10 82

10_81

10_83

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10 82 Quick Notes

10 82 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X7,16,8,17 X3948 X15,3,16,2 X5,15,6,14 X9,5,10,4 X11,18,12,19 X13,20,14,1 X17,10,18,11 X19,12,20,13
Gauss code	-1, 4, -3, 6, -5, 1, -2, 3, -6, 9, -7, 10, -8, 5, -4, 2, -9, 7, -10, 8
Dowker-Thistlethwaite code	6 8 14 16 4 18 20 2 10 12
Conway Notation	[.4.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-3] Hyperbolic Volume 12.4315 A-Polynomial See Data:10 82/A-polynomial

[edit Notes for 10 82'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 [2,4]} Rasmussen s-Invariant -2

[edit Notes for 10 82'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+4 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +4 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^8-4 z^6-4 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	{ 63, -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^3-3 q^2+5 q-7+10 q^{-1} -10 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 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^6-6 a^2 z^6+z^6+4 a^4 z^4-12 a^2 z^4+4 z^4+4 a^4 z^2-8 a^2 z^2+4 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 2 a^3 z^9+2 a z^9+4 a^4 z^8+8 a^2 z^8+4 z^8+4 a^5 z^7-a^3 z^7-2 a z^7+3 z^7 a^{-1} +4 a^6 z^6-8 a^4 z^6-27 a^2 z^6+z^6 a^{-2} -14 z^6+3 a^7 z^5-3 a^5 z^5-4 a^3 z^5-8 a z^5-10 z^5 a^{-1} +a^8 z^4-4 a^6 z^4+10 a^4 z^4+32 a^2 z^4-3 z^4 a^{-2} +14 z^4-4 a^7 z^3-2 a^5 z^3+5 a^3 z^3+10 a z^3+7 z^3 a^{-1} -a^8 z^2-5 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -6 z^2+a^7 z+2 a^5 z-2 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}-q^{18}+q^{16}-2 q^{14}-q^{12}+q^{10}-q^8+4 q^6-q^4+2 q^2- q^{-2} + q^{-4} - 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}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+8 q^{92}+7 q^{90}-22 q^{88}+35 q^{86}-38 q^{84}+35 q^{82}-25 q^{80}+6 q^{78}+16 q^{76}-37 q^{74}+57 q^{72}-60 q^{70}+47 q^{68}-18 q^{66}-20 q^{64}+52 q^{62}-67 q^{60}+53 q^{58}-17 q^{56}-28 q^{54}+53 q^{52}-53 q^{50}+16 q^{48}+36 q^{46}-76 q^{44}+82 q^{42}-56 q^{40}-q^{38}+64 q^{36}-107 q^{34}+116 q^{32}-84 q^{30}+30 q^{28}+38 q^{26}-85 q^{24}+106 q^{22}-88 q^{20}+49 q^{18}+4 q^{16}-52 q^{14}+75 q^{12}-60 q^{10}+23 q^8+28 q^6-66 q^4+68 q^2-36-21 q^{-2} +70 q^{-4} -95 q^{-6} +84 q^{-8} -38 q^{-10} -21 q^{-12} +69 q^{-14} -87 q^{-16} +78 q^{-18} -43 q^{-20} +2 q^{-22} +27 q^{-24} -41 q^{-26} +39 q^{-28} -26 q^{-30} +13 q^{-32} + q^{-34} -7 q^{-36} +7 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46} }

Further Quantum Invariants

Further quantum knot invariants for 10_82.

