4-a^4 z^4+19 a^2 z^4-2 z^4 a^{-2} +9 z^4-2 a^7 z^3+6 a^5 z^3+19 a^3 z^3+17 a z^3+6 z^3 a^{-1} +2 a^6 z^2+a^4 z^2-3 a^2 z^2+z^2 a^{-2} -z^2-a^5 z-3 a^3 z-3 a z-z a^{-1} +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^{20}-2 q^{18}+2 q^{16}-2 q^{14}+2 q^{10}-3 q^8+4 q^6-3 q^4+3 q^2+1- q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8} }
The G2 invariant	$q^{114}-3q^{112}+6q^{110}-10q^{108}+10q^{106}-8q^{104}+q^{102}+16q^{100}-34q^{98}+54q^{96}-64q^{94}+51q^{92}-23q^{90}-27q^{88}+89q^{86}-140q^{84}+173q^{82}-158q^{80}+88q^{78}+24q^{76}-156q^{74}+263q^{72}-304q^{70}+240q^{68}-89q^{66}-104q^{64}+263q^{62}-309q^{60}+233q^{58}-52q^{56}-150q^{54}+265q^{52}-247q^{50}+82q^{48}+155q^{46}-343q^{44}+398q^{42}-275q^{40}+28q^{38}+249q^{36}-454q^{34}+500q^{32}-388q^{30}+147q^{28}+134q^{26}-350q^{24}+441q^{22}-368q^{20}+181q^{18}+51q^{16}-241q^{14}+301q^{12}-226q^{10}+47q^{8}+170q^{6}-305q^{4}+297q^{2}-138-95q^{-2}+304q^{-4}-396q^{-6}+330q^{-8}-150q^{-10}-70q^{-12}+244q^{-14}-312q^{-16}+271q^{-18}-144q^{-20}+6q^{-22}+90q^{-24}-132q^{-26}+116q^{-28}-72q^{-30}+29q^{-32}+6q^{-34}-21q^{-36}+21q^{-38}-16q^{-40}+8q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 10_116.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{15}-3 q^{13}+4 q^{11}-4 q^9+3 q^7-q^5-q^3+4 q-3 q^{-1} +4 q^{-3} -3 q^{-5} + q^{-7} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{42}-3 q^{40}+q^{38}+8 q^{36}-14 q^{34}+2 q^{32}+24 q^{30}-31 q^{28}-5 q^{26}+45 q^{24}-29 q^{22}-23 q^{20}+40 q^{18}-4 q^{16}-29 q^{14}+10 q^{12}+23 q^{10}-16 q^8-22 q^6+35 q^4+5 q^2-42+29 q^{-2} +24 q^{-4} -42 q^{-6} +6 q^{-8} +28 q^{-10} -20 q^{-12} -8 q^{-14} +13 q^{-16} - q^{-18} -3 q^{-20} + q^{-22} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{81}-3 q^{79}+q^{77}+5 q^{75}-2 q^{73}-9 q^{71}+2 q^{69}+22 q^{67}-15 q^{65}-38 q^{63}+33 q^{61}+77 q^{59}-48 q^{57}-149 q^{55}+45 q^{53}+235 q^{51}-311 q^{47}-87 q^{45}+342 q^{43}+192 q^{41}-307 q^{39}-279 q^{37}+212 q^{35}+326 q^{33}-89 q^{31}-316 q^{29}-37 q^{27}+268 q^{25}+143 q^{23}-203 q^{21}-224 q^{19}+133 q^{17}+277 q^{15}-58 q^{13}-322 q^{11}-13 q^9+344 q^7+100 q^5-340 q^3-191 q+301 q^{-1} +274 q^{-3} -210 q^{-5} -329 q^{-7} +100 q^{-9} +326 q^{-11} +19 q^{-13} -269 q^{-15} -106 q^{-17} +172 q^{-19} +139 q^{-21} -80 q^{-23} -117 q^{-25} +10 q^{-27} +74 q^{-29} +20 q^{-31} -34 q^{-33} -16 q^{-35} +8 q^{-37} +8 q^{-39} - q^{-41} -3 q^{-43} + q^{-45} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{195}-3 q^{193}+q^{191}+5 q^{189}-5 q^{187}+3 q^{183}-5 q^{181}-3 q^{179}+5 q^{177}+q^{175}+9 q^{173}+29 q^{171}-q^{169}-70 q^{167}-106 q^{165}-17 q^{163}+167 q^{161}+318 q^{159}+211 q^{157}-325 q^{155}-871 q^{153}-707 q^{151}+400 q^{149}+1775 q^{147}+2007 q^{145}+67 q^{143}-3101 q^{141}-4520 q^{139}-1749 q^{137}+4176 q^{135}+8385 q^{133}+5742 q^{131}-3795 q^{129}-13084 q^{127}-12572 q^{125}+397 q^{123}+16782 q^{121}+21665 q^{119}+7375 q^{117}-17126 q^{115}-30983 q^{113}-19109 q^{111}+12071 q^{109}+37149 q^{107}+32580 q^{105}-1178 q^{103}-37270 q^{101}-44070 q^{99}-13451 q^{97}+30141 q^{95}+49799 q^{93}+28104 q^{91}-17067 q^{89}-47874 q^{87}-38851 q^{85}+1546 q^{83}+38936 q^{81}+43051 q^{79}+12555 q^{77}-25685 q^{75}-40669 q^{73}-22395 q^{71}+11842 q^{69}+33586 q^{67}+26963 q^{65}-228 q^{63}-24705 q^{61}-27356 q^{59}-7719 q^{57}+16750 q^{55}+25582 q^{53}+12202 q^{51}-11167 q^{49}-23796 q^{47}-14599 q^{45}+8113 q^{43}+23487 q^{41}+16673 q^{39}-6753 q^{37}-24957 q^{35}-19826 q^{33}+5346 q^{31}+27597 q^{29}+24904 q^{27}-2384 q^{25}-29959 q^{23}-31463 q^{21}-3438 q^{19}+30076 q^{17}+38397 q^{15}+12300 q^{13}-26372 q^{11}-43500 q^9-23145 q^7+17861 q^5+44481 q^3+33843 q-5191 q^{-1} -39743 q^{-3} -41284 q^{-5} -9416 q^{-7} +28949 q^{-9} +42951 q^{-11} +22656 