10 79

10_78

10_80

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10 79 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X8493 X12,6,13,5 X18,13,19,14 X16,9,17,10 X10,17,11,18 X20,15,1,16 X14,19,15,20 X2837 X4,12,5,11
Gauss code	1, -9, 2, -10, 3, -1, 9, -2, 5, -6, 10, -3, 4, -8, 7, -5, 6, -4, 8, -7
Dowker-Thistlethwaite code	6 8 12 2 16 4 18 20 10 14
Conway Notation	[(3,2)(3,2)]

Three dimensional invariants

Symmetry type Negative amphicheiral Unknotting number $\{2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 12.5403 A-Polynomial See Data:10 79/A-polynomial

[edit Notes for 10 79'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 79's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-3t^{3}+7t^{2}-12t+15-12t^{-1}+7t^{-2}-3t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+5z^{6}+9z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 61, 0 }
Jones polynomial	$-q^{5}+2q^{4}-5q^{3}+8q^{2}-9q+11-9q^{-1}+8q^{-2}-5q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+7z^{6}-5a^{2}z^{4}-5z^{4}a^{-2}+19z^{4}-9a^{2}z^{2}-9z^{2}a^{-2}+23z^{2}-5a^{2}-5a^{-2}+11$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+3a^{3}z^{7}+4az^{7}+4z^{7}a^{-1}+3z^{7}a^{-3}+2a^{4}z^{6}-7a^{2}z^{6}-7z^{6}a^{-2}+2z^{6}a^{-4}-18z^{6}+a^{5}z^{5}-6a^{3}z^{5}-15az^{5}-15z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}-4a^{4}z^{4}+12a^{2}z^{4}+12z^{4}a^{-2}-4z^{4}a^{-4}+32z^{4}-3a^{5}z^{3}+4a^{3}z^{3}+22az^{3}+22z^{3}a^{-1}+4z^{3}a^{-3}-3z^{3}a^{-5}+a^{4}z^{2}-13a^{2}z^{2}-13z^{2}a^{-2}+z^{2}a^{-4}-28z^{2}+2a^{5}z-2a^{3}z-11az-11za^{-1}-2za^{-3}+2za^{-5}+5a^{2}+5a^{-2}+11$
The A2 invariant	$-q^{14}-3q^{10}+5q^{2}+1+5q^{-2}-3q^{-10}-q^{-14}$
The G2 invariant	$q^{80}-q^{78}+3q^{76}-4q^{74}+4q^{72}-3q^{70}-q^{68}+8q^{66}-15q^{64}+21q^{62}-23q^{60}+16q^{58}-4q^{56}-18q^{54}+44q^{52}-63q^{50}+64q^{48}-47q^{46}+q^{44}+45q^{42}-88q^{40}+101q^{38}-82q^{36}+29q^{34}+29q^{32}-79q^{30}+87q^{28}-58q^{26}+4q^{24}+50q^{22}-75q^{20}+60q^{18}-7q^{16}-53q^{14}+105q^{12}-113q^{10}+85q^{8}-15q^{6}-59q^{4}+126q^{2}-143+126q^{-2}-59q^{-4}-15q^{-6}+85q^{-8}-113q^{-10}+105q^{-12}-53q^{-14}-7q^{-16}+60q^{-18}-75q^{-20}+50q^{-22}+4q^{-24}-58q^{-26}+87q^{-28}-79q^{-30}+29q^{-32}+29q^{-34}-82q^{-36}+101q^{-38}-88q^{-40}+45q^{-42}+q^{-44}-47q^{-46}+64q^{-48}-63q^{-50}+44q^{-52}-18q^{-54}-4q^{-56}+16q^{-58}-23q^{-60}+21q^{-62}-15q^{-64}+8q^{-66}-q^{-68}-3q^{-70}+4q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_79.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{9}-3q^{7}+3q^{5}-q^{3}+2q+2q^{-1}-q^{-3}+3q^{-5}-3q^{-7}+q^{-9}-q^{-11}$
