10 85

10_84

10_86

Visit 10 85's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 85's page at Knotilus! Visit 10 85's page at the original Knot Atlas! 10 85 Quick Notes

10 85 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,6,17,5 X18,11,19,12 X14,7,15,8 X8394 X4,9,5,10 X20,13,1,14 X10,17,11,18 X12,19,13,20 X2,16,3,15
Gauss code	1, -10, 5, -6, 2, -1, 4, -5, 6, -8, 3, -9, 7, -4, 10, -2, 8, -3, 9, -7
Dowker-Thistlethwaite code	6 8 16 14 4 18 20 2 10 12
Conway Notation	[.4.20]

Minimum Braid Representative: Length is 10, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-12][0] Hyperbolic Volume 11.7978 A-Polynomial See Data:10 85/A-polynomial

[edit Notes for 10 85's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $4$ Rasmussen s-Invariant 4

[edit Notes for 10 85's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+8t^{2}-10t+11-10t^{-1}+8t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+4z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, -4 }
Jones polynomial	$-q+3-4q^{-1}+7q^{-2}-8q^{-3}+9q^{-4}-9q^{-5}+7q^{-6}-5q^{-7}+3q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}z^{8}-a^{6}z^{6}+6a^{4}z^{6}-a^{2}z^{6}-4a^{6}z^{4}+12a^{4}z^{4}-4a^{2}z^{4}-4a^{6}z^{2}+9a^{4}z^{2}-3a^{2}z^{2}-a^{6}+a^{4}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}+3z^{4}a^{10}-z^{2}a^{10}+5z^{5}a^{9}-4z^{3}a^{9}+za^{9}+6z^{6}a^{8}-7z^{4}a^{8}+z^{2}a^{8}+6z^{7}a^{7}-10z^{5}a^{7}+2z^{3}a^{7}+5z^{8}a^{6}-12z^{6}a^{6}+8z^{4}a^{6}-5z^{2}a^{6}+a^{6}+2z^{9}a^{5}-15z^{5}a^{5}+14z^{3}a^{5}-2za^{5}+8z^{8}a^{4}-32z^{6}a^{4}+37z^{4}a^{4}-14z^{2}a^{4}+a^{4}+2z^{9}a^{3}-5z^{7}a^{3}-4z^{5}a^{3}+11z^{3}a^{3}-2za^{3}+3z^{8}a^{2}-14z^{6}a^{2}+19z^{4}a^{2}-7z^{2}a^{2}-a^{2}+z^{7}a-4z^{5}a+4z^{3}a-za$
The A2 invariant	$-q^{26}+q^{24}-q^{22}+q^{20}-q^{16}+q^{14}-3q^{12}+2q^{10}+2q^{6}+2q^{4}+1-q^{-2}$
The G2 invariant	$q^{148}-2q^{146}+3q^{144}-4q^{142}+2q^{140}-q^{138}-2q^{136}+8q^{134}-11q^{132}+14q^{130}-12q^{128}+6q^{126}+2q^{124}-11q^{122}+20q^{120}-24q^{118}+21q^{116}-14q^{114}+q^{112}+9q^{110}-16q^{108}+22q^{106}-24q^{104}+21q^{102}-17q^{100}+3q^{98}+12q^{96}-26q^{94}+33q^{92}-29q^{90}+14q^{88}+9q^{86}-28q^{84}+33q^{82}-18q^{80}-7q^{78}+34q^{76}-45q^{74}+30q^{72}+5q^{70}-40q^{68}+64q^{66}-65q^{64}+40q^{62}-2q^{60}-38q^{58}+62q^{56}-69q^{54}+51q^{52}-22q^{50}-13q^{48}+37q^{46}-47q^{44}+43q^{42}-20q^{40}-10q^{38}+33q^{36}-39q^{34}+27q^{32}+7q^{30}-38q^{28}+60q^{26}-49q^{24}+21q^{22}+22q^{20}-54q^{18}+68q^{16}-53q^{14}+23q^{12}+9q^{10}-34q^{8}+42q^{6}-33q^{4}+18q^{2}-2-8q^{-2}+9q^{-4}-9q^{-6}+5q^{-8}-2q^{-10}+q^{-12}$

Further Quantum Invariants

Further quantum knot invariants for 10_85.

A1 Invariants.

