10 116

10_115

10_117

Visit 10 116's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 116's page at Knotilus! Visit 10 116's page at the original Knot Atlas! 10 116 Quick Notes

Square depiction.		Triquetra-heart decorative depiction.		Medieval manuscript.
As decorative double heart knot.		Mongolian ornament.

Knot presentations

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

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

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-3] Hyperbolic Volume 15.4239 A-Polynomial See Data:10 116/A-polynomial

[edit Notes for 10 116's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $4$ Rasmussen s-Invariant 2

[edit Notes for 10 116's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 95, -2 }
Jones polynomial	$q^{3}-4q^{2}+8q-11+15q^{-1}-16q^{-2}+15q^{-3}-12q^{-4}+8q^{-5}-4q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+z^{6}+3a^{4}z^{4}-8a^{2}z^{4}+3z^{4}+2a^{4}z^{2}-4a^{2}z^{2}+2z^{2}+1$
Kauffman polynomial (db, data sources)	$3a^{3}z^{9}+3az^{9}+8a^{4}z^{8}+14a^{2}z^{8}+6z^{8}+10a^{5}z^{7}+9a^{3}z^{7}+3az^{7}+4z^{7}a^{-1}+8a^{6}z^{6}-8a^{4}z^{6}-32a^{2}z^{6}+z^{6}a^{-2}-15z^{6}+4a^{7}z^{5}-13a^{5}z^{5}-29a^{3}z^{5}-22az^{5}-10z^{5}a^{-1}+a^{8}z^{4}-8a^{6}z^{4}-a^{4}z^{4}+19a^{2}z^{4}-2z^{4}a^{-2}+9z^{4}-2a^{7}z^{3}+6a^{5}z^{3}+19a^{3}z^{3}+17az^{3}+6z^{3}a^{-1}+2a^{6}z^{2}+a^{4}z^{2}-3a^{2}z^{2}+z^{2}a^{-2}-z^{2}-a^{5}z-3a^{3}z-3az-za^{-1}+1$
The A2 invariant	$q^{20}-2q^{18}+2q^{16}-2q^{14}+2q^{10}-3q^{8}+4q^{6}-3q^{4}+3q^{2}+1-q^{-2}+2q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{114}-3q^{112}+6q^{110}-10q^{108}+10q^{106}-8q^{104}+q^{102}+16q^{100}-34q^{98}+54q^{96}-64q^{94}+51q^{92}-23q^{90}-27q^{88}+89q^{86}-140q^{84}+173q^{82}-158q^{80}+88q^{78}+24q^{76}-156q^{74}+263q^{72}-304q^{70}+240q^{68}-89q^{66}-104q^{64}+263q^{62}-309q^{60}+233q^{58}-52q^{56}-150q^{54}+265q^{52}-247q^{50}+82q^{48}+155q^{46}-343q^{44}+398q^{42}-275q^{40}+28q^{38}+249q^{36}-454q^{34}+500q^{32}-388q^{30}+147q^{28}+134q^{26}-350q^{24}+441q^{22}-368q^{20}+181q^{18}+51q^{16}-241q^{14}+301q^{12}-226q^{10}+47q^{8}+170q^{6}-305q^{4}+297q^{2}-138-95q^{-2}+304q^{-4}-396q^{-6}+330q^{-8}-150q^{-10}-70q^{-12}+244q^{-14}-312q^{-16}+271q^{-18}-144q^{-20}+6q^{-22}+90q^{-24}-132q^{-26}+116q^{-28}-72q^{-30}+29q^{-32}+6q^{-34}-21q^{-36}+21q^{-38}-16q^{-40}+8q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 10_116.

A1 Invariants.