2	$q^{32}-q^{30}+4q^{26}-5q^{24}-4q^{22}+12q^{20}-7q^{18}-14q^{16}+18q^{14}-19q^{10}+13q^{8}+8q^{6}-12q^{4}+2q^{2}+11+2q^{-2}-12q^{-4}+8q^{-6}+13q^{-8}-19q^{-10}+18q^{-14}-14q^{-16}-7q^{-18}+12q^{-20}-4q^{-22}-5q^{-24}+4q^{-26}-q^{-30}+q^{-32}$
3	$-q^{63}+q^{61}-q^{57}-q^{55}+4q^{53}+2q^{51}-7q^{49}-5q^{47}+15q^{45}+14q^{43}-19q^{41}-33q^{39}+20q^{37}+55q^{35}-9q^{33}-76q^{31}-21q^{29}+91q^{27}+46q^{25}-86q^{23}-78q^{21}+70q^{19}+91q^{17}-45q^{15}-98q^{13}+23q^{11}+87q^{9}+8q^{7}-70q^{5}-27q^{3}+52q+52q^{-1}-27q^{-3}-70q^{-5}+8q^{-7}+87q^{-9}+23q^{-11}-98q^{-13}-45q^{-15}+91q^{-17}+70q^{-19}-78q^{-21}-86q^{-23}+46q^{-25}+91q^{-27}-21q^{-29}-76q^{-31}-9q^{-33}+55q^{-35}+20q^{-37}-33q^{-39}-19q^{-41}+14q^{-43}+15q^{-45}-5q^{-47}-7q^{-49}+2q^{-51}+4q^{-53}-q^{-55}-q^{-57}+q^{-61}-q^{-63}$
4	$q^{104}-q^{102}+q^{98}-2q^{96}+2q^{94}-3q^{92}+q^{90}+6q^{88}-7q^{86}-q^{84}-12q^{82}+6q^{80}+33q^{78}+2q^{76}-14q^{74}-66q^{72}-23q^{70}+85q^{68}+91q^{66}+46q^{64}-158q^{62}-192q^{60}+20q^{58}+220q^{56}+307q^{54}-74q^{52}-402q^{50}-304q^{48}+115q^{46}+599q^{44}+302q^{42}-334q^{40}-635q^{38}-276q^{36}+565q^{34}+644q^{32}+28q^{30}-629q^{28}-593q^{26}+245q^{24}+644q^{22}+325q^{20}-350q^{18}-590q^{16}-59q^{14}+404q^{12}+399q^{10}-64q^{8}-411q^{6}-239q^{4}+158q^{2}+385+158q^{-2}-239q^{-4}-411q^{-6}-64q^{-8}+399q^{-10}+404q^{-12}-59q^{-14}-590q^{-16}-350q^{-18}+325q^{-20}+644q^{-22}+245q^{-24}-593q^{-26}-629q^{-28}+28q^{-30}+644q^{-32}+565q^{-34}-276q^{-36}-635q^{-38}-334q^{-40}+302q^{-42}+599q^{-44}+115q^{-46}-304q^{-48}-402q^{-50}-74q^{-52}+307q^{-54}+220q^{-56}+20q^{-58}-192q^{-60}-158q^{-62}+46q^{-64}+91q^{-66}+85q^{-68}-23q^{-70}-66q^{-72}-14q^{-74}+2q^{-76}+33q^{-78}+6q^{-80}-12q^{-82}-q^{-84}-7q^{-86}+6q^{-88}+q^{-90}-3q^{-92}+2q^{-94}-2q^{-96}+q^{-98}-q^{-102}+q^{-104}$
