10 91

10_90

10_92

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10 91 Quick Notes

10 91 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X20,6,1,5 X16,9,17,10 X10,3,11,4 X2,18,3,17 X14,7,15,8 X8,15,9,16 X12,20,13,19 X18,12,19,11 X4,13,5,14
Gauss code	1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5, -9, 8, -2
Dowker-Thistlethwaite code	6 10 20 14 16 18 4 8 2 12
Conway Notation	[.3.2.20]

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 13.487 A-Polynomial See Data:10 91/A-polynomial

[edit Notes for 10 91's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for 10 91's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+9t^{2}-14t+17-14t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+5z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+9q^{2}-11q+13-11q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+13z^{4}-5a^{2}z^{2}-5z^{2}a^{-2}+12z^{2}-2a^{2}-2a^{-2}+5$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+5z^{8}a^{-2}+9z^{8}+4a^{3}z^{7}+az^{7}+2z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-7a^{2}z^{6}-13z^{6}a^{-2}+3z^{6}a^{-4}-26z^{6}+a^{5}z^{5}-6a^{3}z^{5}-7az^{5}-13z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+7a^{2}z^{4}+16z^{4}a^{-2}-6z^{4}a^{-4}+35z^{4}-2a^{5}z^{3}+9az^{3}+18z^{3}a^{-1}+9z^{3}a^{-3}-2z^{3}a^{-5}+2a^{4}z^{2}-7a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-19z^{2}+a^{5}z-4az-6za^{-1}-3za^{-3}+2a^{2}+2a^{-2}+5$
The A2 invariant	$-q^{14}+q^{12}-2q^{10}+q^{8}-q^{4}+4q^{2}-1+4q^{-2}-q^{-4}+q^{-8}-2q^{-10}+q^{-12}-q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-5q^{70}-3q^{68}+16q^{66}-27q^{64}+36q^{62}-38q^{60}+23q^{58}-q^{56}-34q^{54}+70q^{52}-92q^{50}+95q^{48}-66q^{46}+4q^{44}+68q^{42}-130q^{40}+151q^{38}-122q^{36}+46q^{34}+44q^{32}-116q^{30}+138q^{28}-93q^{26}+9q^{24}+77q^{22}-122q^{20}+97q^{18}-19q^{16}-82q^{14}+161q^{12}-175q^{10}+132q^{8}-27q^{6}-94q^{4}+190q^{2}-223+188q^{-2}-92q^{-4}-27q^{-6}+132q^{-8}-176q^{-10}+165q^{-12}-89q^{-14}-10q^{-16}+93q^{-18}-126q^{-20}+87q^{-22}-3q^{-24}-84q^{-26}+137q^{-28}-124q^{-30}+54q^{-32}+40q^{-34}-124q^{-36}+160q^{-38}-142q^{-40}+74q^{-42}+7q^{-44}-77q^{-46}+108q^{-48}-102q^{-50}+72q^{-52}-28q^{-54}-10q^{-56}+32q^{-58}-42q^{-60}+36q^{-62}-23q^{-64}+12q^{-66}-6q^{-70}+7q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_91.