Weight	Invariant
1	$q^{15}-3q^{13}+4q^{11}-4q^{9}+3q^{7}-q^{5}-q^{3}+4q-3q^{-1}+4q^{-3}-3q^{-5}+q^{-7}$
2	$q^{42}-3q^{40}+q^{38}+8q^{36}-14q^{34}+2q^{32}+24q^{30}-31q^{28}-5q^{26}+45q^{24}-29q^{22}-23q^{20}+40q^{18}-4q^{16}-29q^{14}+10q^{12}+23q^{10}-16q^{8}-22q^{6}+35q^{4}+5q^{2}-42+29q^{-2}+24q^{-4}-42q^{-6}+6q^{-8}+28q^{-10}-20q^{-12}-8q^{-14}+13q^{-16}-q^{-18}-3q^{-20}+q^{-22}$
3	$q^{81}-3q^{79}+q^{77}+5q^{75}-2q^{73}-9q^{71}+2q^{69}+22q^{67}-15q^{65}-38q^{63}+33q^{61}+77q^{59}-48q^{57}-149q^{55}+45q^{53}+235q^{51}-311q^{47}-87q^{45}+342q^{43}+192q^{41}-307q^{39}-279q^{37}+212q^{35}+326q^{33}-89q^{31}-316q^{29}-37q^{27}+268q^{25}+143q^{23}-203q^{21}-224q^{19}+133q^{17}+277q^{15}-58q^{13}-322q^{11}-13q^{9}+344q^{7}+100q^{5}-340q^{3}-191q+301q^{-1}+274q^{-3}-210q^{-5}-329q^{-7}+100q^{-9}+326q^{-11}+19q^{-13}-269q^{-15}-106q^{-17}+172q^{-19}+139q^{-21}-80q^{-23}-117q^{-25}+10q^{-27}+74q^{-29}+20q^{-31}-34q^{-33}-16q^{-35}+8q^{-37}+8q^{-39}-q^{-41}-3q^{-43}+q^{-45}$
5	$q^{195}-3q^{193}+q^{191}+5q^{189}-5q^{187}+3q^{183}-5q^{181}-3q^{179}+5q^{177}+q^{175}+9q^{173}+29q^{171}-q^{169}-70q^{167}-106q^{165}-17q^{163}+167q^{161}+318q^{159}+211q^{157}-325q^{155}-871q^{153}-707q^{151}+400q^{149}+1775q^{147}+2007q^{145}+67q^{143}-3101q^{141}-4520q^{139}-1749q^{137}+4176q^{135}+8385q^{133}+5742q^{131}-3795q^{129}-13084q^{127}-12572q^{125}+397q^{123}+16782q^{121}+21665q^{119}+7375q^{117}-17126q^{115}-30983q^{113}-19109q^{111}+12071q^{109}+37149q^{107}+32580q^{105}-1178q^{103}-37270q^{101}-44070q^{99}-13451q^{97}+30141q^{95}+49799q^{93}+28104q^{91}-17067q^{89}-47874q^{87}-38851q^{85}+1546q^{83}+38936q^{81}+43051q^{79}+12555q^{77}-25685q^{75}-40669q^{73}-22395q^{71}+11842q^{69}+33586q^{67}+26963q^{65}-228q^{63}-24705q^{61}-27356q^{59}-7719q^{57}+16750q^{55}+25582q^{53}+12202q^{51}-11167q^{49}-23796q^{47}-14599q^{45}+8113q^{43}+23487q^{41}+16673q^{39}-6753q^{37}-24957q^{35}-19826q^{33}+5346q^{31}+27597q^{29}+24904q^{27}-2384q^{25}-29959q^{23}-31463q^{21}-3438q^{19}+30076q^{17}+38397q^{15}+12300q^{13}-26372q^{11}-43500q^{9}-23145q^{7}+17861q^{5}+44481q^{3}+33843q-5191q^{-1}-39743q^{-3}-41284q^{-5}-9416q^{-7}+28949q^{-9}+42951q^{-11}+22656q^{-13}-14084q^{-15}-37624q^{-17}-30923q^{-19}-1528q^{-21}+26398q^{-23}+32173q^{-25}+13885q^{-27}-12408q^{-29}-26637q^{-31}-20133q^{-33}-305q^{-35}+16821q^{-37}+19690q^{-39}+8625q^{-41}-6486q^{-43}-14513q^{-45}-11289q^{-47}-1173q^{-49}+7594q^{-51}+9530q^{-53}+4849q^{-55}-1924q^{-57}-5782q^{-59}-4950q^{-61}-1205q^{-63}+2236q^{-65}+3313q^{-67}+2010q^{-69}-172q^{-71}-1516q^{-73}-1467q^{-75}-544q^{-77}+359q^{-79}+726q^{-81}+498q^{-83}+40q^{-85}-227q^{-87}-238q^{-89}-101q^{-91}+29q^{-93}+89q^{-95}+55q^{-97}-3q^{-99}-20q^{-101}-14q^{-103}-5q^{-105}+3q^{-107}+8q^{-109}-q^{-111}-3q^{-113}+q^{-115}$
6	$q^{270}-3q^{268}+q^{266}+5q^{264}-5q^{262}+7q^{256}-14q^{254}-11q^{252}+35q^{250}-q^{248}+7q^{246}+5q^{244}-16q^{242}-102q^{240}-95q^{238}+126q^{236}+173q^{234}+233q^{232}+133q^{230}-214q^{228}-787q^{226}-863q^{224}+89q^{222}+1110q^{220}+2071q^{218}+1821q^{216}-293q^{214}-3794q^{212}-5782q^{210}-3268q^{208}+2530q^{206}+9759q^{204}+12704q^{202}+6431q^{200}-8746q^{198}-23691q^{196}-25304q^{194}-8926q^{192}+22026q^{190}+48619q^{188}+47918q^{186}+9554q^{184}-49536q^{182}-90164q^{180}-79255q^{178}-5653q^{176}+94305q^{174}+154491q^{172}+119077q^{170}-10139q^{168}-161421q^{166}-237