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}-2 q^{13}+2 q^{11}-3 q^9+2 q^7+3 q-2 q^{-1} +2 q^{-3} -2 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}-2 q^{40}-q^{38}+5 q^{36}-4 q^{34}-q^{32}+9 q^{30}-10 q^{28}-3 q^{26}+16 q^{24}-11 q^{22}-9 q^{20}+15 q^{18}-2 q^{16}-10 q^{14}+4 q^{12}+9 q^{10}-5 q^8-8 q^6+13 q^4+2 q^2-15+11 q^{-2} +8 q^{-4} -16 q^{-6} +3 q^{-8} +12 q^{-10} -9 q^{-12} -4 q^{-14} +7 q^{-16} - q^{-18} -2 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}-2 q^{79}-q^{77}+2 q^{75}+4 q^{73}-q^{71}-7 q^{69}+3 q^{65}+3 q^{63}+2 q^{61}+2 q^{59}-11 q^{57}-14 q^{55}+17 q^{53}+35 q^{51}-14 q^{49}-57 q^{47}-2 q^{45}+73 q^{43}+26 q^{41}-73 q^{39}-48 q^{37}+52 q^{35}+65 q^{33}-29 q^{31}-63 q^{29}-3 q^{27}+56 q^{25}+27 q^{23}-42 q^{21}-46 q^{19}+30 q^{17}+56 q^{15}-18 q^{13}-66 q^{11}+8 q^9+75 q^7+7 q^5-74 q^3-27 q+73 q^{-1} +48 q^{-3} -52 q^{-5} -69 q^{-7} +27 q^{-9} +73 q^{-11} +4 q^{-13} -64 q^{-15} -31 q^{-17} +44 q^{-19} +42 q^{-21} -20 q^{-23} -38 q^{-25} +27 q^{-29} +9 q^{-31} -14 q^{-33} -7 q^{-35} +4 q^{-37} +4 q^{-39} - q^{-41} -2 q^{-43} + q^{-45} }
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^{132}-2 q^{130}-q^{128}+2 q^{126}+q^{124}+7 q^{122}-7 q^{120}-9 q^{118}-4 q^{116}+33 q^{112}+6 q^{110}-14 q^{108}-35 q^{106}-41 q^{104}+51 q^{102}+62 q^{100}+42 q^{98}-58 q^{96}-154 q^{94}-21 q^{92}+114 q^{90}+213 q^{88}+53 q^{86}-268 q^{84}-263 q^{82}-12 q^{80}+387 q^{78}+371 q^{76}-151 q^{74}-483 q^{72}-370 q^{70}+268 q^{68}+618 q^{66}+214 q^{64}-355 q^{62}-603 q^{60}-105 q^{58}+463 q^{56}+440 q^{54}+22 q^{52}-441 q^{50}-337 q^{48}+87 q^{46}+337 q^{44}+261 q^{42}-118 q^{40}-310 q^{38}-169 q^{36}+154 q^{34}+310 q^{32}+75 q^{30}-260 q^{28}-285 q^{26}+85 q^{24}+355 q^{22}+193 q^{20}-275 q^{18}-423 q^{16}+11 q^{14}+423 q^{12}+378 q^{10}-192 q^8-542 q^6-203 q^4+314 q^2+542+95 q^{-2} -434 q^{-4} -420 q^{-6} -37 q^{-8} +439 q^{-10} +367 q^{-12} -57 q^{-14} -343 q^{-16} -346 q^{-18} +70 q^{-20} +315 q^{-22} +251 q^{-24} -7 q^{-26} -300 q^{-28} -198 q^{-30} +20 q^{-32} +206 q^{-34} +197 q^{-36} -50 q^{-38} -143 q^{-40} -127 q^{-42} +13 q^{-44} +125 q^{-46} +57 q^{-48} -6 q^{-50} -67 q^{-52} -41 q^{-54} +22 q^{-56} +21 q^{-58} +18 q^{-60} -8 q^{-62} -13 q^{-64} + q^{-66} + q^{-68} +4 q^{-70} - q^{-72} -2 q^{-74} + q^{-76} }