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-3q^{7}+3q^{5}-2q^{3}+2q+2q^{-1}-2q^{-3}+3q^{-5}-3q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}-q^{28}+6q^{26}-6q^{24}-5q^{22}+17q^{20}-8q^{18}-17q^{16}+24q^{14}-25q^{10}+17q^{8}+10q^{6}-18q^{4}+q^{2}+14+q^{-2}-17q^{-4}+10q^{-6}+17q^{-8}-25q^{-10}+q^{-12}+24q^{-14}-18q^{-16}-8q^{-18}+18q^{-20}-5q^{-22}-8q^{-24}+6q^{-26}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}+q^{59}-2q^{57}-3q^{55}+3q^{53}+5q^{51}-9q^{49}-7q^{47}+18q^{45}+18q^{43}-28q^{41}-43q^{39}+32q^{37}+77q^{35}-19q^{33}-111q^{31}-16q^{29}+135q^{27}+61q^{25}-134q^{23}-107q^{21}+113q^{19}+136q^{17}-74q^{15}-147q^{13}+33q^{11}+134q^{9}+11q^{7}-114q^{5}-43q^{3}+82q+78q^{-1}-47q^{-3}-108q^{-5}+18q^{-7}+130q^{-9}+22q^{-11}-147q^{-13}-60q^{-15}+144q^{-17}+100q^{-19}-123q^{-21}-127q^{-23}+81q^{-25}+138q^{-27}-33q^{-29}-122q^{-31}-12q^{-33}+92q^{-35}+34q^{-37}-55q^{-39}-38q^{-41}+23q^{-43}+30q^{-45}-7q^{-47}-17q^{-49}+2q^{-51}+7q^{-53}-3q^{-57}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}-q^{100}+2q^{98}-q^{96}+6q^{94}-3q^{92}-q^{90}+3q^{88}-15q^{86}+4q^{84}-9q^{82}+19q^{80}+48q^{78}-16q^{76}-39q^{74}-104q^{72}-6q^{70}+168q^{68}+139q^{66}+21q^{64}-307q^{62}-297q^{60}+114q^{58}+432q^{56}+462q^{54}-242q^{52}-737q^{50}-435q^{48}+355q^{46}+1046q^{44}+398q^{42}-712q^{40}-1097q^{38}-314q^{36}+1080q^{34}+1080q^{32}-83q^{30}-1165q^{28}-957q^{26}+511q^{24}+1151q^{22}+518q^{20}-669q^{18}-1032q^{16}-79q^{14}+741q^{12}+699q^{10}-144q^{8}-745q^{6}-410q^{4}+316q^{2}+684+232q^{-2}-476q^{-4}-699q^{-6}-32q^{-8}+739q^{-10}+644q^{-12}-221q^{-14}-1039q^{-16}-505q^{-18}+672q^{-20}+1106q^{-22}+288q^{-24}-1105q^{-26}-1051q^{-28}+173q^{-30}+1194q^{-32}+915q^{-34}-570q^{-36}-1158q^{-38}-528q^{-40}+619q^{-42}+1068q^{-44}+183q^{-46}-600q^{-48}-745q^{-50}-112q^{-52}+579q^{-54}+439q^{-56}+35q^{-58}-384q^{-60}-324q^{-62}+69q^{-64}+201q^{-66}+194q^{-68}-39q^{-70}-150q^{-72}-56q^{-74}+6q^{-76}+81q^{-78}+24q^{-80}-26q^{-82}-13q^{-84}-15q^{-86}+15q^{-88}+6q^{-90}-5q^{-92}+q^{-94}-3q^{-96}+3q^{-98}