093q^{164}-162820q^{162}+39294q^{160}+253465q^{158}+331973q^{156}+198607q^{154}-86609q^{152}-358165q^{150}-425759q^{148}-220151q^{146}+153632q^{144}+462497q^{142}+497943q^{140}+216834q^{138}-226470q^{136}-548596q^{134}-537459q^{132}-184007q^{130}+293918q^{128}+597089q^{126}+533241q^{124}+134376q^{122}-343209q^{120}-603314q^{118}-485828q^{116}-77432q^{114}+363100q^{112}+565830q^{110}+413877q^{108}+23694q^{106}-356565q^{104}-495097q^{102}-330502q^{100}+18280q^{98}+327737q^{96}+413365q^{94}+246944q^{92}-51593q^{90}-288528q^{88}-331696q^{86}-169592q^{84}+79471q^{82}+254423q^{80}+255081q^{78}+94302q^{76}-110276q^{74}-226943q^{72}-181511q^{70}-16436q^{68}+150236q^{66}+201155q^{64}+102110q^{62}-71168q^{60}-192883q^{58}-166725q^{56}-9565q^{54}+168047q^{52}+227640q^{50}+113475q^{48}-101456q^{46}-263594q^{44}-241514q^{42}-35861q^{40}+223375q^{38}+345426q^{36}+228197q^{34}-66101q^{32}-342119q^{30}-400744q^{28}-186493q^{26}+177177q^{24}+443917q^{22}+426985q^{20}+121024q^{18}-281071q^{16}-515530q^{14}-424093q^{12}-50416q^{10}+364013q^{8}+555283q^{6}+396290q^{4}-11757q^{2}-413870-561277q^{-2}-359454q^{-4}+56402q^{-6}+430841q^{-8}+535645q^{-10}+320664q^{-12}-76469q^{-14}-414114q^{-16}-491758q^{-18}-283193q^{-20}+76626q^{-22}+368060q^{-24}+434189q^{-26}+250787q^{-28}-59231q^{-30}-308042q^{-32}-366896q^{-34}-219154q^{-36}+30632q^{-38}+240637q^{-40}+297555q^{-42}+189115q^{-44}-4583q^{-46}-172632q^{-48}-228234q^{-50}-159928q^{-52}-16237q^{-54}+113485q^{-56}+165638q^{-58}+126700q^{-60}+29925q^{-62}-65450q^{-64}-113292q^{-66}-94457q^{-68}-33711q^{-70}+32149q^{-72}+69798q^{-74}+66122q^{-76}+31266q^{-78}-12271q^{-80}-38961q^{-82}-41889q^{-84}-24097q^{-86}+858q^{-88}+19593q^{-90}+24318q^{-92}+15819q^{-94}+3007q^{-96}-8161q^{-98}-12290q^{-100}-9660q^{-102}-3024q^{-104}+2875q^{-106}+5293q^{-108}+4888q^{-110}+2148q^{-112}-588q^{-114}-2218q^{-116}-2039q^{-118}-1024q^{-120}-52q^{-122}+713q^{-124}+776q^{-126}+460q^{-128}-8q^{-130}-184q^{-132}-211q^{-134}-161q^{-136}-13q^{-138}+60q^{-140}+74q^{-142}+11q^{-144}+q^{-146}-4q^{-148}-19q^{-150}-5q^{-152}+3q^{-154}+8q^{-156}-q^{-158}-3q^{-160}+q^{-162}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}-2q^{18}+2q^{16}-2q^{14}+2q^{10}-3q^{8}+4q^{6}-3q^{4}+3q^{2}+1-q^{-2}+2q^{-4}-2q^{-6}+q^{-8}$
1,1	$q^{60}-6q^{58}+18q^{56}-38q^{54}+73q^{52}-136q^{50}+226q^{48}-336q^{46}+480q^{44}-664q^{42}+864q^{40}-1066q^{38}+1260q^{36}-1404q^{34}+1438q^{32}-1312q^{30}+1001q^{28}-498q^{26}-180q^{24}+966q^{22}-1751q^{20}+2442q^{18}-2948q^{16}+3196q^{14}-3175q^{12}+2872q^{10}-2332q^{8}+1610q^{6}-788q^{4}-10q^{2}+746-1310q^{-2}+1655q^{-4}-1758q^{-6}+1656q^{-8}-1422q^{-10}+1102q^{-12}-774q^{-14}+492q^{-16}-280q^{-18}+143q^{-20}-62q^{-22}+22q^{-24}-6q^{-26}+q^{-28}$
2,0	$q^{52}-2q^{50}+4q^{46}-5q^{44}+6q^{40}-4q^{38}-6q^{36}+2q^{34}+14q^{32}-7q^{30}-12q^{28}+16q^{26}+6q^{24}-18q^{22}-3q^{20}+13q^{18}-9q^{16}-8q^{14}+8q^{12}+7q^{10}-9q^{8}+2q^{6}+19q^{4}-9q^{2}-7+15q^{-2}+2q^{-4}-15q^{-6}-2q^{-8}+10q^{-10}-9q^{-14}+2q^{-16}+7q^{-18}-2q^{-20}-2q^{-22}+q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-3q^{46}+9q^{42}-10q^{40}-5q^{38}+22q^{36}-16q^{34}-15q^{32}+34q^{30}-15q^{28}-21q^{26}+33q^{24}-9q^{22}-17q^{20}+14q^{18}+4q^{16}-5q^{14}-11q^{12}+14q^{10}+9q^{8}-28q^{6}+13q^{4}+22q^{2}-30+10q^{-2}+22q^{-4}-26q^{-6}+8q^{-8}+10q^{-10}-13q^{-12}+5q^{-14}+2q^{-16}-3q^{-18}+q^{-20}$
