10 100

10_99

10_101

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10 100 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X18,6,19,5 X20,13,1,14 X14,7,15,8 X10,3,11,4 X16,9,17,10 X4,11,5,12 X8,15,9,16 X12,19,13,20 X2,18,3,17
Gauss code	1, -10, 5, -7, 2, -1, 4, -8, 6, -5, 7, -9, 3, -4, 8, -6, 10, -2, 9, -3
Dowker-Thistlethwaite code	6 10 18 14 16 4 20 8 2 12
Conway Notation	[3:2:2]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-12][0] Hyperbolic Volume 12.8109 A-Polynomial See Data:10 100/A-polynomial

[edit Notes for 10 100'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 10 100's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+9t^{2}-12t+13-12t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+5z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 65, -4 }
Jones polynomial	$-q+3-5q^{-1}+8q^{-2}-9q^{-3}+11q^{-4}-10q^{-5}+8q^{-6}-6q^{-7}+3q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}z^{8}-a^{6}z^{6}+6a^{4}z^{6}-a^{2}z^{6}-4a^{6}z^{4}+13a^{4}z^{4}-4a^{2}z^{4}-5a^{6}z^{2}+13a^{4}z^{2}-4a^{2}z^{2}-3a^{6}+5a^{4}-a^{2}$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}+3z^{4}a^{10}+6z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+8z^{6}a^{8}-11z^{4}a^{8}+4z^{2}a^{8}+8z^{7}a^{7}-14z^{5}a^{7}+5z^{3}a^{7}-2za^{7}+6z^{8}a^{6}-12z^{6}a^{6}+5z^{4}a^{6}-6z^{2}a^{6}+3a^{6}+2z^{9}a^{5}+4z^{7}a^{5}-27z^{5}a^{5}+26z^{3}a^{5}-8za^{5}+9z^{8}a^{4}-33z^{6}a^{4}+36z^{4}a^{4}-17z^{2}a^{4}+5a^{4}+2z^{9}a^{3}-3z^{7}a^{3}-11z^{5}a^{3}+20z^{3}a^{3}-6za^{3}+3z^{8}a^{2}-13z^{6}a^{2}+17z^{4}a^{2}-7z^{2}a^{2}+a^{2}+z^{7}a-4z^{5}a+5z^{3}a-2za$
The A2 invariant	$-q^{26}+q^{24}-2q^{22}-q^{18}-q^{16}+3q^{14}-q^{12}+4q^{10}+q^{6}+q^{4}-q^{2}+1-q^{-2}$
The G2 invariant	$q^{148}-2q^{146}+3q^{144}-4q^{142}+3q^{140}-2q^{138}-q^{136}+8q^{134}-12q^{132}+16q^{130}-17q^{128}+11q^{126}-4q^{124}-9q^{122}+24q^{120}-33q^{118}+35q^{116}-28q^{114}+15q^{112}+3q^{110}-20q^{108}+39q^{106}-51q^{104}+48q^{102}-36q^{100}+7q^{98}+24q^{96}-52q^{94}+65q^{92}-53q^{90}+17q^{88}+22q^{86}-59q^{84}+59q^{82}-30q^{80}-23q^{78}+66q^{76}-82q^{74}+58q^{72}+2q^{70}-67q^{68}+112q^{66}-116q^{64}+77q^{62}-10q^{60}-57q^{58}+107q^{56}-111q^{54}+88q^{52}-35q^{50}-20q^{48}+66q^{46}-81q^{44}+66q^{42}-23q^{40}-26q^{38}+64q^{36}-68q^{34}+41q^{32}+16q^{30}-68q^{28}+98q^{26}-86q^{24}+34q^{22}+32q^{20}-84q^{18}+103q^{16}-81q^{14}+36q^{12}+12q^{10}-49q^{8}+59q^{6}-47q^{4}+24q^{2}-2-12q^{-2}+13q^{-4}-11q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$

Further Quantum Invariants

Further quantum knot invariants for 10_100.

A1 Invariants.