q^{-13} -14084 q^{-15} -37624 q^{-17} -30923 q^{-19} -1528 q^{-21} +26398 q^{-23} +32173 q^{-25} +13885 q^{-27} -12408 q^{-29} -26637 q^{-31} -20133 q^{-33} -305 q^{-35} +16821 q^{-37} +19690 q^{-39} +8625 q^{-41} -6486 q^{-43} -14513 q^{-45} -11289 q^{-47} -1173 q^{-49} +7594 q^{-51} +9530 q^{-53} +4849 q^{-55} -1924 q^{-57} -5782 q^{-59} -4950 q^{-61} -1205 q^{-63} +2236 q^{-65} +3313 q^{-67} +2010 q^{-69} -172 q^{-71} -1516 q^{-73} -1467 q^{-75} -544 q^{-77} +359 q^{-79} +726 q^{-81} +498 q^{-83} +40 q^{-85} -227 q^{-87} -238 q^{-89} -101 q^{-91} +29 q^{-93} +89 q^{-95} +55 q^{-97} -3 q^{-99} -20 q^{-101} -14 q^{-103} -5 q^{-105} +3 q^{-107} +8 q^{-109} - q^{-111} -3 q^{-113} + q^{-115} }
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^{270}-3 q^{268}+q^{266}+5 q^{264}-5 q^{262}+7 q^{256}-14 q^{254}-11 q^{252}+35 q^{250}-q^{248}+7 q^{246}+5 q^{244}-16 q^{242}-102 q^{240}-95 q^{238}+126 q^{236}+173 q^{234}+233 q^{232}+133 q^{230}-214 q^{228}-787 q^{226}-863 q^{224}+89 q^{222}+1110 q^{220}+2071 q^{218}+1821 q^{216}-293 q^{214}-3794 q^{212}-5782 q^{210}-3268 q^{208}+2530 q^{206}+9759 q^{204}+12704 q^{202}+6431 q^{200}-8746 q^{198}-23691 q^{196}-25304 q^{194}-8926 q^{192}+22026 q^{190}+48619 q^{188}+47918 q^{186}+9554 q^{184}-49536 q^{182}-90164 q^{180}-79255 q^{178}-5653 q^{176}+94305 q^{174}+154491 q^{172}+119077 q^{170}-10139 q^{168}-161421 q^{166}-237093 q^{164}-162820 q^{162}+39294 q^{160}+253465 q^{158}+331973 q^{156}+198607 q^{154}-86609 q^{152}-358165 q^{150}-425759 q^{148}-220151 q^{146}+153632 q^{144}+462497 q^{142}+497943 q^{140}+216834 q^{138}-226470 q^{136}-548596 q^{134}-537459 q^{132}-184007 q^{130}+293918 q^{128}+597089 q^{126}+533241 q^{124}+134376 q^{122}-343209 q^{120}-603314 q^{118}-485828 q^{116}-77432 q^{114}+363100 q^{112}+565830 q^{110}+413877 q^{108}+23694 q^{106}-356565 q^{104}-495097 q^{102}-330502 q^{100}+18280 q^{98}+327737 q^{96}+413365 q^{94}+246944 q^{92}-51593 q^{90}-288528 q^{88}-331696 q^{86}-169592 q^{84}+79471 q^{82}+254423 q^{80}+255081 q^{78}+94302 q^{76}-110276 q^{74}-226943 q^{72}-181511 q^{70}-16436 q^{68}+150236 q^{66}+201155 q^{64}+102110 q^{62}-71168 q^{60}-192883 q^{58}-166725 q^{56}-9565 q^{54}+168047 q^{52}+227640 q^{50}+113475 q^{48}-101456 q^{46}-263594 q^{44}-241514 q^{42}-35861 q^{40}+223375 q^{38}+345426 q^{36}+228197 q^{34}-66101 q^{32}-342119 q^{30}-400744 q^{28}-186493 q^{26}+177177 q^{24}+443917 q^{22}+426985 q^{20}+121024 q^{18}-281071 q^{16}-515530 q^{14}-424093 q^{12}-50416 q^{10}+364013 q^8+555283 q^6+396290 q^4-11757 q^2-413870-561277 q^{-2} -359454 q^{-4} +56402 q^{-6} +430841 q^{-8} +535645 q^{-10} +320664 q^{-12} -76469 q^{-14} -414114 q^{-16} -491758 q^{-18} -283193 q^{-20} +76626 q^{-22} +368060 q^{-24} +434189 q^{-26} +250787 q^{-28} -59231 q^{-30} -308042 q^{-32} -366896 q^{-34} -219154 q^{-36} +30632 q^{-38} +240637 q^{-40} +297555 q^{-42} +189115 q^{-44} -4583 q^{-46} -172632 q^{-48} -228234 q^{-50} -159928 q^{-52} -16237 q^{-54} +113485 q^{-56} +165638 q^{-58} +126700 q^{-60} +29925 q^{-62} -65450 q^{-64} -113292 q^{-66} -94457 q^{-68} -33711 q^{-70} +32149 q^{-72} +69798 q^{-74} +66122 q^{-76} +31266 q^{-78} -12271 q^{-80} -38961 q^{-82} -41889 q^{-84} -24097 q^{-86} +858 q^{-88} +19593 q^{-90} +24318 q^{-92} +15819 q^{-94} +3007 q^{-96} -8161 q^{-98} -12290 q^{-100} -9660 q^{-102} -3024 q^{-104} +2875 q^{-106} +5293 q^{-108} +4888 q^{-110} +2148 q^{-112} -588 q^{-114} -2218 q^{-116} -2039 q^{-118} -1024 q^{-120} -52 q^{-122} +713 q^{-124} +776 q^{-126} +460 q^{-128} -8 q^{-130} -184 q^{-132} -211 q^{-134} -161 q^{-136} -13 q^{-138} +60 q^{-140} +74 q^{-142} +11 q^{-144} + q^{-146} -4 q^{-148} -19 q^{-150} -5 q^{-152} +3 q^{-154} +8 q^{-156} - q^{-158} -3 q^{-160} + q^{-162} }

