10 116

10_115

10_117

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10 116 Quick Notes

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

Knot presentations

Planar diagram presentation	X6271 X16,3,17,4 X14,7,15,8 X8,15,9,16 X10,18,11,17 X18,6,19,5 X20,13,1,14 X12,19,13,20 X2,10,3,9 X4,11,5,12
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]

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

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

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{66}-3q^{64}+3q^{62}-4q^{60}+10q^{58}-14q^{56}+14q^{54}-18q^{52}+27q^{50}-29q^{48}+26q^{46}-32q^{44}+35q^{42}-27q^{40}+19q^{38}-16q^{36}+6q^{34}+11q^{32}-19q^{30}+28q^{28}-43q^{26}+54q^{24}-56q^{22}+60q^{20}-65q^{18}+60q^{16}-52q^{14}+46q^{12}-39q^{10}+24q^{8}-10q^{6}+3q^{4}+9q^{2}-18+30q^{-2}-29q^{-4}+33q^{-6}-34q^{-8}+33q^{-10}-28q^{-12}+23q^{-14}-19q^{-16}+13q^{-18}-8q^{-20}+5q^{-22}-3q^{-24}+q^{-26}$

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

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]=

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

Vassiliev invariants

V2 and V3:

(0, 0)

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 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 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 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} 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} 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} $-32$ 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 -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 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 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 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 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}} $-{\frac {544}{3}}$ 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	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											1

Computer Talk

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

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 \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 116]]
Out[2]=	10
In[3]:=	PD[Knot[10, 116]]
Out[3]=	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[4]:=	GaussCode[Knot[10, 116]]
Out[4]=	GaussCode[1, -9, 2, -10, 6, -1, 3, -4, 9, -5, 10, -8, 7, -3, 4, -2, 5, -6, 8, -7]
In[5]:=	BR[Knot[10, 116]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, -1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[10, 116]][t]
Out[6]=	-4 5 12 19 2 3 4 -21 - t + -- - -- + -- + 19 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 116]][z]
Out[7]=	4 6 8 1 - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 116], Knot[11, Alternating, 7], Knot[11, Alternating, 33], Knot[11, Alternating, 82]}
In[9]:=	{KnotDet[Knot[10, 116]], KnotSignature[Knot[10, 116]]}
Out[9]=	{95, -2}
In[10]:=	J=Jones[Knot[10, 116]][q]
Out[10]=	-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[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 116]}
In[12]:=	A2Invariant[Knot[10, 116]][q]
Out[12]=	-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[13]:=	Kauffman[Knot[10, 116]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[10, 116]], Vassiliev[3][Knot[10, 116]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 116]][q, t]
Out[15]=	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