Weight	Invariant
1	$-q^{19}+2q^{17}-2q^{15}+2q^{13}-2q^{11}+q^{7}-q^{5}+3q^{3}-q+2q^{-1}-q^{-3}$
2	$q^{52}-2q^{50}+4q^{46}-5q^{44}+q^{42}+4q^{40}-8q^{38}+5q^{36}+6q^{34}-10q^{32}+q^{30}+9q^{28}-5q^{26}-6q^{24}+6q^{22}+3q^{20}-8q^{18}-q^{16}+9q^{14}-5q^{12}-6q^{10}+11q^{8}-10q^{4}+9q^{2}+6-9q^{-2}+q^{-4}+6q^{-6}-3q^{-8}-2q^{-10}+q^{-12}$
3	$-q^{99}+2q^{97}-2q^{93}-q^{91}+3q^{89}+4q^{87}-5q^{85}-4q^{83}-q^{81}+5q^{79}+8q^{77}-16q^{73}-12q^{71}+24q^{69}+26q^{67}-21q^{65}-41q^{63}+8q^{61}+46q^{59}+11q^{57}-39q^{55}-28q^{53}+19q^{51}+40q^{49}+4q^{47}-41q^{45}-19q^{43}+33q^{41}+32q^{39}-28q^{37}-35q^{35}+18q^{33}+37q^{31}-15q^{29}-37q^{27}+6q^{25}+40q^{23}+5q^{21}-40q^{19}-20q^{17}+35q^{15}+35q^{13}-20q^{11}-44q^{9}-q^{7}+47q^{5}+20q^{3}-31q-34q^{-1}+16q^{-3}+36q^{-5}+3q^{-7}-27q^{-9}-13q^{-11}+13q^{-13}+15q^{-15}-4q^{-17}-9q^{-19}-2q^{-21}+3q^{-23}+2q^{-25}-q^{-27}$
4	$q^{160}-2q^{158}+2q^{154}-q^{152}+3q^{150}-8q^{148}+8q^{144}+4q^{142}+10q^{140}-28q^{138}-17q^{136}+13q^{134}+33q^{132}+46q^{130}-49q^{128}-78q^{126}-25q^{124}+81q^{122}+148q^{120}-24q^{118}-178q^{116}-152q^{114}+79q^{112}+297q^{110}+122q^{108}-201q^{106}-333q^{104}-80q^{102}+323q^{100}+318q^{98}-12q^{96}-337q^{94}-295q^{92}+91q^{90}+308q^{88}+225q^{86}-81q^{84}-289q^{82}-169q^{80}+65q^{78}+247q^{76}+167q^{74}-99q^{72}-226q^{70}-118q^{68}+137q^{66}+218q^{64}+19q^{62}-188q^{60}-146q^{58}+95q^{56}+211q^{54}+34q^{52}-206q^{50}-166q^{48}+91q^{46}+252q^{44}+99q^{42}-208q^{40}-246q^{38}-12q^{36}+250q^{34}+234q^{32}-74q^{30}-254q^{28}-189q^{26}+84q^{24}+274q^{22}+140q^{20}-79q^{18}-245q^{16}-152q^{14}+101q^{12}+201q^{10}+156q^{8}-72q^{6}-207q^{4}-120q^{2}+32+191q^{-2}+124q^{-4}-39q^{-6}-132q^{-8}-128q^{-10}+33q^{-12}+109q^{-14}+83q^{-16}+q^{-18}-90q^{-20}-54q^{-22}+44q^{-26}+45q^{-28}-7q^{-30}-20q^{-32}-20q^{-34}-3q^{-36}+12q^{-38}+5q^{-40}+2q^{-42}-3q^{-44}-2q^{-46}+q^{-48}$
5	$-q^{235}+2q^{233}-2q^{229}+q^{227}-q^{225}+2q^{223}+4q^{221}-3q^{219}-11q^{217}-4q^{215}+9q^{213}+21q^{211}+18q^{209}-17q^{207}-51q^{205}-44q^{203}+28q^{201}+98q^{199}+84q^{197}-22q^{195}-162q^{193}-174q^{191}+19q^{189}+265q^{187}+288q^{185}+2q^{183}-386q^{181}-488q^{179}-66q^{177}+569q^{175}+775q^{173}+193q^{171}-752q^{169}-1154q^{167}-475q^{165}+857q^{163}+1619q^{161}+928q^{159}-785q^{157}-2007q^{155}-1521q^{153}+399q^{151}+2165q^{149}+2143q^{147}+253q^{145}-1932q^{143}-2546q^{141}-1047q^{139}+1276q^{137}+2531q^{135}+1745q^{133}-357q^{131}-2058q^{129}-2080q^{127}-558q^{125}+1227q^{123}+1975q^{121}+1235q^{119}-325q^{117}-1519q^{115}-1513q^{113}-399q^{111}+914q^{109}+1450q^{107}+818q^{105}-423q^{103}-1220q^{101}-910q^{99}+164q^{97}+985q^{95}+831q^{93}-128q^{91}-927q^{89}-727q^{87}+260q^{85}+1010q^{83}+736q^{81}-370q^{79}-1229q^{77}-907q^{75}+388q^{73}+1447q^{71}+1210q^{69}-207q^{67}-1564q^{65}-1578q^{63}-155q^{61}+1483q^{59}+1896q^{57}+664q^{55}-1165q^{53}-2042q^{51}-1220q^{49}+603q^{47}+1928q^{45}+1677q^{43}+95q^{41}-1500q^{39}-1879q^{37}-811q^{35}+804q^{33}+1735q^{31}+1340q^{29}-3q^{27}-1218q^{25}-1496q^{23}-730q^{21}+473q^{19}+1260q^{17}+1136q^{15}+277q^{13}-665q^{11}-1117q^{9}-817q^{7}-14q^{5}+728q^{3}+944q+553q^{-1}-149q^{-3}-708q^{-5}-767q^{-7}-339q^{-9}+263q^{-11}+643q^{-13}+568q^{-15}+157q^{-17}-309q^{-19}-522q^{-21}-378q^{-23}-11q^{-25}+294q^{-27}+362q^{-29}+207q^{-31}-55q^{-33}-231q^{-35}-221q^{-37}-78q^{-39}+68q^{-41}+139q^{-43}+113q^{-45}+17q^{-47}-55q^{-49}-68q^{-51}-38q^{-53}+26q^{-57}+28q^{-59}+8q^{-61}-5q^{-63}-8q^{-65}-5q^{-67}-2q^{-69}+3q^{-71}+2q^{-73}-q^{-75}$