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^{20}-2 q^{18}+2 q^{16}-2 q^{14}+2 q^{10}-3 q^8+4 q^6-3 q^4+3 q^2+1- q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-6 q^{58}+18 q^{56}-38 q^{54}+73 q^{52}-136 q^{50}+226 q^{48}-336 q^{46}+480 q^{44}-664 q^{42}+864 q^{40}-1066 q^{38}+1260 q^{36}-1404 q^{34}+1438 q^{32}-1312 q^{30}+1001 q^{28}-498 q^{26}-180 q^{24}+966 q^{22}-1751 q^{20}+2442 q^{18}-2948 q^{16}+3196 q^{14}-3175 q^{12}+2872 q^{10}-2332 q^8+1610 q^6-788 q^4-10 q^2+746-1310 q^{-2} +1655 q^{-4} -1758 q^{-6} +1656 q^{-8} -1422 q^{-10} +1102 q^{-12} -774 q^{-14} +492 q^{-16} -280 q^{-18} +143 q^{-20} -62 q^{-22} +22 q^{-24} -6 q^{-26} + q^{-28} }
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^{52}-2 q^{50}+4 q^{46}-5 q^{44}+6 q^{40}-4 q^{38}-6 q^{36}+2 q^{34}+14 q^{32}-7 q^{30}-12 q^{28}+16 q^{26}+6 q^{24}-18 q^{22}-3 q^{20}+13 q^{18}-9 q^{16}-8 q^{14}+8 q^{12}+7 q^{10}-9 q^8+2 q^6+19 q^4-9 q^2-7+15 q^{-2} +2 q^{-4} -15 q^{-6} -2 q^{-8} +10 q^{-10} -9 q^{-14} +2 q^{-16} +7 q^{-18} -2 q^{-20} -2 q^{-22} + q^{-24} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-3 q^{46}+9 q^{42}-10 q^{40}-5 q^{38}+22 q^{36}-16 q^{34}-15 q^{32}+34 q^{30}-15 q^{28}-21 q^{26}+33 q^{24}-9 q^{22}-17 q^{20}+14 q^{18}+4 q^{16}-5 q^{14}-11 q^{12}+14 q^{10}+9 q^8-28 q^6+13 q^4+22 q^2-30+10 q^{-2} +22 q^{-4} -26 q^{-6} +8 q^{-8} +10 q^{-10} -13 q^{-12} +5 q^{-14} +2 q^{-16} -3 q^{-18} + q^{-20} }
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^{25}-2 q^{23}+3 q^{21}-4 q^{19}+3 q^{17}-3 q^{15}+2 q^{13}-q^{11}+q^9+q^7-q^5+3 q^3-2 q+4 q^{-1} -3 q^{-3} +3 q^{-5} -2 q^{-7} + q^{-9} }
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^{78}-6 q^{76}+15 q^{74}-17 q^{72}-3 q^{70}+48 q^{68}-97 q^{66}+98 q^{64}-5 q^{62}-156 q^{60}+291 q^{58}-284 q^{56}+65 q^{54}+294 q^{52}-596 q^{50}+624 q^{48}-260 q^{46}-385 q^{44}+998 q^{42}-1218 q^{40}+835 q^{38}+25 q^{36}-970 q^{34}+1530 q^{32}-1430 q^{30}+773 q^{28}+88 q^{26}-691 q^{24}+807 q^{22}-522 q^{20}+141 q^{18}-44 q^{16}+305 q^{14}-777 q^{12}+1059 q^{10}-833 q^8+121 q^6+822 q^4-1490 q^2+1613-1120 q^{-2} +289 q^{-4} +509 q^{-6} -975 q^{-8} +973 q^{-10} -628 q^{-12} +160 q^{-14} +196 q^{-16} -325 q^{-18} +264 q^{-20} -123 q^{-22} +10 q^{-24} +39 q^{-26} -37 q^{-28} +19 q^{-30} -6 q^{-32} + q^{-34} }