Weight	Invariant
1	$-q^{19}+2q^{17}-3q^{15}+2q^{13}-2q^{11}+q^{9}+2q^{7}-q^{5}+3q^{3}-2q+2q^{-1}-q^{-3}$
2	$q^{52}-2q^{50}+q^{48}+4q^{46}-7q^{44}+3q^{42}+5q^{40}-13q^{38}+8q^{36}+9q^{34}-16q^{32}+q^{30}+13q^{28}-7q^{26}-9q^{24}+9q^{22}+5q^{20}-11q^{18}-q^{16}+14q^{14}-7q^{12}-9q^{10}+17q^{8}-15q^{4}+11q^{2}+7-12q^{-2}+q^{-4}+7q^{-6}-3q^{-8}-2q^{-10}+q^{-12}$
3	$-q^{99}+2q^{97}-q^{95}-2q^{93}+q^{91}+3q^{89}-4q^{85}+3q^{83}-q^{81}-7q^{79}+11q^{77}+16q^{75}-18q^{73}-37q^{71}+24q^{69}+58q^{67}-16q^{65}-76q^{63}-9q^{61}+80q^{59}+36q^{57}-59q^{55}-62q^{53}+28q^{51}+72q^{49}+12q^{47}-72q^{45}-38q^{43}+55q^{41}+59q^{39}-43q^{37}-68q^{35}+26q^{33}+70q^{31}-13q^{29}-73q^{27}-2q^{25}+73q^{23}+26q^{21}-70q^{19}-46q^{17}+57q^{15}+69q^{13}-31q^{11}-78q^{9}-q^{7}+78q^{5}+30q^{3}-54q-50q^{-1}+25q^{-3}+51q^{-5}+q^{-7}-37q^{-9}-16q^{-11}+17q^{-13}+18q^{-15}-4q^{-17}-10q^{-19}-2q^{-21}+3q^{-23}+2q^{-25}-q^{-27}$
4	$q^{160}-2q^{158}+q^{156}+2q^{154}-3q^{152}+3q^{150}-6q^{148}+2q^{146}+5q^{144}-4q^{142}+14q^{140}-19q^{138}-14q^{136}+4q^{134}+19q^{132}+64q^{130}-35q^{128}-93q^{126}-57q^{124}+75q^{122}+225q^{120}+21q^{118}-245q^{116}-278q^{114}+43q^{112}+474q^{110}+296q^{108}-253q^{106}-586q^{104}-271q^{102}+477q^{100}+631q^{98}+116q^{96}-542q^{94}-632q^{92}+28q^{90}+547q^{88}+518q^{86}-43q^{84}-546q^{82}-413q^{80}+58q^{78}+497q^{76}+395q^{74}-145q^{72}-472q^{70}-300q^{68}+242q^{66}+469q^{64}+109q^{62}-362q^{60}-367q^{58}+112q^{56}+432q^{54}+180q^{52}-339q^{50}-399q^{48}+57q^{46}+463q^{44}+313q^{42}-290q^{40}-512q^{38}-152q^{36}+407q^{34}+536q^{32}-12q^{30}-480q^{28}-467q^{26}+67q^{24}+561q^{22}+378q^{20}-114q^{18}-527q^{16}-368q^{14}+190q^{12}+450q^{10}+330q^{8}-161q^{6}-435q^{4}-232q^{2}+102+369q^{-2}+206q^{-4}-105q^{-6}-245q^{-8}-191q^{-10}+76q^{-12}+181q^{-14}+116q^{-16}-16q^{-18}-132q^{-20}-71q^{-22}+7q^{-24}+60q^{-26}+55q^{-28}-9q^{-30}-24q^{-32}-23q^{-34}-3q^{-36}+13q^{-38}+5q^{-40}+2q^{-42}-3q^{-44}-2q^{-46}+q^{-48}$
5	$-q^{235}+2q^{233}-q^{231}-2q^{229}+3q^{227}-q^{225}+4q^{221}-3q^{219}-7q^{217}+6q^{213}+11q^{211}+13q^{209}-12q^{207}-43q^{205}-37q^{203}+33q^{201}+102q^{199}+83q^{197}-46q^{195}-219q^{193}-212q^{191}+75q^{189}+430q^{187}+437q^{185}-71q^{183}-733q^{181}-856q^{179}-57q^{177}+1139q^{175}+1534q^{173}+394q^{171}-1517q^{169}-2419q^{167}-1129q^{165}+1636q^{163}+3441q^{161}+2280q^{159}-1275q^{157}-4213q^{155}-3700q^{153}+200q^{151}+4353q^{149}+5050q^{147}+1432q^{145}-3587q^{143}-5764q^{141}-3220q^{139}+1909q^{137}+5472q^{135}+4635q^{133}+211q^{131}-4147q^{129}-5115q^{127}-2212q^{125}+2127q^{123}+4602q^{121}+3517q^{119}-39q^{117}-3323q^{115}-3943q^{113}-1567q^{111}+1816q^{109}+3594q^{107}+2458q^{105}-569q^{103}-2919q^{101}-2637q^{99}-116q^{97}+2274q^{95}+2430q^{93}+299q^{91}-2001q^{89}-2180q^{87}-114q^{85}+2073q^{83}+2148q^{81}-68q^{79}-2417q^{77}-2