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}+q^{151}-2q^{149}+q^{147}-2q^{145}-6q^{143}-q^{141}+7q^{139}+7q^{137}+14q^{135}+9q^{133}-21q^{131}-42q^{129}-34q^{127}+6q^{125}+67q^{123}+114q^{121}+68q^{119}-86q^{117}-230q^{115}-229q^{113}-10q^{111}+337q^{109}+541q^{107}+312q^{105}-327q^{103}-914q^{101}-900q^{99}-60q^{97}+1158q^{95}+1774q^{93}+986q^{91}-942q^{89}-2613q^{87}-2479q^{85}-132q^{83}+2956q^{81}+4253q^{79}+2154q^{77}-2281q^{75}-5656q^{73}-4835q^{71}+270q^{69}+6021q^{67}+7522q^{65}+2799q^{63}-4910q^{61}-9312q^{59}-6264q^{57}+2355q^{55}+9646q^{53}+9218q^{51}+970q^{49}-8359q^{47}-10865q^{45}-4266q^{43}+5890q^{41}+10936q^{39}+6697q^{37}-2954q^{35}-9629q^{33}-7863q^{31}+307q^{29}+7487q^{27}+7789q^{25}+1633q^{23}-5202q^{21}-6905q^{19}-2664q^{17}+3225q^{15}+5673q^{13}+3104q^{11}-1790q^{9}-4664q^{7}-3236q^{5}+875q^{3}+4027q+3531q^{-1}-167q^{-3}-3868q^{-5}-4244q^{-7}-626q^{-9}+3982q^{-11}+5379q^{-13}+1868q^{-15}-3940q^{-17}-6868q^{-19}-3681q^{-21}+3417q^{-23}+8205q^{-25}+5965q^{-27}-1999q^{-29}-8937q^{-31}-8380q^{-33}-285q^{-35}+8536q^{-37}+10289q^{-39}+3193q^{-41}-6785q^{-43}-11078q^{-45}-6071q^{-47}+3866q^{-49}+10342q^{-51}+8163q^{-53}-435q^{-55}-8081q^{-57}-8823q^{-59}-2710q^{-61}+4870q^{-63}+7921q^{-65}+4688q^{-67}-1549q^{-69}-5752q^{-71}-5199q^{-73}-1054q^{-75}+3147q^{-77}+4391q^{-79}+2384q^{-81}-845q^{-83}-2851q^{-85}-2536q^{-87}-611q^{-89}+1307q^{-91}+1917q^{-93}+1124q^{-95}-203q^{-97}-1060q^{-99}-1016q^{-101}-324q^{-103}+405q^{-105}+645q^{-107}+391q^{-109}-35q^{-111}-300q^{-113}-280q^{-115}-85q^{-117}+108q^{-119}+145q^{-121}+67q^{-123}-18q^{-125}-54q^{-127}-42q^{-129}-4q^{-131}+25q^{-133}+14q^{-135}-2q^{-137}-3q^{-139}-3q^{-141}-3q^{-143}+2q^{-145}+3q^{-147}-3q^{-149}+2q^{-153}-q^{-155}$
6	$q^{216}-2q^{214}-q^{212}+2q^{210}-q^{208}+2q^{206}+2q^{204}+10q^{202}-5q^{200}-17q^{198}-6q^{196}-15q^{194}+22q^{190}+66q^{188}+35q^{186}-26q^{184}-53q^{182}-116q^{180}-105q^{178}-10q^{176}+211q^{174}+268q^{172}+173q^{170}+q^{168}-370q^{166}-611q^{164}-514q^{162}+164q^{160}+824q^{158}+1177q^{156}+1011q^{154}-149q^{152}-1661q^{150}-2616q^{148}-1812q^{146}+264q^{144}+2928q^{142}+4822q^{140}+3798q^{138}-158q^{136}-5401q^{134}-8227q^{132}-6897q^{130}-564q^{128}+8415