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}-2 q^{193}-q^{191}+2 q^{189}+q^{187}+4 q^{185}+q^{183}-9 q^{181}-13 q^{179}+13 q^{175}+27 q^{173}+23 q^{171}-17 q^{169}-57 q^{167}-57 q^{165}+4 q^{163}+80 q^{161}+116 q^{159}+49 q^{157}-106 q^{155}-210 q^{153}-128 q^{151}+111 q^{149}+318 q^{147}+293 q^{145}-55 q^{143}-487 q^{141}-552 q^{139}-82 q^{137}+625 q^{135}+948 q^{133}+433 q^{131}-687 q^{129}-1473 q^{127}-1043 q^{125}+511 q^{123}+1986 q^{121}+1932 q^{119}+64 q^{117}-2277 q^{115}-2977 q^{113}-1068 q^{111}+2116 q^{109}+3853 q^{107}+2369 q^{105}-1348 q^{103}-4236 q^{101}-3661 q^{99}+105 q^{97}+3920 q^{95}+4510 q^{93}+1313 q^{91}-2916 q^{89}-4646 q^{87}-2531 q^{85}+1529 q^{83}+4076 q^{81}+3163 q^{79}-140 q^{77}-2954 q^{75}-3194 q^{73}-944 q^{71}+1738 q^{69}+2712 q^{67}+1518 q^{65}-646 q^{63}-2036 q^{61}-1704 q^{59}-75 q^{57}+1448 q^{55}+1653 q^{53}+461 q^{51}-1101 q^{49}-1624 q^{47}-621 q^{45}+1024 q^{43}+1758 q^{41}+756 q^{39}-1145 q^{37}-2093 q^{35}-1008 q^{33}+1259 q^{31}+2572 q^{29}+1481 q^{27}-1224 q^{25}-3062 q^{23}-2147 q^{21}+894 q^{19}+3360 q^{17}+2922 q^{15}-197 q^{13}-3313 q^{11}-3622 q^9-757 q^7+2788 q^5+3991 q^3+1869 q-1786 q^{-1} -3894 q^{-3} -2782 q^{-5} +481 q^{-7} +3161 q^{-9} +3262 q^{-11} +881 q^{-13} -1975 q^{-15} -3102 q^{-17} -1881 q^{-19} +544 q^{-21} +2305 q^{-23} +2286 q^{-25} +724 q^{-27} -1135 q^{-29} -2005 q^{-31} -1471 q^{-33} -50 q^{-35} +1216 q^{-37} +1571 q^{-39} +892 q^{-41} -286 q^{-43} -1143 q^{-45} -1176 q^{-47} -443 q^{-49} +470 q^{-51} +977 q^{-53} +789 q^{-55} +99 q^{-57} -543 q^{-59} -724 q^{-61} -401 q^{-63} +111 q^{-65} +462 q^{-67} +437 q^{-69} +123 q^{-71} -188 q^{-73} -295 q^{-75} -181 q^{-77} +7 q^{-79} +145 q^{-81} +141 q^{-83} +39 q^{-85} -43 q^{-87} -66 q^{-89} -39 q^{-91} +28 q^{-95} +21 q^{-97} -7 q^{-101} -5 q^{-103} -2 q^{-105} + q^{-107} +4 q^{-109} - q^{-111} -2 q^{-113} + q^{-115} }
6	$q^{270}-2q^{268}-q^{266}+2q^{264}+q^{262}+4q^{260}-2q^{258}-q^{256}-13q^{254}-9q^{252}+13q^{250}+14q^{248}+27q^{246}+10q^{244}-7q^{242}-64q^{240}-67q^{238}-5q^{236}+38q^{234}+109q^{232}+108q^{230}+63q^{228}-126q^{226}-212q^{224}-146q^{222}-33q^{220}+191q^{218}+311q^{216}+273q^{214}-127q^{212}-407q^{210}-397q^{208}-167q^{206}+342q^{204}+709q^{202}+622q^{200}-250q^{198}-1047q^{196}-1190q^{194}-507q^{192}+985q^{190}+2277q^{188}+2202q^{186}+25q^{184}-2767q^{182}-4327q^{180}-3257q^{178}+809q^{176}+5653q^{174}+7774q^{172}+4452q^{170}-2851q^{168}-9942q^{166}-11951q^{164}-5764q^{162}+5982q^{160}+16039q^{158}+16644q^{156}+6076q^{154}-10333q^{152}-22654q^{150}-21292q^{148}-5371q^{146}+16099q^{144}+28920q^{142}+24120q^{140}+3309q^{138}-21544q^{136}-33763q^{134}-24841q^{132}+390q^{130}+25729q^{128}+35371q^{126}+23053q^{124}-3977q^{122}-27889q^{120}-33953q^{118}-18819q^{116}+6776q^{114}+26990q^{112