472q^{75}+35q^{73}+2786q^{71}+3059q^{69}+395q^{67}-2896q^{65}-3777q^{63}-1250q^{61}+2583q^{59}+4333q^{57}+2370q^{55}-1696q^{53}-4460q^{51}-3550q^{49}+316q^{47}+3955q^{45}+4401q^{43}+1337q^{41}-2732q^{39}-4609q^{37}-2884q^{35}+972q^{33}+3967q^{31}+3895q^{29}+934q^{27}-2536q^{25}-3984q^{23}-2509q^{21}+662q^{19}+3157q^{17}+3260q^{15}+1089q^{13}-1621q^{11}-3008q^{9}-2237q^{7}-29q^{5}+1969q^{3}+2440q+1255q^{-1}-592q^{-3}-1834q^{-5}-1729q^{-7}-531q^{-9}+828q^{-11}+1455q^{-13}+1060q^{-15}+89q^{-17}-774q^{-19}-1005q^{-21}-572q^{-23}+124q^{-25}+589q^{-27}+595q^{-29}+259q^{-31}-160q^{-33}-380q^{-35}-312q^{-37}-80q^{-39}+123q^{-41}+196q^{-43}+141q^{-45}+13q^{-47}-75q^{-49}-84q^{-51}-43q^{-53}+2q^{-55}+30q^{-57}+31q^{-59}+8q^{-61}-6q^{-63}-8q^{-65}-5q^{-67}-2q^{-69}+3q^{-71}+2q^{-73}-q^{-75}$
6	$q^{324}-2q^{322}+q^{320}+2q^{318}-3q^{316}+q^{314}-2q^{312}+2q^{310}-3q^{308}+5q^{306}+11q^{304}-17q^{302}-7q^{300}-4q^{298}+6q^{296}+13q^{294}+35q^{292}+29q^{290}-71q^{288}-80q^{286}-42q^{284}+48q^{282}+142q^{280}+196q^{278}+58q^{276}-312q^{274}-428q^{272}-228q^{270}+268q^{268}+756q^{266}+834q^{264}+102q^{262}-1223q^{260}-1797q^{258}-1011q^{256}+978q^{254}+2952q^{252}+3195q^{250}+648q^{248}-3626q^{246}-6114q^{244}-4334q^{242}+1537q^{240}+8114q^{238}+10269q^{236}+4754q^{234}-6257q^{232}-15068q^{230}-14515q^{228}-3008q^{226}+13371q^{224}+23406q^{222}+18261q^{220}-1169q^{218}-22733q^{216}-31591q^{214}-19984q^{212}+6895q^{210}+32245q^{208}+38539q^{206}+19620q^{204}-13403q^{202}-39833q^{200}-42397q^{198}-18056q^{196}+18506q^{194}+44418q^{192}+43076q^{190}+15682q^{188}-20956q^{186}-44742q^{184}-41414q^{182}-13850q^{180}+20803q^{178}+41655q^{176}+37876q^{174}+12847q^{172}-18062q^{170}-36839q^{168}-33796q^{166}-12064q^{164}+14534q^{162}+31365q^{160}+29468q^{158}+11216q^{156}-11970q^{154}-26525q^{152}-24577q^{150}-8839q^{148}+10877q^{146}+22143q^{144}+18940q^{142}+4305q^{140}-11305q^{138}-17551q^{136}-11529q^{134}+1751q^{132}+11894q^{130}+12056q^{128}+2561q^{126}-8290q^{124}-11518q^{122}-4645q^{120}+7030q^{118}+13532q^{116}+9179q^{114}-4082q^{112}-16057q^{110}-16761q^{108}-4271q^{106}+13370q^{104}+23163q^{102}+17274q^{100}-2151q^{98}-22058q^{96}-28251q^{94}-15167q^{92}+9324q^{90}+29120q^{88}+30840q^{86}+11821q^{84}-15989q^{82}-34611q^{80}-31303q^{78}-7901q^{76}+21178q^{74}+37823q^{72}+30855q^{70}+4582q^{68}-24862q^{66}-38978q^{64}-29693q^{62}-2558q^{60}+26144q^{58}+38872q^{56}+28552q^{54}+1666q^{52}-25236q^{50}-37234q^{48}-27629q^{46}-2573q^{44}+22889q^{42}+34537q^{40}+26467q^{38}+4665q^{36}-18893q^{34}-30874q^{32}-25532q^{30}-6956q^{28}+14042q^{26}+26109q^{24}+24115q^{22}+9452q^{20}-8826q^{18}-21131q^{16}-21622q^{14}-11272q^{12}+3679q^{10}+15769q^{8}+18588q^{6}+12113q^{4}+286q^{2}-10275-14854q^{-2}-11857q^{-4}-3231q^{-6}+5658q^{-8}+10894q^{-10}+10205q^{-12}+5009q^{-14}