10 82

10_81

10_83

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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 $1$ Topological 4 genus $1$ Concordance genus $[2,4]$ Rasmussen s-Invariant -2

[edit Notes for 10 82's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+4t^{3}-8t^{2}+12t-13+12t^{-1}-8t^{-2}+4t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-4z^{6}-4z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 63, -2 }
Jones polynomial	$q^{3}-3q^{2}+5q-7+10q^{-1}-10q^{-2}+10q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+z^{6}+4a^{4}z^{4}-12a^{2}z^{4}+4z^{4}+4a^{4}z^{2}-8a^{2}z^{2}+4z^{2}+1$
Kauffman polynomial (db, data sources)	$2a^{3}z^{9}+2az^{9}+4a^{4}z^{8}+8a^{2}z^{8}+4z^{8}+4a^{5}z^{7}-a^{3}z^{7}-2az^{7}+3z^{7}a^{-1}+4a^{6}z^{6}-8a^{4}z^{6}-27a^{2}z^{6}+z^{6}a^{-2}-14z^{6}+3a^{7}z^{5}-3a^{5}z^{5}-4a^{3}z^{5}-8az^{5}-10z^{5}a^{-1}+a^{8}z^{4}-4a^{6}z^{4}+10a^{4}z^{4}+32a^{2}z^{4}-3z^{4}a^{-2}+14z^{4}-4a^{7}z^{3}-2a^{5}z^{3}+5a^{3}z^{3}+10az^{3}+7z^{3}a^{-1}-a^{8}z^{2}-5a^{4}z^{2}-13a^{2}z^{2}+z^{2}a^{-2}-6z^{2}+a^{7}z+2a^{5}z-2az-za^{-1}+1$
The A2 invariant	$q^{20}-q^{18}+q^{16}-2q^{14}-q^{12}+q^{10}-q^{8}+4q^{6}-q^{4}+2q^{2}-q^{-2}+q^{-4}-q^{-6}+q^{-8}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+4q^{106}-2q^{104}-4q^{102}+12q^{100}-18q^{98}+22q^{96}-19q^{94}+8q^{92}+7q^{90}-22q^{88}+35q^{86}-38q^{84}+35q^{82}-25q^{80}+6q^{78}+16q^{76}-37q^{74}+57q^{72}-60q^{70}+47q^{68}-18q^{66}-20q^{64}+52q^{62}-67q^{60}+53q^{58}-17q^{56}-28q^{54}+53q^{52}-53q^{50}+16q^{48}+36q^{46}-76q^{44}+82q^{42}-56q^{40}-q^{38}+64q^{36}-107q^{34}+116q^{32}-84q^{30}+30q^{28}+38q^{26}-85q^{24}+106q^{22}-88q^{20}+49q^{18}+4q^{16}-52q^{14}+75q^{12}-60q^{10}+23q^{8}+28q^{6}-66q^{4}+68q^{2}-36-21q^{-2}+70q^{-4}-95q^{-6}+84q^{-8}-38q^{-10}-21q^{-12}+69q^{-14}-87q^{-16}+78q^{-18}-43q^{-20}+2q^{-22}+27q^{-24}-41q^{-26}+39q^{-28}-26q^{-30}+13q^{-32}+q^{-34}-7q^{-36}+7q^{-38}-7q^{-40}+4q^{-42}-2q^{-44}+q^{-46}$

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	$q^{78}-4q^{76}+8q^{74}-8q^{72}-2q^{70}+21q^{68}-38q^{66}+37q^{64}-10q^{62}-33q^{60}+69q^{58}-72q^{56}+41q^{54}+14q^{52}-65q^{50}+73q^{48}-32q^{46}-51q^{44}+117q^{42}-119q^{40}+44q^{38}+97q^{36}-211q^{34}+254q^{32}-198q^{30}+65q^{28}+68q^{26}-171q^{24}+188q^{22}-159q^{20}+106q^{18}-66q^{16}+69q^{14}-77q^{12}+88q^{10}-36q^{8}-59q^{6}+174q^{4}-246q^{2}+244-160q^{-2}+38q^{-4}+84q^{-6}-159q^{-8}+175q^{-10}-135q^{-12}+57q^{-14}+17q^{-16}-60q^{-18}+64q^{-20}-41q^{-22}+13q^{-24}+6q^{-26}-10q^{-28}+8q^{-30}-4q^{-32}+q^{-34}$

