10 82

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	$q^{15}-2q^{13}+2q^{11}-3q^{9}+2q^{7}+3q-2q^{-1}+2q^{-3}-2q^{-5}+q^{-7}$
2	$q^{42}-2q^{40}-q^{38}+5q^{36}-4q^{34}-q^{32}+9q^{30}-10q^{28}-3q^{26}+16q^{24}-11q^{22}-9q^{20}+15q^{18}-2q^{16}-10q^{14}+4q^{12}+9q^{10}-5q^{8}-8q^{6}+13q^{4}+2q^{2}-15+11q^{-2}+8q^{-4}-16q^{-6}+3q^{-8}+12q^{-10}-9q^{-12}-4q^{-14}+7q^{-16}-q^{-18}-2q^{-20}+q^{-22}$
3	$q^{81}-2q^{79}-q^{77}+2q^{75}+4q^{73}-q^{71}-7q^{69}+3q^{65}+3q^{63}+2q^{61}+2q^{59}-11q^{57}-14q^{55}+17q^{53}+35q^{51}-14q^{49}-57q^{47}-2q^{45}+73q^{43}+26q^{41}-73q^{39}-48q^{37}+52q^{35}+65q^{33}-29q^{31}-63q^{29}-3q^{27}+56q^{25}+27q^{23}-42q^{21}-46q^{19}+30q^{17}+56q^{15}-18q^{13}-66q^{11}+8q^{9}+75q^{7}+7q^{5}-74q^{3}-27q+73q^{-1}+48q^{-3}-52q^{-5}-69q^{-7}+27q^{-9}+73q^{-11}+4q^{-13}-64q^{-15}-31q^{-17}+44q^{-19}+42q^{-21}-20q^{-23}-38q^{-25}+27q^{-29}+9q^{-31}-14q^{-33}-7q^{-35}+4q^{-37}+4q^{-39}-q^{-41}-2q^{-43}+q^{-45}$
4	$q^{132}-2q^{130}-q^{128}+2q^{126}+q^{124}+7q^{122}-7q^{120}-9q^{118}-4q^{116}+33q^{112}+6q^{110}-14q^{108}-35q^{106}-41q^{104}+51q^{102}+62q^{100}+42q^{98}-58q^{96}-154q^{94}-21q^{92}+114q^{90}+213q^{88}+53q^{86}-268q^{84}-263q^{82}-12q^{80}+387q^{78}+371q^{76}-151q^{74}-483q^{72}-370q^{70}+268q^{68}+618q^{66}+214q^{64}-355q^{62}-603q^{60}-105q^{58}+463q^{56}+440q^{54}+22q^{52}-441q^{50}-337q^{48}+87q^{46}+337q^{44}+261q^{42}-118q^{40}-310q^{38}-169q^{36}+154q^{34}+310q^{32}+75q^{30}-260q^{28}-285q^{26}+85q^{24}+355q^{22}+193q^{20}-275q^{18}-423q^{16}+11q^{14}+423q^{12}+378q^{10}-192q^{8}-542q^{6}-203q^{4}+314q^{2}+542+95q^{-2}-434q^{-4}-420q^{-6}-37q^{-8}+439q^{-10}+367q^{-12}-57q^{-14}-343q^{-16}-346q^{-18}+70q^{-20}+315q^{-22}+251q^{-24}-7q^{-26}-300q^{-28}-198q^{-30}+20q^{-32}+206q^{-34}+197q^{-36}-50q^{-38}-143q^{-40}-127q^{-42}+13q^{-44}+125q^{-46}+57q^{-48}-6q^{-50}-67q^{-52}-41q^{-54}+22q^{-56}+21q^{-58}+18q^{-60}-8q^{-62}-13q^{-64}+q^{-66}+q^{-68}+4q^{-70}-q^{-72}-2q^{-74}+q^{-76}$
5	$q^{195}-2q^{193}-q^{191}+2q^{189}+q^{187}+4q^{185}+q^{183}-9q^{181}-13q^{179}+13q^{175}+27q^{173}+23q^{171}-17q^{169}-57q^{167}-57q^{165}+4q^{163}+80q^{161}+116q^{159}+49q^{157}-106q^{155}-210q^{153}-128q^{151}+111q^{149}+318q^{147}+293q^{145}-55q^{143}-487q^{141}-552q^{139}-82q^{137}+625q^{135}+948q^{133}+433q^{131}-687q^{129}-1473q^{127}-1043q^{125}+511q^{123}+1986q^{121}+1932q^{119}+64q^{117}-2277q^{115}-2977q^{113}-1068q^{111}+2116q^{109}+3853q^{107}+2369q^{105}-1348q^{103}-4236q^{101}-3661q^{99}+105q^{97}+3920q^{95}+4510q^{93}+1313q^{91}-2916q^{89}-4646q^{87}-2531q^{85}+1529q^{83}+4076q^{81}+3163q^{79}-140q^{77}-2954q^{75}-3194q^{73}-944q^{71}+1738q^{69}+2712q^{67}+1518q^{65}-646q^{63}-2036q^{61}-1