-1998q^{-16}-7148q^{-18}-8053q^{-20}-5262q^{-22}-392q^{-24}+3769q^{-26}+5760q^{-28}+4633q^{-30}+1598q^{-32}-1512q^{-34}-3457q^{-36}-3445q^{-38}-2010q^{-40}+212q^{-42}+1754q^{-44}+2204q^{-46}+1631q^{-48}+461q^{-50}-634q^{-52}-1264q^{-54}-1064q^{-56}-539q^{-58}+96q^{-60}+519q^{-62}+617q^{-64}+418q^{-66}+51q^{-68}-156q^{-70}-268q^{-72}-221q^{-74}-101q^{-76}+37q^{-78}+103q^{-80}+80q^{-82}+56q^{-84}+5q^{-86}-24q^{-88}-37q^{-90}-16q^{-92}+q^{-94}+q^{-96}+8q^{-98}+5q^{-100}+2q^{-102}-3q^{-104}-2q^{-106}+q^{-108}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}+q^{24}-2q^{22}-q^{18}-q^{16}+3q^{14}-q^{12}+4q^{10}+q^{6}+q^{4}-q^{2}+1-q^{-2}$
1,1	$q^{76}-4q^{74}+8q^{72}-12q^{70}+22q^{68}-36q^{66}+48q^{64}-60q^{62}+81q^{60}-102q^{58}+110q^{56}-116q^{54}+138q^{52}-154q^{50}+156q^{48}-174q^{46}+180q^{44}-170q^{42}+128q^{40}-52q^{38}-45q^{36}+176q^{34}-308q^{32}+416q^{30}-512q^{28}+552q^{26}-548q^{24}+494q^{22}-403q^{20}+292q^{18}-134q^{16}-10q^{14}+147q^{12}-248q^{10}+326q^{8}-348q^{6}+318q^{4}-266q^{2}+202-132q^{-2}+74q^{-4}-38q^{-6}+14q^{-8}-4q^{-10}+q^{-12}$
2,0	$q^{66}-q^{64}+q^{62}+2q^{60}-q^{58}+2q^{56}-q^{54}-2q^{52}-q^{50}+q^{48}-q^{46}-8q^{44}+5q^{40}-2q^{38}-6q^{36}+7q^{34}+4q^{32}-5q^{30}+2q^{26}-4q^{22}+5q^{20}+3q^{18}-3q^{16}+4q^{14}+7q^{12}-3q^{10}-3q^{8}+5q^{6}+3q^{4}-4q^{2}-2+3q^{-2}+q^{-4}-3q^{-6}-q^{-8}+q^{-10}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{62}-2q^{60}-q^{58}+6q^{56}-3q^{54}-6q^{52}+10q^{50}-q^{48}-10q^{46}+10q^{44}+3q^{42}-12q^{40}+7q^{38}+2q^{36}-12q^{34}-2q^{32}-q^{30}-q^{28}-6q^{26}+4q^{24}+13q^{22}-3q^{20}+3q^{18}+14q^{16}-5q^{14}-2q^{12}+9q^{10}-8q^{8}-3q^{6}+6q^{4}-6q^{2}+1+2q^{-2}-2q^{-4}+q^{-6}$
1,0,0	$-q^{33}+q^{31}-3q^{29}+q^{27}-4q^{25}+q^{23}-q^{21}+2q^{19}+2q^{17}+2q^{15}+3q^{13}+3q^{9}-2q^{7}+2q^{5}-2q^{3}+q-q^{-1}$
1,0,1	$q^{100}-4q^{98}+6q^{96}-2q^{94}-7q^{92}+18q^{90}-21q^{88}+6q^{86}+20q^{84}-36q^{82}+34q^{80}-7q^{78}-28q^{76}+43q^{74}-32q^{72}+9q^{70}+22q^{68}-41q^{66}+28q^{64}+11q^{62}-68q^{60}+107q^{58}-97q^{56}+21q^{54}+83q^{52}-169q^{50}+204q^{48}-177q^{46}+101q^{44}-18q^{42}-80q^{40}+121q^{38}-155q^{36}+139q^{34}-126q^{32}+123q^{30}-104q^{28}+106q^{26}-34q^{24}-35q^{22}+145q^{20}-203q^{18}+226q^{16}-176q^{14}+82q^{12}+15q^{10}-96q^{8}+131q^{6}-115q^{4}+69q^{2}-21-17q^{-2}+26q^{-4}-22q^{-6}+12q^{-8}-4q^{-10}+q^{-12}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{76}-q^{74}-2q^{72}+4q^{70}+2q^{68}-7q^{66}+2q^{64}+10q^{62}-4q^{60}-6q^{58}+10q^{56}+4q^{54}-8q^{52}-2q^{50}+7q^{48}-12q^{46}-17q^{44}-8q^{40}-20q^{38}+3q^{36}+15q^{34}-4q^{32}+10q^{30}+23q^{28}+13q^{26}+q^{24}+8q^{22}+7q^{20}-5q^{18}-7q^{16}-4q^{12}-6q^{10}+2q^{8}-2q^{4}+2q^{2}+1-q^{-2}+q^{-4}$
1,0,0,0	$-q^{40}+q^{38}-3q^{36}-3q^{32}-2q^{30}-q^{26}+3q^{24}+q^{22}+5q^{20}+q^{18}+4q^{16}+2q^{12}-q^{8}+q^{6}-2q^{4}+q^{2}-1$