6	$q^{324}-2q^{322}+2q^{318}-q^{316}+q^{314}-4q^{312}+2q^{310}-q^{308}+6q^{306}+11q^{304}-12q^{302}-11q^{300}-18q^{298}+2q^{296}+17q^{294}+47q^{292}+44q^{290}-41q^{288}-75q^{286}-90q^{284}-17q^{282}+75q^{280}+182q^{278}+153q^{276}-76q^{274}-218q^{272}-267q^{270}-81q^{268}+190q^{266}+433q^{264}+310q^{262}-211q^{260}-543q^{258}-523q^{256}+33q^{254}+712q^{252}+1028q^{250}+384q^{248}-974q^{246}-1765q^{244}-1298q^{242}+534q^{240}+2539q^{238}+3158q^{236}+1183q^{234}-2494q^{232}-5070q^{230}-4381q^{228}-69q^{226}+5342q^{224}+8081q^{222}+5240q^{220}-2104q^{218}-9094q^{216}-10797q^{214}-5175q^{212}+4807q^{210}+12834q^{208}+12918q^{206}+4288q^{204}-7748q^{202}-15562q^{200}-13829q^{198}-3187q^{196}+9931q^{194}+17032q^{192}+13631q^{190}+2016q^{188}-10703q^{186}-16838q^{184}-12833q^{182}-1379q^{180}+10291q^{178}+15360q^{176}+11501q^{174}+1425q^{172}-8783q^{170}-13354q^{168}-10152q^{166}-1653q^{164}+6958q^{162}+11125q^{160}+8833q^{158}+1862q^{156}-5568q^{154}-9192q^{152}-7229q^{150}-1383q^{148}+4763q^{146}+7392q^{144}+5244q^{142}+18q^{140}-4554q^{138}-5416q^{136}-2475q^{134}+1912q^{132}+4281q^{130}+3028q^{128}-816q^{126}-3930q^{124}-3547q^{122}+90q^{120}+4183q^{118}+5212q^{116}+2034q^{114}-3488q^{112}-7119q^{110}-5611q^{108}+407q^{106}+6874q^{104}+9026q^{102}+4957q^{100}-3165q^{98}-9823q^{96}-10083q^{94}-3372q^{92}+6017q^{90}+11908q^{88}+10233q^{86}+1578q^{84}-8540q^{82}-13387q^{80}-9661q^{78}+278q^{76}+10299q^{74}+14089q^{72}+9013q^{70}-1786q^{68}-11492q^{66}-14080q^{64}-8233q^{62}+2568q^{60}+11791q^{58}+13804q^{56}+7594q^{54}-2862q^{52}-11230q^{50}-13011q^{48}-7287q^{46}+2356q^{44}+10184q^{42}+11892q^{40}+6984q^{38}-1264q^{36}-8438q^{34}-10547q^{32}-6917q^{30}+35q^{28}+6326q^{26}+8811q^{24}+6784q^{22}+1401q^{20}-4079q^{18}-7044q^{16}-6261q^{14}-2637q^{12}+1832q^{10}+5138q^{8}+5569q^{6}+3479q^{4}-60q^{2}-3124-4569q^{-2}-3845q^{-4}-1314q^{-6}+1442q^{-8}+3385q^{-10}+3544q^{-12}+2218q^{-14}-97q^{-16}-2170q^{-18}-2964q^{-20}-2421q^{-22}-772q^{-24}+969q^{-26}+2230q^{-28}+2212q^{-30}+1189q^{-32}-191q^{-34}-1351q^{-36}-1702q^{-38}-1309q^{-40}-243q^{-42}+673q^{-44}+1138q^{-46}+1032q^{-48}+474q^{-50}-190q^{-52}-698q^{-54}-689q^{-56}-435q^{-58}-31q^{-60}+290q^{-62}+421q^{-64}+329q^{-66}+72q^{-68}-89q^{-70}-195q^{-72}-177q^{-74}-93q^{-76}+20q^{-78}+83q^{-80}+69q^{-82}+51q^{-84}+7q^{-86}-20q^{-88}-34q^{-90}-16q^{-92}+q^{-96}+8q^{-98}+5q^{-100}+2q^{-102}-3q^{-104}-2q^{-106}+q^{-108}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}+q^{24}-q^{22}+q^{20}-q^{16}+q^{14}-3q^{12}+2q^{10}+2q^{6}+2q^{4}+1-q^{-2}$
1,1	$q^{76}-4q^{74}+8q^{72}-12q^{70}+20q^{68}-32q^{66}+42q^{64}-50q^{62}+60q^{60}-72q^{58}+74q^{56}-68q^{54}+70q^{52}-72q^{50}+72q^{48}-76q^{46}+82q^{44}-86q^{42}+70q^{40}-40q^{38}-10q^{36}+84q^{34}-158q^{32}+230q^{30}-287q^{28}+326q^{26}-338q^{24}+310q^{22}-271q^{20}+202q^{18}-120q^{16}+24q^{14}+69q^{12}-136q^{10}+200q^{8}-216q^{6}+213q^{4}-178q^{2}+140-96q^{-2}+54q^{-4}-30q^{-6}+12q^{-8}-4q^{-10}+q^{-12}$
2,0	$q^{66}-q^{64}+q^{60}-q^{58}+q^{56}-2q^{54}-q^{52}+2q^{48}-3q^{44}+3q^{42}+4q^{40}-2q^{38}-6q^{36}+5q^{34}+2q^{32}-3q^{30}+q^{26}-2q^{24}-5q^{22}+q^{20}-q^{18}-3q^{16}+3q^{14}+6q^{12}-q^{10}+q^{8}+7q^{6}+4q^{4}-3q^{2}-1+2q^{-2}-3q^{-6}-q^{-8}+q^{-10}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{62}-2q^{60}-q^{58}+5q^{56}-3q^{54}-4q^{52}+8q^{50}-q^{48}-7q^{46}+7q^{44}+q^{42}-9q^{40}+4q^{38}+2q^{36}-6q^{34}+q^{32}+3q^{30}+2q^{28}-4q^{26}+5q^{22}-8q^{20}-2q^{18}+7q^{16}-3q^{14}+2q^{12}+9q^{10}-2q^{8}+5q^{4}-5q^{2}+1+q^{-2}-2q^{-4}+q^{-6}$
1,0,0	$-q^{33}+q^{31}-2q^{29}+2q^{27}-2q^{25}+2q^{23}-q^{21}-q^{17}-q^{15}+q^{13}+4q^{9}+3q^{5}-q^{3}+q-q^{-1}$
1,0,1	$q^{100}-4q^{98}+6q^{96}-2q^{94}-7q^{92}+17q^{90}-20q^{88}+7q^{86}+16q^{84}-31q^{82}+29q^{80}-13q^{78}-12q^{76}+27q^{74}-24q^{72}+16q^{70}-3q^{68}-4q^{66}-2q^{64}+21q^{62}-41q^{60}+42q^{58}-19q^{56}-32q^{54}+83q^{52}-117q^{50}+117q^{48}-89q^{46}+44q^{44}+9q^{42}-53q^{40}+96q^{38}-114q^{36}+120q^{34}-109q^{32}+76q^{30}-56q^{28}+12q^{26}+7q^{24}-57q^{22}+93q^{20}-110q^{18}+126q^{16}-84q^{14}+59q^{12}+10q^{10}-41q^{8}+75q^{6}-75q^{4}+49q^{2}-24-6q^{-2}+15q^{-4}-16q^{-6}+10q^{-8}-4q^{-10}+q^{-12}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{76}-q^{74}-2q^{72}+3q^{70}+q^{68}-6q^{66}+q^{64}+8q^{62}-2q^{60}-4q^{58}+7q^{56}+3q^{54}-7q^{52}-4q^{50}+4q^{48}-7q^{46}-8q^{44}+6q^{42}+2q^{40}-7q^{38}+6q^{36}+9q^{34}-8q^{32}-3q^{30}+4q^{28}-q^{26}-6q^{24}+2q^{22}+6q^{20}+3q^{18}+2q^{16}+6q^{14}+2q^{12}+2q^{8}-q^{6}-2q^{4}+q^{2}-q^{-2}+q^{-4}$
1,0,0,0	$-q^{40}+q^{38}-2q^{36}+q^{34}-q^{32}+q^{28}-q^{26}+q^{24}-2q^{22}+q^{20}-2q^{18}+2q^{16}+3q^{12}+2q^{10}+q^{8}+2q^{6}-q^{4}+q^{2}-1$