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^{58}-2 q^{56}-2 q^{54}+7 q^{52}-2 q^{50}-11 q^{48}+13 q^{46}+10 q^{44}-23 q^{42}-q^{40}+22 q^{38}-9 q^{36}-22 q^{34}+17 q^{32}+17 q^{30}-23 q^{28}-5 q^{26}+33 q^{24}-10 q^{22}-28 q^{20}+31 q^{18}+6 q^{16}-36 q^{14}+7 q^{12}+25 q^{10}-20 q^8-13 q^6+22 q^4+12 q^2-17+2 q^{-2} +19 q^{-4} -8 q^{-6} -8 q^{-8} +10 q^{-10} - q^{-12} -7 q^{-14} +5 q^{-16} -2 q^{-20} + q^{-22} }

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{30}-2 q^{28}+3 q^{26}-3 q^{24}+q^{22}-2 q^{18}+2 q^{16}-2 q^{14}+3 q^{12}-2 q^{10}+3 q^8-2 q^6+3 q^4-q^2+1+2 q^{-2} -2 q^{-4} +3 q^{-6} -2 q^{-8} + q^{-10} }

B2 Invariants.

Weight

Invariant

0,1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-3 q^{46}+6 q^{44}-11 q^{42}+18 q^{40}-25 q^{38}+32 q^{36}-38 q^{34}+41 q^{32}-40 q^{30}+31 q^{28}-17 q^{26}-q^{24}+21 q^{22}-41 q^{20}+60 q^{18}-74 q^{16}+81 q^{14}-79 q^{12}+70 q^{10}-55 q^8+36 q^6-13 q^4-6 q^2+24-34 q^{-2} +42 q^{-4} -42 q^{-6} +38 q^{-8} -32 q^{-10} +23 q^{-12} -15 q^{-14} +8 q^{-16} -3 q^{-18} + q^{-20} }

1,0

q^{78}-3q^{74}-3q^{72}+3q^{70}+10q^{68}+3q^{66}-14q^{64}-15q^{62}+8q^{60}+27q^{58}+8q^{56}-29q^{54}-28q^{52}+16q^{50}+42q^{48}+4q^{46}-41q^{44}-22q^{42}+30q^{40}+33q^{38}-17q^{36}-36q^{34}+4q^{32}+34q^{30}+5q^{28}-30q^{26}-11q^{24}+25q^{22}+16q^{20}-21q^{18}-20q^{16}+19q^{14}+27q^{12}-13q^{10}-37q^{8}+q^{6}+42q^{4}+20q^{2}-35-36q^{-2}+19q^{-4}+45q^{-6}+3q^{-8}-37q^{-10}-20q^{-12}+22q^{-14}+24q^{-16}-7q^{-18}-18q^{-20}-3q^{-22}+10q^{-24}+5q^{-26}-3q^{-28}-3q^{-30}+q^{-34}

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}-3 q^{64}+3 q^{62}-4 q^{60}+10 q^{58}-14 q^{56}+14 q^{54}-18 q^{52}+27 q^{50}-29 q^{48}+26 q^{46}-32 q^{44}+35 q^{42}-27 q^{40}+19 q^{38}-16 q^{36}+6 q^{34}+11 q^{32}-19 q^{30}+28 q^{28}-43 q^{26}+54 q^{24}-56 q^{22}+60 q^{20}-65 q^{18}+60 q^{16}-52 q^{14}+46 q^{12}-39 q^{10}+24 q^8-10 q^6+3 q^4+9 q^2-18+30 q^{-2} -29 q^{-4} +33 q^{-6} -34 q^{-8} +33 q^{-10} -28 q^{-12} +23 q^{-14} -19 q^{-16} +13 q^{-18} -8 q^{-20} +5 q^{-22} -3 q^{-24} + q^{-26} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+10 q^{106}-8 q^{104}+q^{102}+16 q^{100}-34 q^{98}+54 q^{96}-64 q^{94}+51 q^{92}-23 q^{90}-27 q^{88}+89 q^{86}-140 q^{84}+173 q^{82}-158 q^{80}+88 q^{78}+24 q^{76}-156 q^{74}+263 q^{72}-304 q^{70}+240 q^{68}-89 q^{66}-104 q^{64}+263 q^{62}-309 q^{60}+233 q^{58}-52 q^{56}-150 q^{54}+265 q^{52}-247 q^{50}+82 q^{48}+155 q^{46}-343 q^{44}+398 q^{42}-275 q^{40}+28 q^{38}+249 q^{36}-454 q^{34}+500 q^{32}-388 q^{30}+147 q^{28}+134 q^{26}-350 q^{24}+441 q^{22}-368 q^{20}+181 q^{18}+51 q^{16}-241 q^{14}+301 q^{12}-226 q^{10}+47 q^8+170 q^6-305 q^4+297 q^2-138-95 q^{-2} +304 q^{-4} -396 q^{-6} +330 q^{-8} -150 q^{-10} -70 q^{-12} +244 q^{-14} -312 q^{-16} +271 q^{-18} -144 q^{-20} +6 q^{-22} +90 q^{-24} -132 q^{-26} +116 q^{-28} -72 q^{-30} +29 q^{-32} +6 q^{-34} -21 q^{-36} +21 q^{-38} -16 q^{-40} +8 q^{-42} -3 q^{-44} + q^{-46} }

.

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["10 116"];

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^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }

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^8-3 z^6-2 z^4+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]= { 95, -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^3-4 q^2+8 q-11+15 q^{-1} -16 q^{-2} +15 q^{-3} -12 q^{-4} +8 q^{-5} -4 q^{-6} + q^{-7} }

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

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-8 a^2 z^4+3 z^4+2 a^4 z^2-4 a^2 z^2+2 z^2+1}

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 3 a^3 z^9+3 a z^9+8 a^4 z^8+14 a^2 z^8+6 z^8+10 a^5 z^7+9 a^3 z^7+3 a z^7+4 z^7 a^{-1} +8 a^6 z^6-8 a^4 z^6-32 a^2 z^6+z^6 a^{-2} -15 z^6+4 a^7 z^5-13 a^5 z^5-29 a^3 z^5-22 a z^5-10 z^5 a^{-1} +a^8 z^4-8 a^6 z^4-a^4 z^4+19 a^2 z^4-2 z^4 a^{-2} +9 z^4-2 a^7 z^3+6 a^5 z^3+19 a^3 z^3+17 a z^3+6 z^3 a^{-1} +2 a^6 z^2+a^4 z^2-3 a^2 z^2+z^2 a^{-2} -z^2-a^5 z-3 a^3 z-3 a z-z a^{-1} +1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a7, K11a33, K11a82, ...}

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

Vassiliev invariants

V₂ and V₃:

(0, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 16}

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 -32}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 40}

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{464}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{544}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{200}{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 -40}

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 10 116. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