B2 Invariants.

Weight	Invariant
0,1	$-q^{62}+2q^{60}-3q^{58}+6q^{56}-9q^{54}+10q^{52}-14q^{50}+15q^{48}-16q^{46}+16q^{44}-13q^{42}+8q^{40}-q^{38}-8q^{36}+16q^{34}-24q^{32}+29q^{30}-33q^{28}+34q^{26}-32q^{24}+27q^{22}-17q^{20}+11q^{18}-5q^{14}+14q^{12}-17q^{10}+18q^{8}-17q^{6}+16q^{4}-12q^{2}+9-6q^{-2}+2q^{-4}-q^{-6}$
1,0	$q^{100}-2q^{96}-2q^{94}+q^{92}+6q^{90}+3q^{88}-6q^{86}-8q^{84}+12q^{80}+7q^{78}-8q^{76}-13q^{74}+15q^{70}+8q^{68}-11q^{66}-14q^{64}+5q^{62}+16q^{60}-q^{58}-17q^{56}-7q^{54}+11q^{52}+7q^{50}-10q^{48}-11q^{46}+5q^{44}+10q^{42}-4q^{40}-9q^{38}+5q^{36}+15q^{34}+2q^{32}-13q^{30}-6q^{28}+16q^{26}+16q^{24}-5q^{22}-19q^{20}-q^{18}+18q^{16}+10q^{14}-12q^{12}-15q^{10}+4q^{8}+13q^{6}+q^{4}-9q^{2}-5+5q^{-2}+4q^{-4}-2q^{-6}-2q^{-8}+q^{-12}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{86}-2q^{84}+q^{82}-2q^{80}+6q^{78}-6q^{76}+5q^{74}-8q^{72}+12q^{70}-10q^{68}+10q^{66}-13q^{64}+13q^{62}-10q^{60}+10q^{58}-8q^{56}+4q^{54}-7q^{50}+8q^{48}-20q^{46}+14q^{44}-27q^{42}+21q^{40}-29q^{38}+27q^{36}-22q^{34}+27q^{32}-13q^{30}+22q^{28}-2q^{26}+10q^{24}+2q^{22}-3q^{20}+10q^{18}-12q^{16}+11q^{14}-17q^{12}+14q^{10}-15q^{8}+11q^{6}-11q^{4}+9q^{2}-6+4q^{-2}-2q^{-4}+q^{-6}$