1,0,0	$q^{25}-2q^{23}+3q^{21}-4q^{19}+3q^{17}-3q^{15}+2q^{13}-q^{11}+q^{9}+q^{7}-q^{5}+3q^{3}-2q+4q^{-1}-3q^{-3}+3q^{-5}-2q^{-7}+q^{-9}$
1,0,1	$q^{78}-6q^{76}+15q^{74}-17q^{72}-3q^{70}+48q^{68}-97q^{66}+98q^{64}-5q^{62}-156q^{60}+291q^{58}-284q^{56}+65q^{54}+294q^{52}-596q^{50}+624q^{48}-260q^{46}-385q^{44}+998q^{42}-1218q^{40}+835q^{38}+25q^{36}-970q^{34}+1530q^{32}-1430q^{30}+773q^{28}+88q^{26}-691q^{24}+807q^{22}-522q^{20}+141q^{18}-44q^{16}+305q^{14}-777q^{12}+1059q^{10}-833q^{8}+121q^{6}+822q^{4}-1490q^{2}+1613-1120q^{-2}+289q^{-4}+509q^{-6}-975q^{-8}+973q^{-10}-628q^{-12}+160q^{-14}+196q^{-16}-325q^{-18}+264q^{-20}-123q^{-22}+10q^{-24}+39q^{-26}-37q^{-28}+19q^{-30}-6q^{-32}+q^{-34}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{58}-2q^{56}-2q^{54}+7q^{52}-2q^{50}-11q^{48}+13q^{46}+10q^{44}-23q^{42}-q^{40}+22q^{38}-9q^{36}-22q^{34}+17q^{32}+17q^{30}-23q^{28}-5q^{26}+33q^{24}-10q^{22}-28q^{20}+31q^{18}+6q^{16}-36q^{14}+7q^{12}+25q^{10}-20q^{8}-13q^{6}+22q^{4}+12q^{2}-17+2q^{-2}+19q^{-4}-8q^{-6}-8q^{-8}+10q^{-10}-q^{-12}-7q^{-14}+5q^{-16}-2q^{-20}+q^{-22}$
1,0,0,0	$q^{30}-2q^{28}+3q^{26}-3q^{24}+q^{22}-2q^{18}+2q^{16}-2q^{14}+3q^{12}-2q^{10}+3q^{8}-2q^{6}+3q^{4}-q^{2}+1+2q^{-2}-2q^{-4}+3q^{-6}-2q^{-8}+q^{-10}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-3q^{46}+6q^{44}-11q^{42}+18q^{40}-25q^{38}+32q^{36}-38q^{34}+41q^{32}-40q^{30}+31q^{28}-17q^{26}-q^{24}+21q^{22}-41q^{20}+60q^{18}-74q^{16}+81q^{14}-79q^{12}+70q^{10}-55q^{8}+36q^{6}-13q^{4}-6q^{2}+24-34q^{-2}+42q^{-4}-42q^{-6}+38q^{-8}-32q^{-10}+23q^{-12}-15q^{-14}+8q^{-16}-3q^{-18}+q^{-20}$
1,0	$q^{78}-3q^{74}-3q^{72}+3q^{70}+10q^{68}+3q^{66}-14q^{64}-15q^{62}+8q^{60}+27q^{58}+8q^{56}-29q^{54}-28q^{52}+16q^{50}+42q^{48}+4q^{46}-41q^{44}-22q^{42}+30q^{40}+33q^{38}-17q^{36}-36q^{34}+4q^{32}+34q^{30}+5q^{28}-30q^{26}-11q^{24}+25q^{22}+16q^{20}-21q^{18}-20q^{16}+19q^{14}+27q^{12}-13q^{10}-37q^{8}+q^{6}+42q^{4}+20q^{2}-35-36q^{-2}+19q^{-4}+45q^{-6}+3q^{-8}-37q^{-10}-20q^{-12}+22q^{-14}+24q^{-16}-7q^{-18}-18q^{-20}-3q^{-22}+10q^{-24}+5q^{-26}-3q^{-28}-3q^{-30}+q^{-34}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{66}-3q^{64}+3q^{62}-4q^{60}+10q^{58}-14q^{56}+14q^{54}-18q^{52}+27q^{50}-29q^{48}+26q^{46}-32q^{44}+35q^{42}-27q^{40}+19q^{38}-16q^{36}+6q^{34}+11q^{32}-19q^{30}+28q^{28}-43q^{26}+54q^{24}-56q^{22}+60q^{20}-65q^{18}+60q^{16}-52q^{14}+46q^{12}-39q^{10}+24q^{8}-10q^{6}+3q^{4}+9q^{2}-18+30q^{-2}-29q^{-4}+33q^{-6}-34q^{-8}+33q^{-10}-28q^{-12}+23q^{-14}-19q^{-16}+13q^{-18}-8q^{-20}+5q^{-22}-3q^{-24}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-3q^{112}+6q^{110}-10q^{108}+10q^{106}-8q^{104}+q^{102}+16q^{100}-34q^{98}+54q^{96}-64q^{94}+51q^{92}-23q^{90}-27q^{88}+89q^{86}-140q^{84}+173q^{82}-158q^{80}+88q^{78}+24q^{76}-156q^{74}+263q^{72}-304q^{70}+240q^{68}-89q^{66}-104q^{64}+263q^{62}-309q^{60}+233q^{58}-52q^{56}-150q^{54}+265q^{52}-247q^{50}+82q^{48}+155q^{46}-343q^{44}+398q^{42}-275q^{40}+28q^{38}+249q^{36}-454q^{34}+500q^{32}-388q^{30}+147q^{28}+134q^{26}-350q^{24}+441q^{22}-368q^{20}+181q^{18}+51q^{16}-241q^{14}+301q^{12}-226q^{10}+47q^{8}+170q^{6}-305q^{4}+297q^{2}-138-95q^{-2}+304q^{-4}-396q^{-6}+330q^{-8}-150q^{-10}-70q^{-12}+244q^{-14}-312q^{-16}+271q^{-18}-144q^{-20}+6q^{-22}+90q^{-24}-132q^{-26}+116q^{-28}-72q^{-30}+29q^{-32}+6q^{-34}-21q^{-36}+21q^{-38}-16q^{-40}+8q^{-42}-3q^{-44}+q^{-46}$