}+29781q^{110}+14185q^{108}-8412q^{106}-24021q^{104}-23712q^{102}-9831q^{100}+8388q^{98}+19677q^{96}+17926q^{94}+6138q^{92}-7736q^{90}-15046q^{88}-12885q^{86}-3334q^{84}+6936q^{82}+11631q^{80}+8678q^{78}+578q^{76}-6798q^{74}-9172q^{72}-4928q^{70}+2480q^{68}+7842q^{66}+7132q^{64}+713q^{62}-6470q^{60}-9155q^{58}-4510q^{56}+4319q^{54}+11134q^{52}+9893q^{50}+510q^{48}-10612q^{46}-15276q^{44}-8896q^{42}+4907q^{40}+17207q^{38}+18167q^{36}+5956q^{34}-11771q^{32}-22836q^{30}-18828q^{28}-1546q^{26}+18498q^{24}+26989q^{22}+17637q^{20}-4042q^{18}-23870q^{16}-28776q^{14}-15236q^{12}+8937q^{10}+27543q^{8}+28812q^{6}+11855q^{4}-12251q^{2}-28682-27467q^{-2}-9150q^{-4}+13933q^{-6}+27945q^{-8}+24702q^{-10}+7438q^{-12}-13416q^{-14}-25505q^{-16}-21891q^{-18}-6415q^{-20}+11529q^{-22}+21381q^{-24}+19027q^{-26}+6386q^{-28}-8517q^{-30}-17022q^{-32}-15857q^{-34}-6564q^{-36}+4678q^{-38}+12564q^{-40}+12917q^{-42}+6863q^{-44}-1485q^{-46}-8096q^{-48}-9915q^{-50}-7158q^{-52}-1041q^{-54}+4453q^{-56}+7237q^{-58}+6518q^{-60}+2914q^{-62}-1556q^{-64}-5070q^{-66}-5497q^{-68}-3631q^{-70}-237q^{-72}+2908q^{-74}+4405q^{-76}+3649q^{-78}+1046q^{-80}-1410q^{-82}-3083q^{-84}-2987q^{-86}-1507q^{-88}+567q^{-90}+2011q^{-92}+2086q^{-94}+1344q^{-96}-33q^{-98}-1084q^{-100}-1448q^{-102}-927q^{-104}-103q^{-106}+485q^{-108}+808q^{-110}+590q^{-112}+168q^{-114}-271q^{-116}-379q^{-118}-278q^{-120}-124q^{-122}+95q^{-124}+172q^{-126}+144q^{-128}+24q^{-130}-30q^{-132}-51q^{-134}-57q^{-136}-13q^{-138}+15q^{-140}+27q^{-142}+3q^{-144}+q^{-146}+q^{-148}-8q^{-150}-2q^{-152}+q^{-154}+4q^{-156}-q^{-158}-2q^{-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}-q^{18}+q^{16}-2 q^{14}-q^{12}+q^{10}-q^8+4 q^6-q^4+2 q^2- q^{-2} + q^{-4} - 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}-4 q^{58}+10 q^{56}-20 q^{54}+34 q^{52}-54 q^{50}+78 q^{48}-100 q^{46}+122 q^{44}-140 q^{42}+156 q^{40}-174 q^{38}+187 q^{36}-202 q^{34}+212 q^{32}-202 q^{30}+172 q^{28}-108 q^{26}+8 q^{24}+116 q^{22}-256 q^{20}+384 q^{18}-498 q^{16}+576 q^{14}-604 q^{12}+588 q^{10}-518 q^8+414 q^6-269 q^4+112 q^2+50-190 q^{-2} +294 q^{-4} -350 q^{-6} +358 q^{-8} -328 q^{-10} +272 q^{-12} -202 q^{-14} +136 q^{-16} -86 q^{-18} +48 q^{-20} -22 q^{-22} +10 q^{-24} -4 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}-q^{50}-q^{48}+q^{46}-q^{44}+q^{42}+2 q^{40}-q^{38}-q^{36}+2 q^{34}+4 q^{32}-5 q^{30}-6 q^{28}+4 q^{26}-8 q^{22}+q^{20}+9 q^{18}-q^{16}-q^{14}+4 q^{12}+2 q^{10}-4 q^8+6 q^4-5 q^2-1+7 q^{-2} + q^{-4} -6 q^{-6} + q^{-8} +5 q^{-10} -2 q^{-12} -4 q^{-14} + q^{-16} +3 