B2 Invariants.

Weight	Invariant
0,1	$-q^{62}+2q^{60}-3q^{58}+5q^{56}-7q^{54}+8q^{52}-10q^{50}+11q^{48}-11q^{46}+11q^{44}-9q^{42}+5q^{40}-6q^{36}+12q^{34}-17q^{32}+21q^{30}-24q^{28}+24q^{26}-24q^{24}+19q^{22}-14q^{20}+8q^{18}-q^{16}-3q^{14}+10q^{12}-11q^{10}+14q^{8}-12q^{6}+13q^{4}-9q^{2}+7-5q^{-2}+2q^{-4}-q^{-6}$
1,0	$q^{100}-2q^{96}-2q^{94}+q^{92}+5q^{90}+2q^{88}-5q^{86}-6q^{84}+q^{82}+9q^{80}+5q^{78}-6q^{76}-9q^{74}+10q^{70}+5q^{68}-8q^{66}-10q^{64}+3q^{62}+11q^{60}-11q^{56}-4q^{54}+9q^{52}+7q^{50}-5q^{48}-7q^{46}+4q^{44}+7q^{42}-3q^{40}-8q^{38}+2q^{36}+8q^{34}-q^{32}-12q^{30}-6q^{28}+10q^{26}+11q^{24}-3q^{22}-13q^{20}+14q^{16}+10q^{14}-7q^{12}-10q^{10}+3q^{8}+11q^{6}+q^{4}-7q^{2}-4+4q^{-2}+3q^{-4}-2q^{-6}-2q^{-8}+q^{-12}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{86}-2q^{84}+q^{82}-2q^{80}+5q^{78}-5q^{76}+4q^{74}-6q^{72}+9q^{70}-7q^{68}+7q^{66}-9q^{64}+9q^{62}-7q^{60}+6q^{58}-6q^{56}+2q^{54}-5q^{50}+7q^{48}-12q^{46}+13q^{44}-16q^{42}+18q^{40}-19q^{38}+20q^{36}-18q^{34}+16q^{32}-15q^{30}+11q^{28}-7q^{26}+4q^{24}-q^{22}-q^{20}+9q^{18}-5q^{16}+11q^{14}-9q^{12}+13q^{10}-10q^{8}+9q^{6}-9q^{4}+7q^{2}-5+3q^{-2}-2q^{-4}+q^{-6}$

G2 Invariants.

Weight

Invariant

1,0

$q^{148}-2q^{146}+3q^{144}-4q^{142}+2q^{140}-q^{138}-2q^{136}+8q^{134}-11q^{132}+14q^{130}-12q^{128}+6q^{126}+2q^{124}-11q^{122}+20q^{120}-24q^{118}+21q^{116}-14q^{114}+q^{112}+9q^{110}-16q^{108}+22q^{106}-24q^{104}+21q^{102}-17q^{100}+3q^{98}+12q^{96}-26q^{94}+33q^{92}-29q^{90}+14q^{88}+9q^{86}-28q^{84}+33q^{82}-18q^{80}-7q^{78}+34q^{76}-45q^{74}+30q^{72}+5q^{70}-40q^{68}+64q^{66}-65q^{64}+40q^{62}-2q^{60}-38q^{58}+62q^{56}-69q^{54}+51q^{52}-22q^{50}-13q^{48}+37q^{46}-47q^{44}+43q^{42}-20q^{40}-10q^{38}+33q^{36}-39q^{34}+27q^{32}+7q^{30}-38q^{28}+60q^{26}-49q^{24}+21q^{22}+22q^{20}-54q^{18}+68q^{16}-53q^{14}+23q^{12}+9q^{10}-34q^{8}+42q^{6}-33q^{4}+18q^{2}-2-8q^{-2}+9q^{-4}-9q^{-6}+5q^{-8}-2q^{-10}+q^{-12}$

.