q^{126}+14040q^{124}+12017q^{122}+1420q^{120}-12120q^{118}-21811q^{116}-19675q^{114}-3864q^{112}+17532q^{110}+32243q^{108}+29010q^{106}+7550q^{104}-23624q^{102}-45160q^{100}-41132q^{98}-10684q^{96}+31190q^{94}+58901q^{92}+54223q^{90}+13506q^{88}-40305q^{86}-74157q^{84}-64849q^{82}-13776q^{80}+49834q^{78}+88138q^{76}+72211q^{74}+10264q^{72}-61061q^{70}-97033q^{68}-73330q^{66}-3373q^{64}+71176q^{62}+100362q^{60}+67117q^{58}-7919q^{56}-76638q^{54}-95921q^{52}-54917q^{50}+19905q^{48}+77417q^{46}+83879q^{44}+37456q^{42}-28664q^{40}-71967q^{38}-66850q^{36}-19669q^{34}+33992q^{32}+61163q^{30}+46783q^{28}+5305q^{26}-34721q^{24}-47784q^{22}-28341q^{20}+5852q^{18}+32118q^{16}+33749q^{14}+13526q^{12}-13663q^{10}-28520q^{8}-22157q^{6}-1010q^{4}+20049q^{2}+25575+12761q^{-2}-10407q^{-4}-26842q^{-6}-24716q^{-8}-3554q^{-10}+23004q^{-12}+35485q^{-14}+23930q^{-16}-7415q^{-18}-37620q^{-20}-45703q^{-22}-21294q^{-24}+21842q^{-26}+54957q^{-28}+54079q^{-30}+14814q^{-32}-38933q^{-34}-72716q^{-36}-58950q^{-38}-3884q^{-40}+58180q^{-42}+86227q^{-44}+58431q^{-46}-9267q^{-48}-75600q^{-50}-94134q^{-52}-52644q^{-54}+23575q^{-56}+86248q^{-58}+94811q^{-60}+44229q^{-62}-34894q^{-64}-89869q^{-66}-88817q^{-68}-33787q^{-70}+39509q^{-72}+85832q^{-74}+78917q^{-76}+24663q^{-78}-38975q^{-80}-75574q^{-82}-65579q^{-84}-19254q^{-86}+33738q^{-88}+62390q^{-90}+52059q^{-92}+15336q^{-94}-25610q^{-96}-47167q^{-98}-40662q^{-100}-12962q^{-102}+17501q^{-104}+33234q^{-106}+29980q^{-108}+11451q^{-110}-9670q^{-112}-22494q^{-114}-21164q^{-116}-9504q^{-118}+4152q^{-120}+13708q^{-122}+14400q^{-124}+7972q^{-126}-1388q^{-128}-7710q^{-130}-9017q^{-132}-6002q^{-134}-348q^{-136}+4107q^{-138}+5613q^{-140}+3770q^{-142}+821q^{-144}-1881q^{-146}-3184q^{-148}-2387q^{-150}-647q^{-152}+988q^{-154}+1497q^{-156}+1301q^{-158}+468q^{-160}-464q^{-162}-779q^{-164}-598q^{-166}-136q^{-168}+137q^{-170}+336q^{-172}+284q^{-174}+40q^{-176}-102q^{-178}-129q^{-180}-57q^{-182}-31q^{-184}+38q^{-186}+60q^{-188}+14q^{-190}-11q^{-192}-17q^{-194}+2q^{-196}-10q^{-198}+11q^{-202}-2q^{-206}-3q^{-208}+3q^{-210}-2q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}+q^{12}-2q^{10}+q^{8}-q^{4}+4q^{2}-1+4q^{-2}-q^{-4}+q^{-8}-2q^{-10}