.

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["10 116"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}-2z^{4}+1$

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]=

{ 95, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+8q-11+15q^{-1}-16q^{-2}+15q^{-3}-12q^{-4}+8q^{-5}-4q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+z^{6}+3a^{4}z^{4}-8a^{2}z^{4}+3z^{4}+2a^{4}z^{2}-4a^{2}z^{2}+2z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{3}z^{9}+3az^{9}+8a^{4}z^{8}+14a^{2}z^{8}+6z^{8}+10a^{5}z^{7}+9a^{3}z^{7}+3az^{7}+4z^{7}a^{-1}+8a^{6}z^{6}-8a^{4}z^{6}-32a^{2}z^{6}+z^{6}a^{-2}-15z^{6}+4a^{7}z^{5}-13a^{5}z^{5}-29a^{3}z^{5}-22az^{5}-10z^{5}a^{-1}+a^{8}z^{4}-8a^{6}z^{4}-a^{4}z^{4}+19a^{2}z^{4}-2z^{4}a^{-2}+9z^{4}-2a^{7}z^{3}+6a^{5}z^{3}+19a^{3}z^{3}+17az^{3}+6z^{3}a^{-1}+2a^{6}z^{2}+a^{4}z^{2}-3a^{2}z^{2}+z^{2}a^{-2}-z^{2}-a^{5}z-3a^{3}z-3az-za^{-1}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a7, K11a33, K11a82, ...}