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 85"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{4}-4t^{3}+8t^{2}-10t+11-10t^{-1}+8t^{-2}-4t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+4z^{6}+4z^{4}+2z^{2}+1$

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]=

{ 57, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q+3-4q^{-1}+7q^{-2}-8q^{-3}+9q^{-4}-9q^{-5}+7q^{-6}-5q^{-7}+3q^{-8}-q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{4}z^{8}-a^{6}z^{6}+6a^{4}z^{6}-a^{2}z^{6}-4a^{6}z^{4}+12a^{4}z^{4}-4a^{2}z^{4}-4a^{6}z^{2}+9a^{4}z^{2}-3a^{2}z^{2}-a^{6}+a^{4}+a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}+3z^{4}a^{10}-z^{2}a^{10}+5z^{5}a^{9}-4z^{3}a^{9}+za^{9}+6z^{6}a^{8}-7z^{4}a^{8}+z^{2}a^{8}+6z^{7}a^{7}-10z^{5}a^{7}+2z^{3}a^{7}+5z^{8}a^{6}-12z^{6}a^{6}+8z^{4}a^{6}-5z^{2}a^{6}+a^{6}+2z^{9}a^{5}-15z^{5}a^{5}+14z^{3}a^{5}-2za^{5}+8z^{8}a^{4}-32z^{6}a^{4}+37z^{4}a^{4}-14z^{2}a^{4}+a^{4}+2z^{9}a^{3}-5z^{7}a^{3}-4z^{5}a^{3}+11z^{3}a^{3}-2za^{3}+3z^{8}a^{2}-14z^{6}a^{2}+19z^{4}a^{2}-7z^{2}a^{2}-a^{2}+z^{7}a-4z^{5}a+4z^{3}a-za$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(2, -3)

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 $8$ $-24$ $32$ ${\frac {172}{3}}$ $-{\frac {52}{3}}$ $-192$ $-208$ $32$ $8$ ${\frac {256}{3}}$ $288$ ${\frac {1376}{3}}$ $-{\frac {416}{3}}$ ${\frac {12631}{15}}$ ${\frac {316}{15}}$ $-{\frac {1436}{45}}$ $-{\frac {55}{9}}$ $-{\frac {569}{15}}$