q^{-18} - q^{-20} - 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}-2 q^{46}+4 q^{42}-6 q^{40}-q^{38}+9 q^{36}-7 q^{34}-4 q^{32}+15 q^{30}-5 q^{28}-8 q^{26}+12 q^{24}-5 q^{22}-9 q^{20}+2 q^{18}+q^{16}-q^{14}-3 q^{12}+9 q^{10}+6 q^8-10 q^6+6 q^4+7 q^2-12+3 q^{-2} +8 q^{-4} -9 q^{-6} +4 q^{-8} +4 q^{-10} -5 q^{-12} +3 q^{-14} -2 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}-q^{23}+2 q^{21}-3 q^{19}+q^{17}-3 q^{15}+q^{13}+2 q^9+2 q^7+2 q^3-2 q+2 q^{-1} -2 q^{-3} +2 q^{-5} - 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}-4 q^{76}+8 q^{74}-8 q^{72}-2 q^{70}+21 q^{68}-38 q^{66}+37 q^{64}-10 q^{62}-33 q^{60}+69 q^{58}-72 q^{56}+41 q^{54}+14 q^{52}-65 q^{50}+73 q^{48}-32 q^{46}-51 q^{44}+117 q^{42}-119 q^{40}+44 q^{38}+97 q^{36}-211 q^{34}+254 q^{32}-198 q^{30}+65 q^{28}+68 q^{26}-171 q^{24}+188 q^{22}-159 q^{20}+106 q^{18}-66 q^{16}+69 q^{14}-77 q^{12}+88 q^{10}-36 q^8-59 q^6+174 q^4-246 q^2+244-160 q^{-2} +38 q^{-4} +84 q^{-6} -159 q^{-8} +175 q^{-10} -135 q^{-12} +57 q^{-14} +17 q^{-16} -60 q^{-18} +64 q^{-20} -41 q^{-22} +13 q^{-24} +6 q^{-26} -10 q^{-28} +8 q^{-30} -4 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}-q^{56}-q^{54}+3 q^{52}-2 q^{50}-6 q^{48}+4 q^{46}+4 q^{44}-9 q^{42}+q^{40}+12 q^{38}+2 q^{36}-6 q^{34}+7 q^{32}+6 q^{30}-12 q^{28}-8 q^{26}+6 q^{24}-8 q^{22}-12 q^{20}+13 q^{18}+5 q^{16}-7 q^{14}+7 q^{12}+13 q^{10}-6 q^8-5 q^6+5 q^4+3 q^2-6+7 q^{-4} - q^{-6} +3 q^{-10} - q^{-12} -2 q^{-14} +2 q^{-16} - q^{-18} - 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}-q^{28}+2 q^{26}-2 q^{24}-q^{20}-2 q^{18}+q^{16}-q^{14}+3 q^{12}+3 q^8-q^6+2 q^4-q^2+ q^{-2} - q^{-4} +2 q^{-6} - 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}-2 q^{46}+4 q^{44}-6 q^{42}+8 q^{40}-11 q^{38}+13 q^{36}-15 q^{34}+16 q^{32}-15 q^{30}+11 q^{28}-6 q^{26}-2 q^{24}+9 q^{22}-17 q^{20}+24 q^{18}-29 q^{16}+33 q^{14}-31 q^{12}+29 q^{10}-22 q^8+16 q^6-6 q^4-q^2+8-13 q^{-2} +16 q^{-4} -17 q^{-6} +16 q^{-8} -14 q^{-10} +11 q^{-12} -7 q^{-14} +4 q^{-16} -2 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}-2 q^{74}-2 q^{72}+2 q^{70}+5 q^{68}-7 q^{64}-6 q^{62}+5 q^{60}+11 q^{58}+2 q^{56}-11 q^{54}-10 q^{52}+7 q^{50}+17 q^{48}+2 q^{46}-15 q^{44}-9 q^{42}+11 q^{40}+12 q^{38}-7 q^{36}-15 q^{34}+12 q^{30}+q^{28}-12 q^{26}-5 q^{24}+10 q^{22}+7 q^{20}-6 q^{18}-6 q^{16}+9 q^{14}+11 q^{12}-4 q^{10}-14 q^8+q^6+16 q^4+8 q^2-14-16 q^{-2} +6 q^{-4} +18 q^{-6} +3 q^{-8} -15 q^{-10} -9 q^{-12} +9 q^{-14} +11 q^{-16} -2 q^{-18} -8 q^{-20} - q^{-22} +5 q^{-24} +2 q^{-26} -2 q^{-28} -2 