5

1

4

1

6

3

-3

-1

9

5

4

-3

8

7

-1

-5

7

8

-1

-7

5

8

3

-9

3

7

-4

-11

1

5

4

-13

3

-3

-15

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=-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 {\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=-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}^{3}\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=-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}^{5}\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=-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}^{7}\oplus{\mathbb Z}_2^{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}^{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 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}^{8}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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}^{8}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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}^{7}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{9}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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^{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}^{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 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}\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=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}	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^{10}-4 q^9+2 q^8+15 q^7-25 q^6-10 q^5+63 q^4-47 q^3-58 q^2+129 q-42-129 q^{-1} +176 q^{-2} -12 q^{-3} -186 q^{-4} +182 q^{-5} +27 q^{-6} -199 q^{-7} +143 q^{-8} +52 q^{-9} -155 q^{-10} +80 q^{-11} +46 q^{-12} -81 q^{-13} +30 q^{-14} +20 q^{-15} -26 q^{-16} +8 q^{-17} +4 q^{-18} -4 q^{-19} + q^{-20} }
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^{21}-4 q^{20}+2 q^{19}+9 q^{18}+q^{17}-28 q^{16}-16 q^{15}+63 q^{14}+55 q^{13}-92 q^{12}-143 q^{11}+100 q^{10}+274 q^9-59 q^8-421 q^7-63 q^6+562 q^5+248 q^4-647 q^3-492 q^2+681 q+732-620 q^{-1} -984 q^{-2} +532 q^{-3} +1172 q^{-4} -376 q^{-5} -1341 q^{-6} +223 q^{-7} +1436 q^{-8} -41 q^{-9} -1485 q^{-10} -134 q^{-11} +1457 q^{-12} +305 q^{-13} -1360 q^{-14} -439 q^{-15} +1178 q^{-16} +532 q^{-17} -945 q^{-18} -553 q^{-19} +687 q^{-20} +504 q^{-21} -446 q^{-22} -403 q^{-23} +258 q^{-24} +280 q^{-25} -135 q^{-26} -168 q^{-27} +68 q^{-28} +86 q^{-29} -34 q^{-30} -43 q^{-31} +24 q^{-32} +15 q^{-33} -11 q^{-34} -6 q^{-35} +4 q^{-36} +4 q^{-37} -4 q^{-38} + q^{-39} }
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^{36}-4 q^{35}+2 q^{34}+9 q^{33}-5 q^{32}-2 q^{31}-34 q^{30}+8 q^{29}+74 q^{28}+24 q^{27}+4 q^{26}-216 q^{25}-122 q^{24}+215 q^{23}+300 q^{22}+343 q^{21}-528 q^{20}-786 q^{19}-131 q^{18}+662 q^{17}+1639 q^{16}-66 q^{15}-1661 q^{14}-1744 q^{13}-203 q^{12}+3313 q^{11}+2083 q^{10}-1021 q^9-3791 q^8-3267 q^7+3334 q^6+4829 q^5+2117 q^4-4099 q^3-7212 q^2+751 q+5964+6391 q^{-1} -1879 q^{-2} -9881 q^{-3} -3125 q^{-4} +4841 q^{-5} +9831 q^{-6} +1610 q^{-7} -10619 q^{-8} -6624 q^{-9} +2511 q^{-10} +11806 q^{-11} +4979 q^{-12} -10029 q^{-13} -9201 q^{-14} -107 q^{-15} +12510 q^{-16} +7828 q^{-17} -8382 q^{-18} -10782 q^{-19} -2928 q^{-20} +11658 q^{-21} +9912 q^{-22} -5380 q^{-23} -10643 q^{-24} -5643 q^{-25} +8647 q^{-26} +10226 q^{-27} -1497 q^{-28} -8040 q^{-29} -6883 q^{-30} +4236 q^{-31} +7928 q^{-32} +1345 q^{-33} -3948 q^{-34} -5594 q^{-35} +740 q^{-36} +4182 q^{-37} +1805 q^{-38} -811 q^{-39} -2941 q^{-40} -448 q^{-41} +1360 q^{-42} +862 q^{-43} +249 q^{-44} -983 q^{-45} -269 q^{-46} +269 q^{-47} +145 q^{-48} +198 q^{-49} -232 q^{-50} -28 q^{-51} +55 q^{-52} -27 q^{-53} +55 q^{-54} -52 q^{-55} +11 q^{-56} +19 q^{-57} -16 q^{-58} +9 q^{-59} -10 q^{-60} +4 q^{-61} +4 q^{-62} -4 q^{-63} + q^{-64} }