G2 Invariants.

Weight

Invariant

1,0

$q^{148}-2q^{146}+3q^{144}-4q^{142}+3q^{140}-2q^{138}-q^{136}+8q^{134}-12q^{132}+16q^{130}-17q^{128}+11q^{126}-4q^{124}-9q^{122}+24q^{120}-33q^{118}+35q^{116}-28q^{114}+15q^{112}+3q^{110}-20q^{108}+39q^{106}-51q^{104}+48q^{102}-36q^{100}+7q^{98}+24q^{96}-52q^{94}+65q^{92}-53q^{90}+17q^{88}+22q^{86}-59q^{84}+59q^{82}-30q^{80}-23q^{78}+66q^{76}-82q^{74}+58q^{72}+2q^{70}-67q^{68}+112q^{66}-116q^{64}+77q^{62}-10q^{60}-57q^{58}+107q^{56}-111q^{54}+88q^{52}-35q^{50}-20q^{48}+66q^{46}-81q^{44}+66q^{42}-23q^{40}-26q^{38}+64q^{36}-68q^{34}+41q^{32}+16q^{30}-68q^{28}+98q^{26}-86q^{24}+34q^{22}+32q^{20}-84q^{18}+103q^{16}-81q^{14}+36q^{12}+12q^{10}-49q^{8}+59q^{6}-47q^{4}+24q^{2}-2-12q^{-2}+13q^{-4}-11q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$

.

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["10 100"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-4t^{3}+9t^{2}-12t+13-12t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+4z^{6}+5z^{4}+4z^{2}+1$

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]=

{ 65, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q+3-5q^{-1}+8q^{-2}-9q^{-3}+11q^{-4}-10q^{-5}+8q^{-6}-6q^{-7}+3q^{-8}-q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{4}z^{8}-a^{6}z^{6}+6a^{4}z^{6}-a^{2}z^{6}-4a^{6}z^{4}+13a^{4}z^{4}-4a^{2}z^{4}-5a^{6}z^{2}+13a^{4}z^{2}-4a^{2}z^{2}-3a^{6}+5a^{4}-a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}+3z^{4}a^{10}+6z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+8z^{6}a^{8}-11z^{4}a^{8}+4z^{2}a^{8}+8z^{7}a^{7}-14z^{5}a^{7}+5z^{3}a^{7}-2za^{7}+6z^{8}a^{6}-12z^{6}a^{6}+5z^{4}a^{6}-6z^{2}a^{6}+3a^{6}+2z^{9}a^{5}+4z^{7}a^{5}-27z^{5}a^{5}+26z^{3}a^{5}-8za^{5}+9z^{8}a^{4}-33z^{6}a^{4}+36z^{4}a^{4}-17z^{2}a^{4}+5a^{4}+2z^{9}a^{3}-3z^{7}a^{3}-11z^{5}a^{3}+20z^{3}a^{3}-6za^{3}+3z^{8}a^{2}-13z^{6}a^{2}+17z^{4}a^{2}-7z^{2}a^{2}+a^{2}+z^{7}a-4z^{5}a+5z^{3}a-2za$