q^{-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}-2 q^{64}+2 q^{62}-3 q^{60}+5 q^{58}-7 q^{56}+6 q^{54}-8 q^{52}+11 q^{50}-11 q^{48}+10 q^{46}-11 q^{44}+15 q^{42}-9 q^{40}+7 q^{38}-5 q^{36}+2 q^{34}+4 q^{32}-10 q^{30}+9 q^{28}-19 q^{26}+20 q^{24}-23 q^{22}+24 q^{20}-25 q^{18}+26 q^{16}-19 q^{14}+21 q^{12}-15 q^{10}+11 q^8-5 q^6+2 q^4+2 q^2-6+11 q^{-2} -11 q^{-4} +13 q^{-6} -13 q^{-8} +15 q^{-10} -12 q^{-12} +10 q^{-14} -9 q^{-16} +7 q^{-18} -4 q^{-20} +2 q^{-22} -2 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}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+8 q^{92}+7 q^{90}-22 q^{88}+35 q^{86}-38 q^{84}+35 q^{82}-25 q^{80}+6 q^{78}+16 q^{76}-37 q^{74}+57 q^{72}-60 q^{70}+47 q^{68}-18 q^{66}-20 q^{64}+52 q^{62}-67 q^{60}+53 q^{58}-17 q^{56}-28 q^{54}+53 q^{52}-53 q^{50}+16 q^{48}+36 q^{46}-76 q^{44}+82 q^{42}-56 q^{40}-q^{38}+64 q^{36}-107 q^{34}+116 q^{32}-84 q^{30}+30 q^{28}+38 q^{26}-85 q^{24}+106 q^{22}-88 q^{20}+49 q^{18}+4 q^{16}-52 q^{14}+75 q^{12}-60 q^{10}+23 q^8+28 q^6-66 q^4+68 q^2-36-21 q^{-2} +70 q^{-4} -95 q^{-6} +84 q^{-8} -38 q^{-10} -21 q^{-12} +69 q^{-14} -87 q^{-16} +78 q^{-18} -43 q^{-20} +2 q^{-22} +27 q^{-24} -41 q^{-26} +39 q^{-28} -26 q^{-30} +13 q^{-32} + q^{-34} -7 q^{-36} +7 q^{-38} -7 q^{-40} +4 q^{-42} -2 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 82"];

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+4 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +4 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^8-4 z^6-4 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]=

{ 63, -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^3-3 q^2+5 q-7+10 q^{-1} -10 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 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^6-6 a^2 z^6+z^6+4 a^4 z^4-12 a^2 z^4+4 z^4+4 a^4 z^2-8 a^2 z^2+4 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 2 a^3 z^9+2 a z^9+4 a^4 z^8+8 a^2 z^8+4 z^8+4 a^5 z^7-a^3 z^7-2 a z^7+3 z^7 a^{-1} +4 a^6 z^6-8 a^4 z^6-27 a^2 z^6+z^6 a^{-2} -14 z^6+3 a^7 z^5-3 a^5 z^5-4 a^3 z^5-8 a z^5-10 z^5 a^{-1} +a^8 z^4-4 a^6 z^4+10 a^4 z^4+32 a^2 z^4-3 z^4 a^{-2} +14 z^4-4 a^7 z^3-2 a^5 z^3+5 a^3 z^3+10 a z^3+7 z^3 a^{-1} -a^8 z^2-5 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -6 z^2+a^7 z+2 a^5 z-2 a z-z a^{-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 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 -96} 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 -64} 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 176} 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{208}{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{32}{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{352}{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 -32}