+q^{-12}-q^{-14}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+52q^{36}-86q^{34}+134q^{32}-200q^{30}+273q^{28}-356q^{26}+454q^{24}-534q^{22}+577q^{20}-576q^{18}+508q^{16}-358q^{14}+124q^{12}+156q^{10}-458q^{8}+750q^{6}-996q^{4}+1162q^{2}-1226+1198q^{-2}-1050q^{-4}+834q^{-6}-548q^{-8}+246q^{-10}+58q^{-12}-320q^{-14}+494q^{-16}-598q^{-18}+623q^{-20}-584q^{-22}+494q^{-24}-390q^{-26}+293q^{-28}-202q^{-30}+128q^{-32}-80q^{-34}+46q^{-36}-22q^{-38}+10q^{-40}-4q^{-42}+q^{-44}$
2,0	$q^{38}-q^{36}-q^{34}+2q^{32}-2q^{30}-3q^{28}+4q^{26}+3q^{24}-4q^{22}-3q^{20}+7q^{18}+3q^{16}-12q^{14}+q^{12}+8q^{10}-7q^{8}-6q^{6}+8q^{4}+4q^{2}-4+5q^{-2}+9q^{-4}-3q^{-6}-5q^{-8}+10q^{-10}+q^{-12}-11q^{-14}+4q^{-16}+6q^{-18}-5q^{-20}-6q^{-22}+3q^{-24}+3q^{-26}-4q^{-28}-2q^{-30}+3q^{-32}-q^{-36}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+4q^{28}-7q^{26}+2q^{24}+9q^{22}-15q^{20}+2q^{18}+15q^{16}-21q^{14}-q^{12}+16q^{10}-14q^{8}-2q^{6}+13q^{4}+3q^{2}+2q^{-2}+13q^{-4}-3q^{-6}-14q^{-8}+15q^{-10}-q^{-12}-21q^{-14}+15q^{-16}+2q^{-18}-15q^{-20}+10q^{-22}+2q^{-24}-6q^{-26}+4q^{-28}-2q^{-32}+q^{-34}$
1,0,0	$-q^{17}+q^{15}-3q^{13}+2q^{11}-3q^{9}+2q^{7}-q^{5}+3q^{3}+2q+2q^{-1}+3q^{-3}-q^{-5}+2q^{-7}-3q^{-9}+2q^{-11}-3q^{-13}+q^{-15}-q^{-17}$
1,0,1	$q^{56}-4q^{54}+10q^{52}-14q^{50}+7q^{48}+16q^{46}-49q^{44}+71q^{42}-56q^{40}-13q^{38}+107q^{36}-182q^{34}+175q^{32}-43q^{30}-161q^{28}+356q^{26}-413q^{24}+264q^{22}+51q^{20}-409q^{18}+607q^{16}-567q^{14}+273q^{12}+91q^{10}-359q^{8}+410q^{6}-263q^{4}+82q^{2}+28+54q^{-2}-201q^{-4}+340q^{-6}-272q^{-8}+25q^{-10}+331q^{-12}-591q^{-14}+606q^{-16}-387q^{-18}+6q^{-20}+308q^{-22}-460q^{-24}+390q^{-26}-188q^{-28}-33q^{-30}+176q^{-32}-192q^{-34}+126q^{-36}-29q^{-38}-42q^{-40}+61q^{-42}-46q^{-44}+18q^{-46}+4q^{-48}-10q^{-50}+8q^{-52}-4q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}-q^{38}+4q^{34}-2q^{32}-3q^{30}+8q^{28}-2q^{26}-11q^{24}+5q^{22}+6q^{20}-16q^{18}-9q^{16}+9q^{14}-2q^{12}-17q^{10}+7q^{8}+21q^{6}-8q^{4}+5q^{2}+29+4q^{-2}-11q^{-4}+18q^{-6}+4q^{-8}-19q^{-10}-5q^{-12}+8q^{-14}-8q^{-16}-15q^{-18}+7q^{-20}+7q^{-22}-8q^{-24}-q^{-26}+8q^{-28}-3q^{-30}-2q^{-32}+3q^{-34}-q^{-36}-q^{-38}+q^{-40}$