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^{55}-4 q^{54}+2 q^{53}+9 q^{52}-5 q^{51}-8 q^{50}-8 q^{49}-10 q^{48}+19 q^{47}+67 q^{46}+29 q^{45}-68 q^{44}-138 q^{43}-147 q^{42}+30 q^{41}+334 q^{40}+487 q^{39}+160 q^{38}-505 q^{37}-1050 q^{36}-893 q^{35}+285 q^{34}+1831 q^{33}+2342 q^{32}+798 q^{31}-2127 q^{30}-4334 q^{29}-3460 q^{28}+999 q^{27}+6200 q^{26}+7571 q^{25}+2554 q^{24}-6270 q^{23}-12227 q^{22}-9117 q^{21}+2976 q^{20}+15598 q^{19}+17665 q^{18}+4795 q^{17}-15096 q^{16}-26243 q^{15}-16852 q^{14}+9094 q^{13}+31894 q^{12}+31088 q^{11}+3192 q^{10}-32018 q^9-44778 q^8-20301 q^7+25193 q^6+54628 q^5+39932 q^4-11723 q^3-58780 q^2-58666 q-6675+56169 q^{-1} +74484 q^{-2} +27311 q^{-3} -48142 q^{-4} -85286 q^{-5} -47681 q^{-6} +35814 q^{-7} +91612 q^{-8} +65983 q^{-9} -22045 q^{-10} -93607 q^{-11} -81195 q^{-12} +7789 q^{-13} +93116 q^{-14} +93558 q^{-15} +5243 q^{-16} -90914 q^{-17} -103446 q^{-18} -17383 q^{-19} +87985 q^{-20} +111762 q^{-21} +28669 q^{-22} -84100 q^{-23} -118820 q^{-24} -40095 q^{-25} +78788 q^{-26} +124391 q^{-27} +52038 q^{-28} -70720 q^{-29} -127652 q^{-30} -64564 q^{-31} +59151 q^{-32} +127042 q^{-33} +76515 q^{-34} -43529 q^{-35} -121029 q^{-36} -86308 q^{-37} +24914 q^{-38} +108768 q^{-39} +91499 q^{-40} -5289 q^{-41} -90533 q^{-42} -90423 q^{-43} -12476 q^{-44} +68371 q^{-45} +82476 q^{-46} +25518 q^{-47} -45362 q^{-48} -68728 q^{-49} -32134 q^{-50} +24779 q^{-51} +51857 q^{-52} +32318 q^{-53} -9270 q^{-54} -34970 q^{-55} -27565 q^{-56} -299 q^{-57} +20677 q^{-58} +20444 q^{-59} +4587 q^{-60} -10469 q^{-61} -13275 q^{-62} -5182 q^{-63} +4292 q^{-64} +7475 q^{-65} +4075 q^{-66} -1180 q^{-67} -3738 q^{-68} -2539 q^{-69} +83 q^{-70} +1550 q^{-71} +1304 q^{-72} +239 q^{-73} -570 q^{-74} -599 q^{-75} -149 q^{-76} +175 q^{-77} +197 q^{-78} +75 q^{-79} -24 q^{-80} -63 q^{-81} -42 q^{-82} +24 q^{-83} +13 q^{-84} -14 q^{-85} +12 q^{-86} +6 q^{-87} -12 q^{-88} +4 q^{-89} +5 q^{-90} -10 q^{-91} +4 q^{-92} +4 q^{-93} -4 q^{-94} + q^{-95} }
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^{78}-4 q^{77}+2 q^{76}+9 q^{75}-5 q^{74}-8 q^{73}-14 q^{72}+16 q^{71}+q^{70}+12 q^{69}+72 q^{68}-19 q^{67}-81 q^{66}-162 q^{65}-34 q^{64}+28 q^{63}+188 q^{62}+540 q^{61}+297 q^{60}-144 q^{59}-927 q^{58}-1006 q^{57}-987 q^{56}+9 q^{55}+2170 q^{54}+3033 q^{53}+2596 q^{52}-522 q^{51}-3424 q^{50}-6886 q^{49}-6627 q^{48}-460 q^{47}+7162 q^{46}+13764 q^{45}+12290 q^{44}+5075 q^{43}-11611 q^{42}-25362 q^{41}-25415 q^{40}-10630 q^{39}+16692 q^{38}+38980 q^{37}+48612 q^{36}+23245 q^{35}-21686 q^{34}-63064 q^{33}-76490 q^{32}-44054 q^{31}+20145 q^{30}+96454 q^{29}+118620 q^{28}+75089 q^{27}-24126 q^{26}-128643 q^{25}-173776 q^{24}-123546 q^{23}+28148 q^{22}+174222 q^{21}+243138 q^{20}+169572 q^{19}-20203 q^{18}-230694 q^{17}-333551 q^{16}-221638 q^{15}+26480 q^{14}+301992 q^{13}+418383 q^{12}+289815 q^{11}-47292 q^{10}-399680 q^9-513072 q^8-333339 q^7+93427 q^6+496027 q^5+627460 q^4+349841 q^3-184699 q^2-617876 q-707778-322429 q^{-1} +294204 q^{-2} +774867 q^{-3} +751954 q^{-4} +223348 q^{-5} -458883 q^{-6} -899048 q^{-7} -736858 q^{-8} -79473 q^{-9} +683430 q^{-10} +986413 q^{-11} +625443 q^{-12} -152922 q^{-13} -882116 q^{-14} -1003598 q^{-15} -443143 q^{-16} +469179 q^{-17} +1045038 q^{-18} +901461 q^{-19} +141376 q^{-20} -764887 q^{-21} -1125649 q^{-22} -702379 q^{-23} +263526 q^{-24} +1022958 q^{-25} +1063599 q^{-26} +356307 q^{-27} -650722 q^{-28} -1185242 q^{-29} -879991 q^{-30} +106366 q^{-31} +996800 q^{-32} +1185314 q^{-33} +529585 q^{-34} -551677 q^{-35} -1236161 q^{-36} -1046663 q^{-37} -58709 q^{-38} +951368 q^{-39} +1301981 q^{-40} +734163 q^{-41} -390898 q^{-42} -1236819 q^{-43} -1221615 q^{-44} -307772 q^{-45} +789264 q^{-46} +1345149 q^{-47} +971098 q^{-48} -92361 q^{-49} -1070398 q^{-50} -1307243 q^{-51} -617229 q^{-52} +440482 q^{-53} +1180554 q^{-54} +1109630 q^{-55} +287898 q^{-56} -680215 q^{-57} -1155290 q^{-58} -819959 q^{-59} -50 q^{-60} +772158 q^{-61} +992144 q^{-62} +548003 q^{-63} -202630 q^{-64} -756425 q^{-65} -756111 q^{-66} -303221 q^{-67} +294233 q^{-68} +638692 q^{-69} +536866 q^{-70} +119496 q^{-71} -313121 q^{-72} -475002 q^{-73} -338667 q^{-74} -14632 q^{-75} +264909 q^{-76} +331258 q^{-77} +187090 q^{-78} -41565 q^{-79} -189786 q^{-80} -205301 q^{-81} -93140 q^{-82} +50738 q^{-83} +129144 q^{-84} +112817 q^{-85} +34107 q^{-86} -38504 q^{-87} -76085 q^{-88} -57726 q^{-89} -9448 q^{-90} +29186 q^{-91} +39215 q^{-92} +23198 q^{-93} +2124 q^{-94} -16995 q^{-95} -19362 q^{-96} -8747 q^{-97} +2593 q^{-98} +8263 q^{-99} +6820 q^{-100} +3737 q^{-101} -2050 q^{-102} -4185 q^{-103} -2474 q^{-104} -352 q^{-105} +1034 q^{-106} +1022 q^{-107} +1223 q^{-108} -62 q^{-109} -684 q^{-110} -360 q^{-111} -102 q^{-112} +73 q^{-113} + q^{-114} +271 q^{-115} +14 q^{-116} -111 q^{-117} -13 q^{-118} -2 q^{-119} +13 q^{-120} -46 q^{-121} +50 q^{-122} +7 q^{-123} -25 q^{-124} +8 q^{-125} +5 q^{-127} -10 q^{-128} +4 q^{-129} +4 q^{-130} -4 q^{-131} + q^{-132} }