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 82. 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										2		-2
3									3	1		2
1								4	2			-2
-1							6	3				3
-3						5	5					0
-5					5	5						0
-7				3	5							2
-9			2	5								-3
-11		1	3									2
-13		2										-2
-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, 82]]
Out[2]=	10
In[3]:=	PD[Knot[10, 82]]
Out[3]=	PD[X[1, 6, 2, 7], X[7, 16, 8, 17], X[3, 9, 4, 8], X[15, 3, 16, 2], X[5, 15, 6, 14], X[9, 5, 10, 4], X[11, 18, 12, 19], X[13, 20, 14, 1], X[17, 10, 18, 11], X[19, 12, 20, 13]]
In[4]:=	GaussCode[Knot[10, 82]]
Out[4]=	GaussCode[-1, 4, -3, 6, -5, 1, -2, 3, -6, 9, -7, 10, -8, 5, -4, 2, -9, 7, -10, 8]
In[5]:=	BR[Knot[10, 82]]
Out[5]=	BR[3, {-1, -1, -1, -1, 2, -1, 2, -1, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 82]][t]
Out[6]=	-4 4 8 12 2 3 4 -13 - t + -- - -- + -- + 12 t - 8 t + 4 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 82]][z]
Out[7]=	4 6 8 1 - 4 z - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 82]}
In[9]:=	{KnotDet[Knot[10, 82]], KnotSignature[Knot[10, 82]]}
Out[9]=	{63, -2}
In[10]:=	J=Jones[Knot[10, 82]][q]
Out[10]=	-7 3 5 8 10 10 10 2 3 -7 + q - -- + -- - -- + -- - -- + -- + 5 q - 3 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, 82]}
In[12]:=	A2Invariant[Knot[10, 82]][q]
Out[12]=	-20 -18 -16 2 -12 -10 -8 4 -4 2 2 q - q + q - --- - q + q - q + -- - q + -- - q + 14 6 2 q q q 4 6 8 q - q + q
In[13]:=	Kauffman[Knot[10, 82]][a, z]
Out[13]=	2 z 5 7 2 z 2 2 4 2 1 - - - 2 a z + 2 a z + a z - 6 z + -- - 13 a z - 5 a z - a 2 a 3 4 8 2 7 z 3 3 3 5 3 7 3 4 3 z a z + ---- + 10 a z + 5 a z - 2 a z - 4 a z + 14 z - ---- + a 2 a 5 2 4 4 4 6 4 8 4 10 z 5 3 5 32 a z + 10 a z - 4 a z + a z - ----- - 8 a z - 4 a z - a 6 5 5 7 5 6 z 2 6 4 6 6 6 3 a z + 3 a z - 14 z + -- - 27 a z - 8 a z + 4 a z + 2 a 7 3 z 7 3 7 5 7 8 2 8 4 8 9 ---- - 2 a z - a z + 4 a z + 4 z + 8 a z + 4 a z + 2 a z + a 3 9 2 a z
In[14]:=	{Vassiliev[2][Knot[10, 82]], Vassiliev[3][Knot[10, 82]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 82]][q, t]
Out[15]=	5 6 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 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 5 5 5 5 3 t 2 3 2 ----- + ----- + ---- + ---- + --- + 4 q t + 2 q t + 3 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 + 2 q t + q t