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

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 $0$ $0$ $0$ $16$ $16$ $0$ $-32$ $-32$ $0$ $0$ $0$ $0$ $0$ $40$ ${\frac {464}{3}}$ $-{\frac {544}{3}}$ ${\frac {200}{3}}$ $-40$

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 10 116. 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                   3   -3 3                 5 1   4 1               6 3     -3 -1             9 5       4 -3           8 7         -1 -5         7 8           -1 -7       5 8             3 -9     3 7               -4 -11   1 5                 4 -13   3                   -3 -15 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{10}-4q^{9}+2q^{8}+15q^{7}-25q^{6}-10q^{5}+63q^{4}-47q^{3}-58q^{2}+129q-42-129q^{-1}+176q^{-2}-12q^{-3}-186q^{-4}+182q^{-5}+27q^{-6}-199q^{-7}+143q^{-8}+52q^{-9}-155q^{-10}+80q^{-11}+46q^{-12}-81q^{-13}+30q^{-14}+20q^{-15}-26q^{-16}+8q^{-17}+4q^{-18}-4q^{-19}+q^{-20}$
3	$q^{21}-4q^{20}+2q^{19}+9q^{18}+q^{17}-28q^{16}-16q^{15}+63q^{14}+55q^{13}-92q^{12}-143q^{11}+100q^{10}+274q^{9}-59q^{8}-421q^{7}-63q^{6}+562q^{5}+248q^{4}-647q^{3}-492q^{2}+681q+732-620q^{-1}-984q^{-2}+532q^{-3}+1172q^{-4}-376q^{-5}-1341q^{-6}+223q^{-7}+1436q^{-8}-41q^{-9}-1485q^{-10}-134q^{-11}+1457q^{-12}+305q^{-13}-1360q^{-14}-439q^{-15}+1178q^{-16}+532q^{-17}-945q^{-18}-553q^{-19}+687q^{-20}+504q^{-21}-446q^{-22}-403q^{-23}+258q^{-24}+280q^{-25}-135q^{-26}-168q^{-27}+68q^{-28}+86q^{-29}-34q^{-30}-43q^{-31}+24q^{-32}+15q^{-33}-11q^{-34}-6q^{-35}+4q^{-36}+4q^{-37}-4q^{-38}+q^{-39}$
4	$q^{36}-4q^{35}+2q^{34}+9q^{33}-5q^{32}-2q^{31}-34q^{30}+8q^{29}+74q^{28}+24q^{27}+4q^{26}-216q^{25}-122q^{24}+215q^{23}+300q^{22}+343q^{21}-528q^{20}-786q^{19}-131q^{18}+662q^{17}+1639q^{16}-66q^{15}-1661q^{14}-1744q^{13}-203q^{12}+3313q^{11}+2083q^{10}-1021q^{9}-3791q^{8}-3267q^{7}+3334q^{6}+4829q^{5}+2117q^{4}-4099q^{3}-7212q^{2}+751q+5964+6391q^{-1}-1879q^{-2}-9881q^{-3}-3125q^{-4}+4841q^{-5}+9831q^{-6}+1610q^{-7}-10619q^{-8}-6624q^{-9}+2511q^{-10}+11806q^{-11}+4979q^{-12}-10029q^{-13}-9201q^{-14}-107q^{-15}+12510q^{-16}+7828q^{-17}-8382q^{-18}-10782q^{-19}-2928q^{-20}+11658q^{-21}+9912q^{-22}-5380q^{-23}-10643q^{-24}-5643q^{-25}+8647q^{-26}+10226q^{-27}-1497q^{-28}-8040q^{-29}-6883q^{-30}+4236q^{-31}+7928q^{-32}+1345q^{-33}-3948q^{-34}-5594q^{-35}+740q^{-36}+4182q^{-37}+1805q^{-38}-811q^{-39}-2941q^{-40}-448q^{-41}+1360q^{-42}+862q^{-43}+249q^{-44}-983q^{-45}-269q^{-46}+269q^{-47}+145q^{-48}+198q^{-49}-232q^{-50}-28q^{-51}+55q^{-52}-27q^{-53}+55q^{-54}-52q^{-55}+11q^{-56}+19q^{-57}-16q^{-58}+9q^{-59}-10q^{-60}+4q^{-61}+4q^{-62}-4q^{-63}+q^{-64}$
5	$q^{55}-4q^{54}+2q^{53}+9q^{52}-5q^{51}-8q^{50}-8q^{49}-10q^{48}+19q^{47}+67q^{46}+29q^{45}-68q^{44}-138q^{43}-147q^{42}+30q^{41}+334q^{40}+487q^{39}+160q^{38}-505q^{37}-1050q^{36}-893q^{35}+285q^{34}+1831q^{33}+2342q^{32}+798q^{31}-2127q^{30}-4334q^{29}-3460q^{28}+999q^{27}+6200q^{26}+7571q^{25}+2554q^{24}-6270q^{23}-12227q^{22}-9117q^{21}+2976q^{20}+15598q^{19}+17665q^{18}+4795q^{17}-15096q^{16}-26243q^{15}-16852q^{14}+9094q^{13}+31894q^{12}+31088q^{11}+3192q^{10}-32018q^{9}-44778q^{8}-20301q^{7}+25193q^{6}+54628q^{5}+39932q^{4}-11723q^{3}-58780q^{2}-58666q-6675+56169q^{-1}+74484q^{-2}+27311q^{-3}-48142q^{-4}-85286q^{-5}-47681q^{-6}+35814q^{-7}+91612q^{-8}+65983q^{-9}-22045q^{-10}-93607q^{-11}-81195q^{-12}+7789q^{-13}+93116q^{-14}+93558q^{-15}+5243q^{-16}-90914q^{-17}-103446q^{-18}-17383q^{-19}+87985q^{-20}+111762q^{-21}+28669q^{-22}-84100q^{-23}-118820q^{-24}-40095q^{-25}+78788q^{-26}+124391q^{-27}+52038q^{-28}-70720q^{-29}-127652q^{-30}-64564q^{-31}+59151q^{-32}+127042q^{-33}+76515q^{-34}-43529q^{-35}-121029q^{-36}-86308q^{-37}+24914q^{-38}+108768q^{-39}+91499q^{-40}-5289q^{-41}-90533q^{-42}-90423q^{-43}-12476q^{-44}+68371q^{-45}+82476q^{-46}+25518q^{-47}-45362q^{-48}-68728q^{-49}-32134q^{-50}+24779q^{-51}+51857q^{-52}+32318q^{-53}-9270q^{-54}-34970q^{-55}-27565q^{-56}-299q^{-57}+20677q^{-58}+20444q^{-59}+4587q^{-60}-10469q^{-61}-13275q^{-62}-5182q^{-63}+4292q^{-64}+7475q^{-65}+4075q^{-66}-1180q^{-67}-3738q^{-68}-2539q^{-69}+83q^{-70}+1550q^{-71}+1304q^{-72}+239q^{-73}-570q^{-74}-599q^{-75}-149q^{-76}+175q^{-77}+197q^{-78}+75q^{-79}-24q^{-80}-63q^{-81}-42q^{-82}+24q^{-83}+13q^{-84}-14q^{-85}+12q^{-86}+6q^{-87}-12q^{-88}+4q^{-89}+5q^{-90}-10q^{-91}+4q^{-92}+4q^{-93}-4q^{-94}+q^{-95}$
6	$q^{78}-4q^{77}+2q^{76}+9q^{75}-5q^{74}-8q^{73}-14q^{72}+16q^{71}+q^{70}+12q^{69}+72q^{68}-19q^{67}-81q^{66}-162q^{65}-34q^{64}+28q^{63}+188q^{62}+540q^{61}+297q^{60}-144q^{59}-927q^{58}-1006q^{57}-987q^{56}+9q^{55}+2170q^{54}+3033q^{53}+2596q^{52}-522q^{51}-3424q^{50}-6886q^{49}-6627q^{48}-460q^{47}+7162q^{46}+13764q^{45}+12290q^{44}+5075q^{43}-11611q^{42}-25362q^{41}-25415q^{40}-10630q^{39}+16692q^{38}+38980q^{37}+48612q^{36}+23245q^{35}-21686q^{34}-63064q^{33}-76490q^{32}-44054q^{31}+20145q^{30}+96454q^{29}+118620q^{28}+75089q^{27}-24126q^{26}-128643q^{25}-173776q^{24}-123546q^{23}+28148q^{22}+174222q^{21}+243138q^{20}+169572q^{19}-20203q^{18}-230694q^{17}-333551q^{16}-221638q^{15}+26480q^{14}+301992q^{13}+418383q^{12}+289815q^{11}-47292q^{10}-399680q^{9}-513072q^{8}-333339q^{7}+93427q^{6}+496027q^{5}+627460q^{4}+349841q^{3}-184699q^{2}-617876q-707778-322429q^{-1}+294204q^{-2}+774867q^{-3}+751954q^{-4}+223348q^{-5}-458883q^{-6}-899048q^{-7}-736858q^{-8}-79473q^{-9}+683430q^{-10}+986413