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	$q^{30}-q^{28}+2q^{26}-2q^{24}-q^{20}-2q^{18}+q^{16}-q^{14}+3q^{12}+3q^{8}-q^{6}+2q^{4}-q^{2}+q^{-2}-q^{-4}+2q^{-6}-q^{-8}+q^{-10}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-2q^{46}+4q^{44}-6q^{42}+8q^{40}-11q^{38}+13q^{36}-15q^{34}+16q^{32}-15q^{30}+11q^{28}-6q^{26}-2q^{24}+9q^{22}-17q^{20}+24q^{18}-29q^{16}+33q^{14}-31q^{12}+29q^{10}-22q^{8}+16q^{6}-6q^{4}-q^{2}+8-13q^{-2}+16q^{-4}-17q^{-6}+16q^{-8}-14q^{-10}+11q^{-12}-7q^{-14}+4q^{-16}-2q^{-18}+q^{-20}$
1,0	$q^{78}-2q^{74}-2q^{72}+2q^{70}+5q^{68}-7q^{64}-6q^{62}+5q^{60}+11q^{58}+2q^{56}-11q^{54}-10q^{52}+7q^{50}+17q^{48}+2q^{46}-15q^{44}-9q^{42}+11q^{40}+12q^{38}-7q^{36}-15q^{34}+12q^{30}+q^{28}-12q^{26}-5q^{24}+10q^{22}+7q^{20}-6q^{18}-6q^{16}+9q^{14}+11q^{12}-4q^{10}-14q^{8}+q^{6}+16q^{4}+8q^{2}-14-16q^{-2}+6q^{-4}+18q^{-6}+3q^{-8}-15q^{-10}-9q^{-12}+9q^{-14}+11q^{-16}-2q^{-18}-8q^{-20}-q^{-22}+5q^{-24}+2q^{-26}-2q^{-28}-2q^{-30}+q^{-34}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-2q^{64}+2q^{62}-3q^{60}+5q^{58}-7q^{56}+6q^{54}-8q^{52}+11q^{50}-11q^{48}+10q^{46}-11q^{44}+15q^{42}-9q^{40}+7q^{38}-5q^{36}+2q^{34}+4q^{32}-10q^{30}+9q^{28}-19q^{26}+20q^{24}-23q^{22}+24q^{20}-25q^{18}+26q^{16}-19q^{14}+21q^{12}-15q^{10}+11q^{8}-5q^{6}+2q^{4}+2q^{2}-6+11q^{-2}-11q^{-4}+13q^{-6}-13q^{-8}+15q^{-10}-12q^{-12}+10q^{-14}-9q^{-16}+7q^{-18}-4q^{-20}+2q^{-22}-2q^{-24}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-2q^{112}+4q^{110}-6q^{108}+4q^{106}-2q^{104}-4q^{102}+12q^{100}-18q^{98}+22q^{96}-19q^{94}+8q^{92}+7q^{90}-22q^{88}+35q^{86}-38q^{84}+35q^{82}-25q^{80}+6q^{78}+16q^{76}-37q^{74}+57q^{72}-60q^{70}+47q^{68}-18q^{66}-20q^{64}+52q^{62}-67q^{60}+53q^{58}-17q^{56}-28q^{54}+53q^{52}-53q^{50}+16q^{48}+36q^{46}-76q^{44}+82q^{42}-56q^{40}-q^{38}+64q^{36}-107q^{34}+116q^{32}-84q^{30}+30q^{28}+38q^{26}-85q^{24}+106q^{22}-88q^{20}+49q^{18}+4q^{16}-52q^{14}+75q^{12}-60q^{10}+23q^{8}+28q^{6}-66q^{4}+68q^{2}-36-21q^{-2}+70q^{-4}-95q^{-6}+84q^{-8}-38q^{-10}-21q^{-12}+69q^{-14}-87q^{-16}+78q^{-18}-43q^{-20}+2q^{-22}+27q^{-24}-41q^{-26}+39q^{-28}-26q^{-30}+13q^{-32}+q^{-34}-7q^{-36}+7q^{-38}-7q^{-40}+4q^{-42}-2q^{-44}+q^{-46}$

.

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]=

$-t^{4}+4t^{3}-8t^{2}+12t-13+12t^{-1}-8t^{-2}+4t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-4z^{6}-4z^{4}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 63, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{3}-3q^{2}+5q-7+10q^{-1}-10q^{-2}+10q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+z^{6}+4a^{4}z^{4}-12a^{2}z^{4}+4z^{4}+4a^{4}z^{2}-8a^{2}z^{2}+4z^{2}+1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$2a^{3}z^{9}+2az^{9}+4a^{4}z^{8}+8a^{2}z^{8}+4z^{8}+4a^{5}z^{7}-a^{3}z^{7}-2az^{7}+3z^{7}a^{-1}+4a^{6}z^{6}-8a^{4}z^{6}-27a^{2}z^{6}+z^{6}a^{-2}-14z^{6}+3a^{7}z^{5}-3a^{5}z^{5}-4a^{3}z^{5}-8az^{5}-10z^{5}a^{-1}+a^{8}z^{4}-4a^{6}z^{4}+10a^{4}z^{4}+32a^{2}z^{4}-3z^{4}a^{-2}+14z^{4}-4a^{7}z^{3}-2a^{5}z^{3}+5a^{3}z^{3}+10az^{3}+7z^{3}a^{-1}-a^{8}z^{2}-5a^{4}z^{2}-13a^{2}z^{2}+z^{2}a^{-2}-6z^{2}+a^{7}z+2a^{5}z-2az-za^{-1}+1$

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 $0$ $0$ $0$ $32$ $32$ $0$ $-96$ $-32$ $-64$ $0$ $0$ $0$ $0$ $176$ ${\frac {208}{3}}$ $-{\frac {32}{3}}$ ${\frac {352}{3}}$ $-32$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

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

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