704q^{59}-75q^{57}+1448q^{55}+1653q^{53}+461q^{51}-1101q^{49}-1624q^{47}-621q^{45}+1024q^{43}+1758q^{41}+756q^{39}-1145q^{37}-2093q^{35}-1008q^{33}+1259q^{31}+2572q^{29}+1481q^{27}-1224q^{25}-3062q^{23}-2147q^{21}+894q^{19}+3360q^{17}+2922q^{15}-197q^{13}-3313q^{11}-3622q^{9}-757q^{7}+2788q^{5}+3991q^{3}+1869q-1786q^{-1}-3894q^{-3}-2782q^{-5}+481q^{-7}+3161q^{-9}+3262q^{-11}+881q^{-13}-1975q^{-15}-3102q^{-17}-1881q^{-19}+544q^{-21}+2305q^{-23}+2286q^{-25}+724q^{-27}-1135q^{-29}-2005q^{-31}-1471q^{-33}-50q^{-35}+1216q^{-37}+1571q^{-39}+892q^{-41}-286q^{-43}-1143q^{-45}-1176q^{-47}-443q^{-49}+470q^{-51}+977q^{-53}+789q^{-55}+99q^{-57}-543q^{-59}-724q^{-61}-401q^{-63}+111q^{-65}+462q^{-67}+437q^{-69}+123q^{-71}-188q^{-73}-295q^{-75}-181q^{-77}+7q^{-79}+145q^{-81}+141q^{-83}+39q^{-85}-43q^{-87}-66q^{-89}-39q^{-91}+28q^{-95}+21q^{-97}-7q^{-101}-5q^{-103}-2q^{-105}+q^{-107}+4q^{-109}-q^{-111}-2q^{-113}+q^{-115}$
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	$q^{20}-q^{18}+q^{16}-2q^{14}-q^{12}+q^{10}-q^{8}+4q^{6}-q^{4}+2q^{2}-q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,1	$q^{60}-4q^{58}+10q^{56}-20q^{54}+34q^{52}-54q^{50}+78q^{48}-100q^{46}+122q^{44}-140q^{42}+156q^{40}-174q^{38}+187q^{36}-202q^{34}+212q^{32}-202q^{30}+172q^{28}-108q^{26}+8q^{24}+116q^{22}-256q^{20}+384q^{18}-498q^{16}+576q^{14}-604q^{12}+588q^{10}-518q^{8}+414q^{6}-269q^{4}+112q^{2}+50-190q^{-2}+294q^{-4}-350q^{-6}+358q^{-8}-328q^{-10}+272q^{-12}-202q^{-14}+136q^{-16}-86q^{-18}+48q^{-20}-22q^{-22}+10q^{-24}-4q^{-26}+q^{-28}$
2,0	$q^{52}-q^{50}-q^{48}+q^{46}-q^{44}+q^{42}+2q^{40}-q^{38}-q^{36}+2q^{34}+4q^{32}-5q^{30}-6q^{28}+4q^{26}-8q^{22}+q^{20}+9q^{18}-q^{16}-q^{14}+4q^{12}+2q^{10}-4q^{8}+6q^{4}-5q^{2}-1+7q^{-2}+q^{-4}-6q^{-6}+q^{-8}+5q^{-10}-2q^{-12}-4q^{-14}+q^{-16}+3q^{-18}-q^{-20}-q^{-22}+q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-2q^{46}+4q^{42}-6q^{40}-q^{38}+9q^{36}-7q^{34}-4q^{32}+15q^{30}-5q^{28}-8q^{26}+12q^{24}-5q^{22}-9q^{20}+2q^{18}+q^{16}-q^{14}-3q^{12}+9q^{10}+6q^{8}-10q^{6}+6q^{4}+7q^{2}-12+3q^{-2}+8q^{-4}-9q^{-6}+4q^{-8}+4q^{-10}-5q^{-12}+3q^{-14}-2q^{-18}+q^{-20}$
1,0,0	$q^{25}-q^{23}+2q^{21}-3q^{19}+q^{17}-3q^{15}+q^{13}+2q^{9}+2q^{7}+2q^{3}-2q+2q^{-1}-2q^{-3}+2q^{-5}-q^{-7}+q^{-9}$
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	$q^{58}-q^{56}-q^{54}+3q^{52}-2q^{50}-6q^{48}+4q^{46}+4q^{44}-9q^{42}+q^{40}+12q^{38}+2q^{36}-6q^{34}+7q^{32}+6q^{30}-12q^{28}-8q^{26}+6q^{24}-8q^{22}-12q^{20}+13q^{18}+5q^{16}-7q^{14}+7q^{12}+13q^{10}-6q^{8}-5q^{6}+5q^{4}+3q^{2}-6+7q^{-4}-q^{-6}+3q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-q^{-18}-q^{-20}+q^{-22}$
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}