q^{-11}+625443q^{-12}-152922q^{-13}-882116q^{-14}-1003598q^{-15}-443143q^{-16}+469179q^{-17}+1045038q^{-18}+901461q^{-19}+141376q^{-20}-764887q^{-21}-1125649q^{-22}-702379q^{-23}+263526q^{-24}+1022958q^{-25}+1063599q^{-26}+356307q^{-27}-650722q^{-28}-1185242q^{-29}-879991q^{-30}+106366q^{-31}+996800q^{-32}+1185314q^{-33}+529585q^{-34}-551677q^{-35}-1236161q^{-36}-1046663q^{-37}-58709q^{-38}+951368q^{-39}+1301981q^{-40}+734163q^{-41}-390898q^{-42}-1236819q^{-43}-1221615q^{-44}-307772q^{-45}+789264q^{-46}+1345149q^{-47}+971098q^{-48}-92361q^{-49}-1070398q^{-50}-1307243q^{-51}-617229q^{-52}+440482q^{-53}+1180554q^{-54}+1109630q^{-55}+287898q^{-56}-680215q^{-57}-1155290q^{-58}-819959q^{-59}-50q^{-60}+772158q^{-61}+992144q^{-62}+548003q^{-63}-202630q^{-64}-756425q^{-65}-756111q^{-66}-303221q^{-67}+294233q^{-68}+638692q^{-69}+536866q^{-70}+119496q^{-71}-313121q^{-72}-475002q^{-73}-338667q^{-74}-14632q^{-75}+264909q^{-76}+331258q^{-77}+187090q^{-78}-41565q^{-79}-189786q^{-80}-205301q^{-81}-93140q^{-82}+50738q^{-83}+129144q^{-84}+112817q^{-85}+34107q^{-86}-38504q^{-87}-76085q^{-88}-57726q^{-89}-9448q^{-90}+29186q^{-91}+39215q^{-92}+23198q^{-93}+2124q^{-94}-16995q^{-95}-19362q^{-96}-8747q^{-97}+2593q^{-98}+8263q^{-99}+6820q^{-100}+3737q^{-101}-2050q^{-102}-4185q^{-103}-2474q^{-104}-352q^{-105}+1034q^{-106}+1022q^{-107}+1223q^{-108}-62q^{-109}-684q^{-110}-360q^{-111}-102q^{-112}+73q^{-113}+q^{-114}+271q^{-115}+14q^{-116}-111q^{-117}-13q^{-118}-2q^{-119}+13q^{-120}-46q^{-121}+50q^{-122}+7q^{-123}-25q^{-124}+8q^{-125}+5q^{-127}-10q^{-128}+4q^{-129}+4q^{-130}-4q^{-131}+q^{-132}$
7	$q^{105}-4q^{104}+2q^{103}+9q^{102}-5q^{101}-8q^{100}-14q^{99}+10q^{98}+27q^{97}-6q^{96}+17q^{95}+24q^{94}-32q^{93}-81q^{92}-140q^{91}-35q^{90}+180q^{89}+203q^{88}+317q^{87}+277q^{86}-69q^{85}-501q^{84}-1187q^{83}-1177q^{82}-299q^{81}+772q^{80}+2326q^{79}+3310q^{78}+2759q^{77}+771q^{76}-3595q^{75}-7603q^{74}-8625q^{73}-6386q^{72}+1164q^{71}+10906q^{70}+18808q^{69}+21313q^{68}+11831q^{67}-7272q^{66}-28685q^{65}-44958q^{64}-42440q^{63}-18261q^{62}+22324q^{61}+68514q^{60}+92746q^{59}+78711q^{58}+22968q^{57}-64103q^{56}-142103q^{55}-173979q^{54}-132631q^{53}-10319q^{52}+146476q^{51}+273463q^{50}+304034q^{49}+189710q^{48}-38424q^{47}-302773q^{46}-488677q^{45}-475811q^{44}-237719q^{43}+165868q^{42}+581175q^{41}+796327q^{40}+682179q^{39}+222203q^{38}-440031q^{37}-1007750q^{36}-1207684q^{35}-869708q^{34}-49172q^{33}+922337q^{32}+1625024q^{31}+1670699q^{30}+916490q^{29}-379123q^{28}-1699282q^{27}-2405172q^{26}-2047942q^{25}-674546q^{24}+1219794q^{23}+2785688q^{22}+3199093q^{21}+2139708q^{20}-94472q^{19}-2556796q^{18}-4048701q^{17}-3761567q^{16}-1601328q^{15}+1577925q^{14}+4295576q^{13}+5201831q^{12}+3630757q^{11}+115205q^{10}-3754692q^{9}-6129725q^{8}-5656063q^{7}-2320647q^{6}+2410339q^{5}+6319177q^{4}+7335380q^{3}+4724899q^{2}-414462q-5700197-8413324q^{-1}-6996260q^{-2}-1962291q^{-3}+4368608q^{-4}+8770447q^{-5}+8859134q^{-6}+4415087q^{-7}-2531357q^{-8}-8438363q^{-9}-10161317q^{-10}-6671885q^{-11}+457160q^{-12}+7560095q^{-13}+10866184q^{-14}+8550235q^{-15}+1610856q^{-16}-6340984q^{-17}-11055527q^{-18}-9971490q^{-19}-3476329q^{-20}+4993754q^{-21}+10863732q^{-22}+10947851q^{-23}+5035675q^{-24}-3692443q^{-25}-10458859q^{-26}-11562618q^{-27}-6254716q^{-28}+2558510q^{-29}+9987193q^{-30}+11928935q^{-31}+7172709q^{-32}-1642486q^{-33}-9565849q^{-34}-12173638q^{-35}-7870129q^{-36}+937575q^{-37}+9258553q^{-38}+12405864q^{-39}+8457565q^{-40}-379459q^{-41}-9074976q^{-42}-12704262q^{-43}-9050877q^{-44}-138945q^{-45}+8964002q^{-46}+13098623q^{-47}+9750846q^{-48}+748305q^{-49}-8817560q^{-50}-13551342q^{-51}-10613901q^{-52}-1581260q^{-53}+8482179q^{-54}+13955752q^{-55}+11626559q^{-56}+2728723q^{-57}-7793083q^{-58}-14137150q^{-59}-12681350q^{-60}-4204675q^{-61}+6614112q^{-62}+13889997q^{-63}+13587239q^{-64}+5911388q^{-65}-4900343q^{-66}-13033080q^{-67}-14093722q^{-68}-7635703q^{-69}+2731817q^{-70}+11473738q^{-71}+13965385q^{-72}+9089906q^{-73}-331461q^{-74}-9267914q^{-75}-13056008q^{-76}-9977978q^{-77}-1977404q^{-78}+6628537q^{-79}+11374372q^{-80}+10093465q^{-81}+3853496q^{-82}-3892291q^{-83}-9107156q^{-84}-9388717q^{-85}-5030815q^{-86}+1430079q^{-87}+6572543q^{-88}+7997555q^{-89}+5404833q^{-90}+459641q^{-91}-4138335q^{-92}-6200227q^{-93}-5051850q^{-94}-1624008q^{-95}+2112926q^{-96}+4328336q^{-97}+4192792q^{-98}+2089008q^{-99}-667941q^{-100}-2673341q^{-101}-3113040q^{-102}-2017173q^{-103}-178950q^{-104}+1411284q^{-105}+2065397q^{-106}+1634346q^{-107}+539213q^{-108}-583518q^{-109}-1217100q^{-110}-1153869q^{-111}-580276q^{-112}+130472q^{-113}+627756q^{-114}+718788q^{-115}+464786q^{-116}+62828q^{-117}-273890q^{-118}-397793q^{-119}-312094q^{-120}-108131q^{-121}+94482q^{-122}+195147q^{-123}+181237q^{-124}+90431q^{-125}-18052q^{-126}-84181q^{-127}-94081q^{-128}-58733q^{-129}-4964q^{-130}+32043q^{-131}+43440q^{-132}+31897q^{-133}+8031q^{-134}-9964q^{-135}-18275q^{-136}-15802q^{-137}-5541q^{-138}+2703q^{-139}+7231q^{-140}+6857q^{-141}+2599q^{-142}-342q^{-143}-2372q^{-144}-2880q^{-145}-1317q^{-146}-110q^{-147}+1004q^{-148}+1178q^{-149}+264q^{-150}+57q^{-151}-160q^{-152}-377q^{-153}-178q^{-154}-128q^{-155}+142q^{-156}+217q^{-157}-46q^{-158}-9q^{-159}+10q^{-160}-24q^{-161}-3q^{-162}-46q^{-163}+18q^{-164}+45q^{-165}-24q^{-166}-5q^{-167}+4q^{-168}+5q^{-170}-10q^{-171}+4q^{-172}+4q^{-173}-4q^{-174}+q^{-175}$