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^{105}-4 q^{104}+2 q^{103}+9 q^{102}-5 q^{101}-8 q^{100}-14 q^{99}+10 q^{98}+27 q^{97}-6 q^{96}+17 q^{95}+24 q^{94}-32 q^{93}-81 q^{92}-140 q^{91}-35 q^{90}+180 q^{89}+203 q^{88}+317 q^{87}+277 q^{86}-69 q^{85}-501 q^{84}-1187 q^{83}-1177 q^{82}-299 q^{81}+772 q^{80}+2326 q^{79}+3310 q^{78}+2759 q^{77}+771 q^{76}-3595 q^{75}-7603 q^{74}-8625 q^{73}-6386 q^{72}+1164 q^{71}+10906 q^{70}+18808 q^{69}+21313 q^{68}+11831 q^{67}-7272 q^{66}-28685 q^{65}-44958 q^{64}-42440 q^{63}-18261 q^{62}+22324 q^{61}+68514 q^{60}+92746 q^{59}+78711 q^{58}+22968 q^{57}-64103 q^{56}-142103 q^{55}-173979 q^{54}-132631 q^{53}-10319 q^{52}+146476 q^{51}+273463 q^{50}+304034 q^{49}+189710 q^{48}-38424 q^{47}-302773 q^{46}-488677 q^{45}-475811 q^{44}-237719 q^{43}+165868 q^{42}+581175 q^{41}+796327 q^{40}+682179 q^{39}+222203 q^{38}-440031 q^{37}-1007750 q^{36}-1207684 q^{35}-869708 q^{34}-49172 q^{33}+922337 q^{32}+1625024 q^{31}+1670699 q^{30}+916490 q^{29}-379123 q^{28}-1699282 q^{27}-2405172 q^{26}-2047942 q^{25}-674546 q^{24}+1219794 q^{23}+2785688 q^{22}+3199093 q^{21}+2139708 q^{20}-94472 q^{19}-2556796 q^{18}-4048701 q^{17}-3761567 q^{16}-1601328 q^{15}+1577925 q^{14}+4295576 q^{13}+5201831 q^{12}+3630757 q^{11}+115205 q^{10}-3754692 q^9-6129725 q^8-5656063 q^7-2320647 q^6+2410339 q^5+6319177 q^4+7335380 q^3+4724899 q^2-414462 q-5700197-8413324 q^{-1} -6996260 q^{-2} -1962291 q^{-3} +4368608 q^{-4} +8770447 q^{-5} +8859134 q^{-6} +4415087 q^{-7} -2531357 q^{-8} -8438363 q^{-9} -10161317 q^{-10} -6671885 q^{-11} +457160 q^{-12} +7560095 q^{-13} +10866184 q^{-14} +8550235 q^{-15} +1610856 q^{-16} -6340984 q^{-17} -11055527 q^{-18} -9971490 q^{-19} -3476329 q^{-20} +4993754 q^{-21} +10863732 q^{-22} +10947851 q^{-23} +5035675 q^{-24} -3692443 q^{-25} -10458859 q^{-26} -11562618 q^{-27} -6254716 q^{-28} +2558510 q^{-29} +9987193 q^{-30} +11928935 q^{-31} +7172709 q^{-32} -1642486 q^{-33} -9565849 q^{-34} -12173638 q^{-35} -7870129 q^{-36} +937575 q^{-37} +9258553 q^{-38} +12405864 q^{-39} +8457565 q^{-40} -379459 q^{-41} -9074976 q^{-42} -12704262 q^{-43} -9050877 q^{-44} -138945 q^{-45} +8964002 q^{-46} +13098623 q^{-47} +9750846 q^{-48} +748305 q^{-49} -8817560 q^{-50} -13551342 q^{-51} -10613901 q^{-52} -1581260 q^{-53} +8482179 q^{-54} +13955752 q^{-55} +11626559 q^{-56} +2728723 q^{-57} -7793083 q^{-58} -14137150 q^{-59} -12681350 q^{-60} -4204675 q^{-61} +6614112 q^{-62} +13889997 q^{-63} +13587239 q^{-64} +5911388 q^{-65} -4900343 q^{-66} -13033080 q^{-67} -14093722 q^{-68} -7635703 q^{-69} +2731817 q^{-70} +11473738 q^{-71} +13965385 q^{-72} +9089906 q^{-73} -331461 q^{-74} -9267914 q^{-75} -13056008 q^{-76} -9977978 q^{-77} -1977404 q^{-78} +6628537 q^{-79} +11374372 q^{-80} +10093465 q^{-81} +3853496 q^{-82} -3892291 q^{-83} -9107156 q^{-84} -9388717 q^{-85} -5030815 q^{-86} +1430079 q^{-87} +6572543 q^{-88} +7997555 q^{-89} +5404833 q^{-90} +459641 q^{-91} -4138335 q^{-92} -6200227 q^{-93} -5051850 q^{-94} -1624008 q^{-95} +2112926 q^{-96} +4328336 q^{-97} +4192792 q^{-98} +2089008 q^{-99} -667941 q^{-100} -2673341 q^{-101} -3113040 q^{-102} -2017173 q^{-103} -178950 q^{-104} +1411284 q^{-105} +2065397 q^{-106} +1634346 q^{-107} +539213 q^{-108} -583518 q^{-109} -1217100 q^{-110} -1153869 q^{-111} -580276 q^{-112} +130472 q^{-113} +627756 q^{-114} +718788 q^{-115} +464786 q^{-116} +62828 q^{-117} -273890 q^{-118} -397793 q^{-119} -312094 q^{-120} -108131 q^{-121} +94482 q^{-122} +195147 q^{-123} +181237 q^{-124} +90431 q^{-125} -18052 q^{-126} -84181 q^{-127} -94081 q^{-128} -58733 q^{-129} -4964 q^{-130} +32043 q^{-131} +43440 q^{-132} +31897 q^{-133} +8031 q^{-134} -9964 q^{-135} -18275 q^{-136} -15802 q^{-137} -5541 q^{-138} +2703 q^{-139} +7231 q^{-140} +6857 q^{-141} +2599 q^{-142} -342 q^{-143} -2372 q^{-144} -2880 q^{-145} -1317 q^{-146} -110 q^{-147} +1004 q^{-148} +1178 q^{-149} +264 q^{-150} +57 q^{-151} -160 q^{-152} -377 q^{-153} -178 q^{-154} -128 q^{-155} +142 q^{-156} +217 q^{-157} -46 q^{-158} -9 q^{-159} +10 q^{-160} -24 q^{-161} -3 q^{-162} -46 q^{-163} +18 q^{-164} +45 q^{-165} -24 q^{-166} -5 q^{-167} +4 q^{-168} +5 q^{-170} -10 q^{-171} +4 q^{-172} +4 q^{-173} -4 q^{-174} + q^{-175} }