1,0,0,0	$-q^{20}+q^{18}-3q^{16}+q^{14}-2q^{12}-q^{10}+q^{8}-q^{6}+4q^{4}+q^{2}+5+q^{-2}+4q^{-4}-q^{-6}+q^{-8}-q^{-10}-2q^{-12}+q^{-14}-3q^{-16}+q^{-18}-q^{-20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+8q^{28}-11q^{26}+16q^{24}-21q^{22}+23q^{20}-24q^{18}+21q^{16}-15q^{14}+7q^{12}+4q^{10}-16q^{8}+28q^{6}-37q^{4}+45q^{2}-46+46q^{-2}-37q^{-4}+29q^{-6}-16q^{-8}+5q^{-10}+7q^{-12}-15q^{-14}+21q^{-16}-24q^{-18}+23q^{-20}-22q^{-22}+16q^{-24}-12q^{-26}+8q^{-28}-4q^{-30}+2q^{-32}-q^{-34}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+6q^{46}-q^{44}-9q^{42}-6q^{40}+9q^{38}+15q^{36}-4q^{34}-21q^{32}-9q^{30}+19q^{28}+20q^{26}-11q^{24}-26q^{22}-3q^{20}+23q^{18}+11q^{16}-17q^{14}-15q^{12}+11q^{10}+17q^{8}-4q^{6}-15q^{4}+4q^{2}+18+4q^{-2}-15q^{-4}-4q^{-6}+16q^{-8}+11q^{-10}-15q^{-12}-17q^{-14}+10q^{-16}+23q^{-18}-2q^{-20}-26q^{-22}-12q^{-24}+20q^{-26}+20q^{-28}-9q^{-30}-22q^{-32}-4q^{-34}+16q^{-36}+9q^{-38}-6q^{-40}-9q^{-42}+6q^{-46}+2q^{-48}-2q^{-50}-2q^{-52}+q^{-56}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{46}-2q^{44}+3q^{42}-4q^{40}+7q^{38}-9q^{36}+10q^{34}-13q^{32}+17q^{30}-19q^{28}+17q^{26}-18q^{24}+17q^{22}-16q^{20}+5q^{18}-6q^{16}-2q^{14}+8q^{12}-19q^{10}+22q^{8}-23q^{6}+39q^{4}-30q^{2}+40-30q^{-2}+39q^{-4}-24q^{-6}+22q^{-8}-20q^{-10}+8q^{-12}-3q^{-14}-6q^{-16}+4q^{-18}-16q^{-20}+17q^{-22}-18q^{-24}+18q^{-26}-19q^{-28}+18q^{-30}-13q^{-32}+11q^{-34}-9q^{-36}+7q^{-38}-4q^{-40}+2q^{-42}-2q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-5q^{70}-3q^{68}+16q^{66}-27q^{64}+36q^{62}-38q^{60}+23q^{58}-q^{56}-34q^{54}+70q^{52}-92q^{50}+95q^{48}-66q^{46}+4q^{44}+68q^{42}-130q^{40}+151q^{38}-122q^{36}+46q^{34}+44q^{32}-116q^{30}+138q^{28}-93q^{26}+9q^{24}+77q^{22}-122q^{20}+97q^{18}-19q^{16}-82q^{14}+161q^{12}-175q^{10}+132q^{8}-27q^{6}-94q^{4}+190q^{2}-223+188q^{-2}-92q^{-4}-27q^{-6}+132q^{-8}-176q^{-10}+165q^{-12}-89q^{-14}-10q^{-16}+93q^{-18}-126q^{-20}+87q^{-22}-3q^{-24}-84q^{-26}+137q^{-28}-124q^{-30}+54q^{-32}+40q^{-34}-124q^{-36}+160q^{-38}-142q^{-40}+74q^{-42}+7q^{-44}-77q^{-46}+108q^{-48}-102q^{-50}+72q^{-52}-28q^{-54}-10q^{-56}+32q^{-58}-42q^{-60}+36q^{-62}-23q^{-64}+12q^{-66}-6q^{-70}+7q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

.

