10 116

10_115

10_117

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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]

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]=

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

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

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, 116]]
Out[2]=	10
In[3]:=	PD[Knot[10, 116]]
Out[3]=	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[4]:=	GaussCode[Knot[10, 116]]
Out[4]=	GaussCode[1, -9, 2, -10, 6, -1, 3, -4, 9, -5, 10, -8, 7, -3, 4, -2, 5, -6, 8, -7]
In[5]:=	BR[Knot[10, 116]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, -1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[10, 116]][t]
Out[6]=	-4 5 12 19 2 3 4 -21 - t + -- - -- + -- + 19 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 116]][z]
Out[7]=	4 6 8 1 - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 116], Knot[11, Alternating, 7], Knot[11, Alternating, 33], Knot[11, Alternating, 82]}
In[9]:=	{KnotDet[Knot[10, 116]], KnotSignature[Knot[10, 116]]}
Out[9]=	{95, -2}
In[10]:=	J=Jones[Knot[10, 116]][q]
Out[10]=	-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[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 116]}
In[12]:=	A2Invariant[Knot[10, 116]][q]
Out[12]=	-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[13]:=	Kauffman[Knot[10, 116]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[10, 116]], Vassiliev[3][Knot[10, 116]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 116]][q, t]
Out[15]=	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