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, 116]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 3, 17, 4], X[14, 7, 15, 8], X[8, 15, 9, 16], X[10, 18, 11, 17], X[18, 6, 19, 5], X[20, 13, 1, 14], X[12, 19, 13, 20], X[2, 10, 3, 9], X[4, 11, 5, 12]]
In[3]:=	GaussCode[Knot[10, 116]]
Out[3]=	GaussCode[1, -9, 2, -10, 6, -1, 3, -4, 9, -5, 10, -8, 7, -3, 4, -2, 5, -6, 8, -7]
In[4]:=	DTCode[Knot[10, 116]]
Out[4]=	DTCode[6, 16, 18, 14, 2, 4, 20, 8, 10, 12]
In[5]:=	br = BR[Knot[10, 116]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 116]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 116]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 116]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 116]][t]
Out[10]=	-4 5 12 19 2 3 4 -21 - t + -- - -- + -- + 19 t - 12 t + 5 t - t 3 2 t t t
In[11]:=	Conway[Knot[10, 116]][z]
Out[11]=	4 6 8 1 - 2 z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 116], Knot[11, Alternating, 7], Knot[11, Alternating, 33], Knot[11, Alternating, 82]}
In[13]:=	{KnotDet[Knot[10, 116]], KnotSignature[Knot[10, 116]]}
Out[13]=	{95, -2}
In[14]:=	Jones[Knot[10, 116]][q]
Out[14]=	-7 4 8 12 15 16 15 2 3 -11 + q - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 116]}
In[16]:=	A2Invariant[Knot[10, 116]][q]
Out[16]=	-20 2 2 2 2 3 4 3 3 2 4 1 + q - --- + --- - --- + --- - -- + -- - -- + -- - q + 2 q - 18 16 14 10 8 6 4 2 q q q q q q q q 6 8 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 116]][a, z]
Out[17]=	2 2 2 4 2 4 2 4 4 4 6 1 + 2 z - 4 a z + 2 a z + 3 z - 8 a z + 3 a z + z - 2 6 4 6 2 8 5 a z + a z - a z
In[18]:=	Kauffman[Knot[10, 116]][a, z]
Out[18]=	2 z 3 5 2 z 2 2 4 2 6 2 1 - - - 3 a z - 3 a z - a z - z + -- - 3 a z + a z + 2 a z + a 2 a 3 4 6 z 3 3 3 5 3 7 3 4 2 z ---- + 17 a z + 19 a z + 6 a z - 2 a z + 9 z - ---- + a 2 a 5 2 4 4 4 6 4 8 4 10 z 5 3 5 19 a z - a z - 8 a z + a z - ----- - 22 a z - 29 a z - a 6 5 5 7 5 6 z 2 6 4 6 6 6 13 a z + 4 a z - 15 z + -- - 32 a z - 8 a z + 8 a z + 2 a 7 4 z 7 3 7 5 7 8 2 8 4 8 ---- + 3 a z + 9 a z + 10 a z + 6 z + 14 a z + 8 a z + a 9 3 9 3 a z + 3 a z
In[19]:=	{Vassiliev[2][Knot[10, 116]], Vassiliev[3][Knot[10, 116]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 116]][q, t]
Out[20]=	7 9 1 3 1 5 3 7 5 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 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 8 7 8 8 5 t 2 3 2 ----- + ----- + ---- + ---- + --- + 6 q t + 3 q t + 5 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 + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 116], 2][q]
Out[21]=	-20 4 4 8 26 20 30 81 46 80 -42 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - 19 18 17 16 15 14 13 12 11 q q q q q q q q q 155 52 143 199 27 182 186 12 176 129 --- + -- + --- - --- + -- + --- - --- - -- + --- - --- + 129 q - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 10 58 q - 47 q + 63 q - 10 q - 25 q + 15 q + 2 q - 4 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

10_115 10_117