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 $s=$-4 is the signature of 10 85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                     1 -1 1                   2   2 -1                 2 1   -1 -3               5 2     3 -5             4 3       -1 -7           5 4         1 -9         4 4           0 -11       3 5             -2 -13     2 4               2 -15   1 3                 -2 -17   2                   2 -19 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{5}-3q^{4}-q^{3}+10q^{2}-8q-11+25q^{-1}-5q^{-2}-30q^{-3}+35q^{-4}+6q^{-5}-47q^{-6}+36q^{-7}+20q^{-8}-57q^{-9}+29q^{-10}+31q^{-11}-54q^{-12}+17q^{-13}+32q^{-14}-40q^{-15}+9q^{-16}+21q^{-17}-24q^{-18}+8q^{-19}+8q^{-20}-12q^{-21}+5q^{-22}+2q^{-23}-3q^{-24}+q^{-25}$
3	$-q^{12}+3q^{11}+q^{10}-5q^{9}-8q^{8}+8q^{7}+20q^{6}-7q^{5}-34q^{4}-6q^{3}+50q^{2}+26q-54-56q^{-1}+53q^{-2}+77q^{-3}-27q^{-4}-104q^{-5}+10q^{-6}+101q^{-7}+28q^{-8}-104q^{-9}-45q^{-10}+81q^{-11}+73q^{-12}-69q^{-13}-79q^{-14}+38q^{-15}+95q^{-16}-17q^{-17}-98q^{-18}-15q^{-19}+102q^{-20}+43q^{-21}-97q^{-22}-67q^{-23}+80q^{-24}+88q^{-25}-61q^{-26}-88q^{-27}+33q^{-28}+77q^{-29}-11q^{-30}-53q^{-31}-5q^{-32}+28q^{-33}+9q^{-34}-6q^{-35}-7q^{-36}-8q^{-37}+5q^{-38}+10q^{-39}+q^{-40}-11q^{-41}-q^{-42}+7q^{-43}-2q^{-45}-2q^{-46}+3q^{-47}-q^{-48}$
4	$q^{22}-3q^{21}-q^{20}+5q^{19}+3q^{18}+8q^{17}-18q^{16}-18q^{15}+5q^{14}+16q^{13}+60q^{12}-19q^{11}-62q^{10}-49q^{9}-20q^{8}+151q^{7}+63q^{6}-36q^{5}-125q^{4}-181q^{3}+147q^{2}+156q+127-58q^{-1}-340q^{-2}-5q^{-3}+69q^{-4}+262q^{-5}+170q^{-6}-295q^{-7}-105q^{-8}-184q^{-9}+169q^{-10}+336q^{-11}-76q^{-12}+29q^{-13}-374q^{-14}-104q^{-15}+271q^{-16}+104q^{-17}+337q^{-18}-358q^{-19}-366q^{-20}+37q^{-21}+142q^{-22}+644q^{-23}-205q^{-24}-527q^{-25}-220q^{-26}+102q^{-27}+884q^{-28}-28q^{-29}-643q^{-30}-461q^{-31}+60q^{-32}+1091q^{-33}+171q^{-34}-724q^{-35}-716q^{-36}-48q^{-37}+1218q^{-38}+437q^{-39}-644q^{-40}-898q^{-41}-282q^{-42}+1098q^{-43}+645q^{-44}-338q^{-45}-815q^{-46}-499q^{-47}+712q^{-48}+603q^{-49}-13q^{-50}-485q^{-51}-494q^{-52}+309q^{-53}+350q^{-54}+119q^{-55}-162q^{-56}-319q^{-57}+91q^{-58}+119q^{-59}+93q^{-60}-8q^{-61}-147q^{-62}+24q^{-63}+13q^{-64}+40q^{-65}+21q^{-66}-52q^{-67}+11q^{-68}-7q^{-69}+10q^{-70}+10q^{-71}-14q^{-72}+5q^{-73}-3q^{-74}+2q^{-75}+2q^{-76}-3q^{-77}+q^{-78}$
5	$-q^{35}+3q^{34}+q^{33}-5q^{32}-3q^{31}-3q^{30}+2q^{29}+16q^{28}+21q^{27}-7q^{26}-29q^{25}-41q^{24}-28q^{23}+29q^{22}+93q^{21}+89q^{20}-3q^{19}-112q^{18}-174q^{17}-114q^{16}+83q^{15}+265q^{14}+259q^{13}+43q^{12}-242q^{11}-419q^{10}-284q^{9}+121q^{8}+472q^{7}+509q^{6}+169q^{5}-344q^{4}-664q^{3}-481q^{2}+44q+568+728q^{-1}+358q^{-2}-273q^{-3}-697q^{-4}-698q^{-5}-235q^{-6}+428q^{-7}+810q^{-8}+669q^{-9}+162q^{-10}-574q^{-11}-1022q^{-12}-775q^{-13}+44q^{-14}+947q^{-15}+1377q^{-16}+769q^{-17}-627q^{-18}-1706q^{-19}-1571q^{-20}-121q^{-21}+1756q^{-22}+2364q^{-23}+955q^{-24}-1455q^{-25}-2896q^{-26}-1944q^{-27}+934q^{-28}+3241q^{-29}+2784q^{-30}-223q^{-31}-3309q^{-32}-3582q^{-33}-489q^{-34}+3255q^{-35}+4141q^{-36}+1194q^{-37}-3072q^{-38}-4641q^{-39}-1784q^{-40}+2933q^{-41}+5000q^{-42}+2300q^{-43}-2798q^{-44}-5391q^{-45}-2771q^{-46}+2733q^{-47}+5799q^{-48}+3259q^{-49}-2644q^{-50}-6212q^{-51}-3845q^{-52}+2423q^{-53}+6596q^{-54}+4500q^{-55}-2012q^{-56}-6748q^{-57}-5158q^{-58}+1309q^{-59}+6590q^{-60}+5694q^{-61}-452q^{-62}-6008q^{-63}-5906q^{-64}-476q^{-65}+5068q^{-66}+5716q^{-67}+1249q^{-68}-3906q^{-69}-5120q^{-70}-1731q^{-71}+2745q^{-72}+4217q^{-73}+1863q^{-74}-1721q^{-75}-3230q^{-76}-1709q^{-77}+979q^{-78}+2297q^{-79}+1377q^{-80}-499q^{-81}-1517q^{-82}-1018q^{-83}+217q^{-84}+965q^{-85}+698q^{-86}-97q^{-87}-572q^{-88}-436q^{-89}+11q^{-90}+330q^{-91}+276q^{-92}+5q^{-93}-184q^{-94}-150q^{-95}-12q^{-96}+84q^{-97}+83q^{-98}+17q^{-99}-44q^{-100}-44q^{-101}+2q^{-102}+14q^{-103}+11q^{-104}+10q^{-105}-10q^{-106}-9q^{-107}+5q^{-108}+2q^{-109}-2q^{-110}+3q^{-111}-2q^{-112}-2q^{-113}+3q^{-114}-q^{-115}$
6	$q^{51}-3q^{50}-q^{49}+5q^{48}+3q^{47}+3q^{46}-7q^{45}-19q^{43}-19q^{42}+19q^{41}+30q^{40}+47q^{39}+11q^{38}+14q^{37}-82q^{36}-132q^{35}-65q^{34}+12q^{33}+153q^{32}+172q^{31}+271q^{30}+10q^{29}-263q^{28}-386q^{27}-392q^{26}-101q^{25}+163q^{24}+779q^{23}+674q^{22}+295q^{21}-280q^{20}-857q^{19}-1017q^{18}-903q^{17}+386q^{16}+1025q^{15}+1455q^{14}+1100q^{13}+166q^{12}-999q^{11}-2164q^{10}-1355q^{9}-624q^{8}+912q^{7}+1894q^{6}+2239q^{5}+1316q^{4}-838q^{3}-1514q^{2}-2567q-1844-637q^{-1}+1515q^{-2}+2761q^{-3}+2226q^{-4}+2025q^{-5}-477q^{-6}-2275q^{-7}-3943q^{-8}-2954q^{-9}-863q^{-10}+1443q^{-11}+4990q^{-12}+5003q^{-13}+3108q^{-14}-1916q^{-15}-5439q^{-16}-7154q^{-17}-5509q^{-18}+1360q^{-19}+7112q^{-20}+10282q^{-21}+6332q^{-22}-531q^{-23}-8862q^{-24}-13337q^{-25}-8283q^{-26}+1388q^{-27}+12063q^{-28}+14700q^{-29}+9925q^{-30}-2652q^{-31}-15350q^{-32}-17506q^{-33}-9413q^{-34}+6216q^{-35}+17288q^{-36}+19631q^{-37}+8147q^{-38}-10274q^{-39}-21296q^{-40}