Vassiliev invariants

V2 and V3:

(4, -7)

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 $16$ $-56$ $128$ ${\frac {776}{3}}$ ${\frac {64}{3}}$ $-896$ $-{\frac {4304}{3}}$ $-{\frac {512}{3}}$ $-216$ ${\frac {2048}{3}}$ $1568$ ${\frac {12416}{3}}$ ${\frac {1024}{3}}$ ${\frac {118262}{15}}$ ${\frac {224}{5}}$ ${\frac {128048}{45}}$ ${\frac {1306}{9}}$ ${\frac {4502}{15}}$

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 10 100. 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                 3 1   -2 -3               5 2     3 -5             5 4       -1 -7           6 4         2 -9         4 5           1 -11       4 6             -2 -13     2 4               2 -15   1 4                 -3 -17   2                   2 -19 1                     -1

Integral Khovanov Homology (db, data source)

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

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, 100]]
Out[2]=	10
In[3]:=	PD[Knot[10, 100]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 6, 19, 5], X[20, 13, 1, 14], X[14, 7, 15, 8], X[10, 3, 11, 4], X[16, 9, 17, 10], X[4, 11, 5, 12], X[8, 15, 9, 16], X[12, 19, 13, 20], X[2, 18, 3, 17]]
In[4]:=	GaussCode[Knot[10, 100]]
Out[4]=	GaussCode[1, -10, 5, -7, 2, -1, 4, -8, 6, -5, 7, -9, 3, -4, 8, -6, 10, -2, 9, -3]
In[5]:=	BR[Knot[10, 100]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, -1, 2, -1, -1, 2}]
In[6]:=	alex = Alexander[Knot[10, 100]][t]
Out[6]=	-4 4 9 12 2 3 4 13 + t - -- + -- - -- - 12 t + 9 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 100]][z]
Out[7]=	2 4 6 8 1 + 4 z + 5 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 100]}
In[9]:=	{KnotDet[Knot[10, 100]], KnotSignature[Knot[10, 100]]}
Out[9]=	{65, -4}
In[10]:=	J=Jones[Knot[10, 100]][q]
Out[10]=	-9 3 6 8 10 11 9 8 5 3 - q + -- - -- + -- - -- + -- - -- + -- - - - q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 100]}
In[12]:=	A2Invariant[Knot[10, 100]][q]
Out[12]=	-26 -24 2 -18 -16 3 -12 4 -6 -4 1 - q + q - --- - q - q + --- - q + --- + q + q - 22 14 10 q q q -2 2 q - q
In[13]:=	Kauffman[Knot[10, 100]][a, z]
Out[13]=	2 4 6 3 5 7 9 a + 5 a + 3 a - 2 a z - 6 a z - 8 a z - 2 a z + 2 a z - 2 2 4 2 6 2 8 2 3 3 3 7 a z - 17 a z - 6 a z + 4 a z + 5 a z + 20 a z + 5 3 7 3 9 3 11 3 2 4 4 4 26 a z + 5 a z - 5 a z + a z + 17 a z + 36 a z + 6 4 8 4 10 4 5 3 5 5 5 5 a z - 11 a z + 3 a z - 4 a z - 11 a z - 27 a z - 7 5 9 5 2 6 4 6 6 6 8 6 14 a z + 6 a z - 13 a z - 33 a z - 12 a z + 8 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 a z - 3 a z + 4 a z + 8 a z + 3 a z + 9 a z + 6 a z + 3 9 5 9 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[10, 100]], Vassiliev[3][Knot[10, 100]]}
Out[14]=	{0, -7}
In[15]:=	Kh[Knot[10, 100]][q, t]
Out[15]=	4 5 1 2 1 4 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 4 6 4 5 6 4 5 2 t 3 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