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 91"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-4t^{3}+9t^{2}-14t+17-14t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 73, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-6q^{3}+9q^{2}-11q+13-11q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+13z^{4}-5a^{2}z^{2}-5z^{2}a^{-2}+12z^{2}-2a^{2}-2a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+5z^{8}a^{-2}+9z^{8}+4a^{3}z^{7}+az^{7}+2z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-7a^{2}z^{6}-13z^{6}a^{-2}+3z^{6}a^{-4}-26z^{6}+a^{5}z^{5}-6a^{3}z^{5}-7az^{5}-13z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+7a^{2}z^{4}+16z^{4}a^{-2}-6z^{4}a^{-4}+35z^{4}-2a^{5}z^{3}+9az^{3}+18z^{3}a^{-1}+9z^{3}a^{-3}-2z^{3}a^{-5}+2a^{4}z^{2}-7a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-19z^{2}+a^{5}z-4az-6za^{-1}-3za^{-3}+2a^{2}+2a^{-2}+5$

Vassiliev invariants

V2 and V3:

(2, 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 $8$ $0$ $32$ ${\frac {28}{3}}$ $-{\frac {76}{3}}$ $0$ $32$ $-32$ $64$ ${\frac {256}{3}}$ $0$ ${\frac {224}{3}}$ $-{\frac {608}{3}}$ ${\frac {511}{15}}$ ${\frac {812}{5}}$ $-{\frac {13556}{45}}$ $-{\frac {175}{9}}$ $-{\frac {1649}{15}}$

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=$0 is the signature of 10 91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	2	3	4	5	χ
11											1	-1
9										2		2
7									4	1		-3
5								5	2			3
3							6	4				-2
1						7	5					2
-1					5	7						2
-3				4	6							-2
-5			2	5								3
-7		1	4									-3
-9		2										2
-11	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, 91]]
Out[2]=	10
In[3]:=	PD[Knot[10, 91]]
Out[3]=	PD[X[6, 2, 7, 1], X[20, 6, 1, 5], X[16, 9, 17, 10], X[10, 3, 11, 4], X[2, 18, 3, 17], X[14, 7, 15, 8], X[8, 15, 9, 16], X[12, 20, 13, 19], X[18, 12, 19, 11], X[4, 13, 5, 14]]
In[4]:=	GaussCode[Knot[10, 91]]
Out[4]=	GaussCode[1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5, -9, 8, -2]
In[5]:=	BR[Knot[10, 91]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, 2, 2, -1, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 91]][t]
Out[6]=	-4 4 9 14 2 3 4 17 + t - -- + -- - -- - 14 t + 9 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 91]][z]
Out[7]=	2 4 6 8 1 + 2 z + 5 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 91]}
In[9]:=	{KnotDet[Knot[10, 91]], KnotSignature[Knot[10, 91]]}
Out[9]=	{73, 0}
In[10]:=	J=Jones[Knot[10, 91]][q]
Out[10]=	-5 3 6 9 11 2 3 4 5 13 - q + -- - -- + -- - -- - 11 q + 9 q - 6 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 43], Knot[10, 91]}
In[12]:=	A2Invariant[Knot[10, 91]][q]
Out[12]=	-14 -12 2 -8 -4 4 2 4 8 10 -1 - q + q - --- + q - q + -- + 4 q - q + q - 2 q + 10 2 q q 12 14 q - q
In[13]:=	Kauffman[Knot[10, 91]][a, z]
Out[13]=	2 2 2 2 3 z 6 z 5 2 z 9 z 5 + -- + 2 a - --- - --- - 4 a z + a z - 19 z + -- - ---- - 2 3 a 4 2 a a a a 3 3 3 2 2 4 2 2 z 9 z 18 z 3 5 3 4 7 a z + 2 a z - ---- + ---- + ----- + 9 a z - 2 a z + 35 z - 5 3 a a a 4 4 5 5 5 6 z 16 z 2 4 4 4 z 12 z 13 z 5 ---- + ----- + 7 a z - 6 a z + -- - ----- - ----- - 7 a z - 4 2 5 3 a a a a a 6 6 7 3 5 5 5 6 3 z 13 z 2 6 4 6 5 z 6 a z + a z - 26 z + ---- - ----- - 7 a z + 3 a z + ---- + 4 2 3 a a a 7 8 9 2 z 7 3 7 8 5 z 2 8 2 z 9 ---- + a z + 4 a z + 9 z + ---- + 4 a z + ---- + 2 a z a 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 91]], Vassiliev[3][Knot[10, 91]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 91]][q, t]
Out[15]=	7 1 2 1 4 2 5 4 - + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 6 5 3 3 2 5 2 5 3 7 3 ---- + --- + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + 3 q t q t 7 4 9 4 11 5 q t + 2 q t + q t