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[10, 116]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 3, 17, 4], X[14, 7, 15, 8], X[8, 15, 9, 16], X[10, 18, 11, 17], X[18, 6, 19, 5], X[20, 13, 1, 14], X[12, 19, 13, 20], X[2, 10, 3, 9], X[4, 11, 5, 12]]
In[3]:=	GaussCode[Knot[10, 116]]
Out[3]=	GaussCode[1, -9, 2, -10, 6, -1, 3, -4, 9, -5, 10, -8, 7, -3, 4, -2, 5, -6, 8, -7]
In[4]:=	DTCode[Knot[10, 116]]
Out[4]=	DTCode[6, 16, 18, 14, 2, 4, 20, 8, 10, 12]
In[5]:=	br = BR[Knot[10, 116]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 116]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 116]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 116]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 116]][t]
Out[10]=	-4 5 12 19 2 3 4 -21 - t + -- - -- + -- + 19 t - 12 t + 5 t - t 3 2 t t t
In[11]:=	Conway[Knot[10, 116]][z]
Out[11]=	4 6 8 1 - 2 z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 116], Knot[11, Alternating, 7], Knot[11, Alternating, 33], Knot[11, Alternating, 82]}
In[13]:=	{KnotDet[Knot[10, 116]], KnotSignature[Knot[10, 116]]}
Out[13]=	{95, -2}
In[14]:=	Jones[Knot[10, 116]][q]
Out[14]=	-7 4 8 12 15 16 15 2 3 -11 + q - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 116]}
In[16]:=	A2Invariant[Knot[10, 116]][q]
Out[16]=	-20 2 2 2 2 3 4 3 3 2 4 1 + q - --- + --- - --- + --- - -- + -- - -- + -- - q + 2 q - 18 16 14 10 8 6 4 2 q q q q q q q q 6 8 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 116]][a, z]
Out[17]=	2 2 2 4 2 4 2 4 4 4 6 1 + 2 z - 4 a z + 2 a z + 3 z - 8 a z + 3 a z + z - 2 6 4 6 2 8 5 a z + a z - a z
In[18]:=	Kauffman[Knot[10, 116]][a, z]
Out[18]=	2 z 3 5 2 z 2 2 4 2 6 2 1 - - - 3 a z - 3 a z - a z - z + -- - 3 a z + a z + 2 a z + a 2 a 3 4 6 z 3 3 3 5 3 7 3 4 2 z ---- + 17 a z + 19 a z + 6 a z - 2 a z + 9 z - ---- + a 2 a 5 2 4 4 4 6 4 8 4 10 z 5 3 5 19 a z - a z - 8 a z + a z - ----- - 22 a z - 29 a z - a 6 5 5 7 5 6 z 2 6 4 6 6 6 13 a z + 4 a z - 15 z + -- - 32 a z - 8 a z + 8 a z + 2 a 7 4 z 7 3 7 5 7 8 2 8 4 8 ---- + 3 a z + 9 a z + 10 a z + 6 z + 14 a z + 8 a z + a 9 3 9 3 a z + 3 a z
In[19]:=	{Vassiliev[2][Knot[10, 116]], Vassiliev[3][Knot[10, 116]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 116]][q, t]
Out[20]=	7 9 1 3 1 5 3 7 5 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 8 7 8 8 5 t 2 3 2 ----- + ----- + ---- + ---- + --- + 6 q t + 3 q t + 5 q t + 7 2 5 2 5 3 q q t q t q t q t 3 3 5 3 7 4 q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 116], 2][q]
Out[21]=	-20 4 4 8 26 20 30 81 46 80 -42 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - 19 18 17 16 15 14 13 12 11 q q q q q q q q q 155 52 143 199 27 182 186 12 176 129 --- + -- + --- - --- + -- + --- - --- - -- + --- - --- + 129 q - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 10 58 q - 47 q + 63 q - 10 q - 25 q + 15 q + 2 q - 4 q + q

See/edit the Rolfsen_Splice_Template.

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10_115

10_117

10 116

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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