-19434q^{-41}-3723q^{-42}+13562q^{-43}+24478q^{-44}+18265q^{-45}-1619q^{-46}-19621q^{-47}-25325q^{-48}-13112q^{-49}+6851q^{-50}+24738q^{-51}+24923q^{-52}+6503q^{-53}-15552q^{-54}-27477q^{-55}-19579q^{-56}+833q^{-57}+23230q^{-58}+28554q^{-59}+12025q^{-60}-12374q^{-61}-28506q^{-62}-23672q^{-63}-2804q^{-64}+22847q^{-65}+31668q^{-66}+15869q^{-67}-11121q^{-68}-30875q^{-69}-28059q^{-70}-5745q^{-71}+23709q^{-72}+36240q^{-73}+21095q^{-74}-8973q^{-75}-33504q^{-76}-34205q^{-77}-11591q^{-78}+21746q^{-79}+39864q^{-80}+28525q^{-81}-2002q^{-82}-31212q^{-83}-38372q^{-84}-20202q^{-85}+13247q^{-86}+36662q^{-87}+33096q^{-88}+8206q^{-89}-21136q^{-90}-34513q^{-91}-25271q^{-92}+1577q^{-93}+25208q^{-94}+29091q^{-95}+14341q^{-96}-8417q^{-97}-22898q^{-98}-21870q^{-99}-5524q^{-100}+12090q^{-101}+18449q^{-102}+12608q^{-103}-603q^{-104}-10862q^{-105}-13240q^{-106}-5608q^{-107}+4063q^{-108}+8467q^{-109}+6986q^{-110}+1100q^{-111}-3872q^{-112}-5896q^{-113}-2767q^{-114}+1324q^{-115}+3056q^{-116}+2674q^{-117}+411q^{-118}-1296q^{-119}-2219q^{-120}-792q^{-121}+705q^{-122}+1051q^{-123}+842q^{-124}-56q^{-125}-505q^{-126}-861q^{-127}-148q^{-128}+389q^{-129}+372q^{-130}+286q^{-131}-76q^{-132}-173q^{-133}-340q^{-134}-25q^{-135}+146q^{-136}+101q^{-137}+100q^{-138}-27q^{-139}-31q^{-140}-111q^{-141}+4q^{-142}+39q^{-143}+9q^{-144}+27q^{-145}-12q^{-146}+3q^{-147}-26q^{-148}+7q^{-149}+9q^{-150}-6q^{-151}+7q^{-152}-5q^{-153}+2q^{-154}-3q^{-155}+2q^{-156}+2q^{-157}-3q^{-158}+q^{-159}$
7	$-q^{70}+3q^{69}+q^{68}-5q^{67}-3q^{66}-3q^{65}+7q^{64}+5q^{63}+3q^{62}+17q^{61}+7q^{60}-20q^{59}-35q^{58}-50q^{57}-13q^{56}+25q^{55}+38q^{54}+116q^{53}+116q^{52}+52q^{51}-56q^{50}-230q^{49}-261q^{48}-203q^{47}-114q^{46}+197q^{45}+457q^{44}+582q^{43}+535q^{42}+16q^{41}-471q^{40}-865q^{39}-1151q^{38}-777q^{37}-84q^{36}+861q^{35}+1815q^{34}+1801q^{33}+1174q^{32}+25q^{31}-1678q^{30}-2620q^{29}-2858q^{28}-1894q^{27}+485q^{26}+2471q^{25}+3902q^{24}+4055q^{23}+2110q^{22}-453q^{21}-3466q^{20}-5587q^{19}-4903q^{18}-2842q^{17}+747q^{16}+4617q^{15}+6277q^{14}+6402q^{13}+3683q^{12}-1023q^{11}-4680q^{10}-7562q^{9}-7618q^{8}-4450q^{7}-526q^{6}+4785q^{5}+8514q^{4}+8854q^{3}+7412q^{2}+2202q-4088-8969q^{-1}-12639q^{-2}-11119q^{-3}-5366q^{-4}+2518q^{-5}+12335q^{-6}+17733q^{-7}+16934q^{-8}+10085q^{-9}-3917q^{-10}-17504q^{-11}-25955q^{-12}-25579q^{-13}-11961q^{-14}+7902q^{-15}+27206q^{-16}+38331q^{-17}+31816q^{-18}+10973q^{-19}-17614q^{-20}-43212q^{-21}-49882q^{-22}-35180q^{-23}-2576q^{-24}+36261q^{-25}+60095q^{-26}+59166q^{-27}+30374q^{-28}-17217q^{-29}-58822q^{-30}-76777q^{-31}-59759q^{-32}-11146q^{-33}+44334q^{-34}+83425q^{-35}+85352q^{-36}+44119q^{-37}-19072q^{-38}-77567q^{-39}-101889q^{-40}-75846q^{-41}-13346q^{-42}+59692q^{-43}+107305q^{-44}+102017q^{-45}+47803q^{-46}-33171q^{-47}-101364q^{-48}-119374q^{-49}-80075q^{-50}+1679q^{-51}+86002q^{-52}+127346q^{-53}+107015q^{-54}+30731q^{-55}-64144q^{-56}-126624q^{-57}-127154q^{-58}-61142q^{-59}+39081q^{-60}+119166q^{-61}+140265q^{-62}+87514q^{-63}-13413q^{-64}-107274q^{-65}-147457q^{-66}-108925q^{-67}-10396q^{-68}+93350q^{-69}+149735q^{-70}+125223q^{-71}+31168q^{-72}-79427q^{-73}-149185q^{-74}-136940q^{-75}-47606q^{-76}+67430q^{-77}+147319q^{-78}+144941q^{-79}+59757q^{-80}-58609q^{-81}-146290q^{-82}-151040q^{-83}-68037q^{-84}+53918q^{-85}+147755q^{-86}+157082q^{-87}+74037q^{-88}-52635q^{-89}-152675q^{-90}-165536q^{-91}-80528q^{-92}+53128q^{-93}+160931q^{-94}+177729q^{-95}+90155q^{-96}-51946q^{-97}-169922q^{-98}-193615q^{-99}-105553q^{-100}+45129q^{-101}+176076q^{-102}+210911q^{-103}+126678q^{-104}-29931q^{-105}-174293q^{-106}-224712q^{-107}-151314q^{-108}+5176q^{-109}+161036q^{-110}+230137q^{-111}+174279q^{-112}+26051q^{-113}-135260q^{-114}-222260q^{-115}-189501q^{-116}-58771q^{-117}+99457q^{-118}+200049q^{-119}+192166q^{-120}+86054q^{-121}-59395q^{-122}-165784q^{-123}-180156q^{-124}-102449q^{-125}+21856q^{-126}+124846q^{-127}+155432q^{-128}+105699q^{-129}+7300q^{-130}-84200q^{-131}-123006q^{-132}-96593q^{-133}-24998q^{-134}+49352q^{-135}+88772q^{-136}+79299q^{-137}+31753q^{-138}-23758q^{-139}-58207q^{-140}-58780q^{-141}-30032q^{-142}+7861q^{-143}+34258q^{-144}+39237q^{-145}+23717q^{-146}+277q^{-147}-17836q^{-148}-23622q^{-149}-16148q^{-150}-2941q^{-151}+8017q^{-152}+12527q^{-153}+9393q^{-154}+2875q^{-155}-2800q^{-156}-5722q^{-157}-4630q^{-158}-1814q^{-159}+661q^{-160}+2141q^{-161}+1611q^{-162}+655q^{-163}+75q^{-164}-423q^{-165}-165q^{-166}-24q^{-167}-122q^{-168}-57q^{-169}-378q^{-170}-366q^{-171}+17q^{-172}+183q^{-173}+479q^{-174}+383q^{-175}+13q^{-176}-99q^{-177}-332q^{-178}-295q^{-179}-78q^{-180}+20q^{-181}+237q^{-182}+211q^{-183}+25q^{-184}-14q^{-185}-106q^{-186}-96q^{-187}-20q^{-188}-27q^{-189}+63q^{-190}+62q^{-191}-2q^{-192}+6q^{-193}-29q^{-194}-17q^{-195}+7q^{-196}-11q^{-197}+12q^{-198}+9q^{-199}-6q^{-200}+6q^{-201}-6q^{-202}-4q^{-203}+5q^{-204}-2q^{-205}+3q^{-206}-2q^{-207}-2q^{-208}+3q^{-209}-q^{-210}$

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 85]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 6, 17, 5], X[18, 11, 19, 12], X[14, 7, 15, 8], X[8, 3, 9, 4], X[4, 9, 5, 10], X[20, 13, 1, 14], X[10, 17, 11, 18], X[12, 19, 13, 20], X[2, 16, 3, 15]]
In[3]:=	GaussCode[Knot[10, 85]]
Out[3]=	GaussCode[1, -10, 5, -6, 2, -1, 4, -5, 6, -8, 3, -9, 7, -4, 10, -2, 8, -3, 9, -7]
In[4]:=	DTCode[Knot[10, 85]]
Out[4]=	DTCode[6, 8, 16, 14, 4, 18, 20, 2, 10, 12]
In[5]:=	br = BR[Knot[10, 85]]
Out[5]=	BR[3, {-1, -1, -1, -1, 2, -1, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 85]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 85]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 85]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 85]][t]
Out[10]=	-4 4 8 10 2 3 4 11 + t - -- + -- - -- - 10 t + 8 t - 4 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 85]][z]
Out[11]=	2 4 6 8 1 + 2 z + 4 z + 4 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 85]}
In[13]:=	{KnotDet[Knot[10, 85]], KnotSignature[Knot[10, 85]]}
Out[13]=	{57, -4}
In[14]:=	Jones[Knot[10, 85]][q]
Out[14]=	-9 3 5 7 9 9 8 7 4 3 - q + -- - -- + -- - -- + -- - -- + -- - - - q 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 85]}
In[16]:=	A2Invariant[Knot[10, 85]][q]
Out[16]=	-26 -24 -22 -20 -16 -14 3 2 2 2 2 1 - q + q - q + q - q + q - --- + --- + -- + -- - q 12 10 6 4 q q q q
In[17]:=	HOMFLYPT[Knot[10, 85]][a, z]
Out[17]=	2 4 6 2 2 4 2 6 2 2 4 4 4 a + a - a - 3 a z + 9 a z - 4 a z - 4 a z + 12 a z - 6 4 2 6 4 6 6 6 4 8 4 a z - a z + 6 a z - a z + a z
In[18]:=	Kauffman[Knot[10, 85]][a, z]
Out[18]=	2 4 6 3 5 9 2 2 4 2 -a + a + a - a z - 2 a z - 2 a z + a z - 7 a z - 14 a z - 6 2 8 2 10 2 3 3 3 5 3 7 3 5 a z + a z - a z + 4 a z + 11 a z + 14 a z + 2 a z - 9 3 11 3 2 4 4 4 6 4 8 4 4 a z + a z + 19 a z + 37 a z + 8 a z - 7 a z + 10 4 5 3 5 5 5 7 5 9 5 3 a z - 4 a z - 4 a z - 15 a z - 10 a z + 5 a z - 2 6 4 6 6 6 8 6 7 3 7 7 7 14 a z - 32 a z - 12 a z + 6 a z + a z - 5 a z + 6 a z + 2 8 4 8 6 8 3 9 5 9 3 a z + 8 a z + 5 a z + 2 a z + 2 a z
In[19]:=	{Vassiliev[2][Knot[10, 85]], Vassiliev[3][Knot[10, 85]]}
Out[19]=	{2, -3}
In[20]:=	Kh[Knot[10, 85]][q, t]
Out[20]=	3 5 1 2 1 3 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 q q q t q t q t q t q t q t 3 5 4 4 5 4 4 2 t 2 t ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + --- + 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q 2 t 2 3 3 -- + 2 q t + q t q
In[21]:=	ColouredJones[Knot[10, 85], 2][q]
Out[21]=	-25 3 2 5 12 8 8 24 21 9 -11 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - 24 23 22 21 20 19 18 17 16 q q q q q q q q q 40 32 17 54 31 29 57 20 36 47 6 35 --- + --- + --- - --- + --- + --- - -- + -- + -- - -- + -- + -- - 15 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q q 30 5 25 2 3 4 5 -- - -- + -- - 8 q + 10 q - q - 3 q + q 3 2 q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

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