5	$-q^{155}+q^{153}-q^{149}+2q^{147}+q^{145}-3q^{143}+q^{139}-2q^{137}+5q^{135}+9q^{133}-5q^{131}-12q^{129}-13q^{127}-9q^{125}+17q^{123}+47q^{121}+36q^{119}-20q^{117}-84q^{115}-110q^{113}-31q^{111}+120q^{109}+232q^{107}+174q^{105}-82q^{103}-376q^{101}-436q^{99}-124q^{97}+417q^{95}+797q^{93}+587q^{91}-230q^{89}-1087q^{87}-1252q^{85}-365q^{83}+1081q^{81}+1983q^{79}+1368q^{77}-589q^{75}-2447q^{73}-2568q^{71}-506q^{69}+2363q^{67}+3692q^{65}+1995q^{63}-1644q^{61}-4278q^{59}-3550q^{57}+293q^{55}+4214q^{53}+4767q^{51}+1252q^{49}-3436q^{47}-5320q^{45}-2715q^{43}+2227q^{41}+5174q^{39}+3668q^{37}-874q^{35}-4452q^{33}-4047q^{31}-264q^{29}+3401q^{27}+3869q^{25}+1072q^{23}-2321q^{21}-3380q^{19}-1467q^{17}+1398q^{15}+2745q^{13}+1651q^{11}-705q^{9}-2228q^{7}-1719q^{5}+218q^{3}+1879q+1879q^{-1}+218q^{-3}-1719q^{-5}-2228q^{-7}-705q^{-9}+1651q^{-11}+2745q^{-13}+1398q^{-15}-1467q^{-17}-3380q^{-19}-2321q^{-21}+1072q^{-23}+3869q^{-25}+3401q^{-27}-264q^{-29}-4047q^{-31}-4452q^{-33}-874q^{-35}+3668q^{-37}+5174q^{-39}+2227q^{-41}-2715q^{-43}-5320q^{-45}-3436q^{-47}+1252q^{-49}+4767q^{-51}+4214q^{-53}+293q^{-55}-3550q^{-57}-4278q^{-59}-1644q^{-61}+1995q^{-63}+3692q^{-65}+2363q^{-67}-506q^{-69}-2568q^{-71}-2447q^{-73}-589q^{-75}+1368q^{-77}+1983q^{-79}+1081q^{-81}-365q^{-83}-1252q^{-85}-1087q^{-87}-230q^{-89}+587q^{-91}+797q^{-93}+417q^{-95}-124q^{-97}-436q^{-99}-376q^{-101}-82q^{-103}+174q^{-105}+232q^{-107}+120q^{-109}-31q^{-111}-110q^{-113}-84q^{-115}-20q^{-117}+36q^{-119}+47q^{-121}+17q^{-123}-9q^{-125}-13q^{-127}-12q^{-129}-5q^{-131}+9q^{-133}+5q^{-135}-2q^{-137}+q^{-139}-3q^{-143}+q^{-145}+2q^{-147}-q^{-149}+q^{-153}-q^{-155}$
6	$q^{216}-q^{214}+q^{210}-2q^{208}-q^{206}+6q^{202}-2q^{200}-5q^{198}+3q^{196}-5q^{194}-4q^{192}+2q^{190}+23q^{188}+9q^{186}-14q^{184}-8q^{182}-31q^{180}-33q^{178}-5q^{176}+79q^{174}+90q^{172}+41q^{170}-133q^{166}-221q^{164}-192q^{162}+77q^{160}+318q^{158}+424q^{156}+397q^{154}-38q^{152}-612q^{150}-1014q^{148}-724q^{146}+18q^{144}+1001q^{142}+1846q^{140}+1590q^{138}+215q^{136}-1824q^{134}-3053q^{132}-2931q^{130}-871q^{128}+2618q^{126}+5189q^{124}+5175q^{122}+1689q^{120}-3379q^{118}-7953q^{116}-8542q^{114}-3498q^{112}+4686q^{110}+11716q^{108}+12553q^{106}+6128q^{104}-6006q^{102}-16477q^{100}-17997q^{98}-8512q^{96}+7925q^{94}+21510q^{92}+24060q^{90}+10906q^{88}-10682q^{86}-27701q^{84}-29020q^{82}-12009q^{80}+13861q^{78}+33801q^{76}+32854q^{74}+11153q^{72}-18682q^{70}-37998q^{68}-33953q^{66}-8484q^{64}+23696q^{62}+40371q^{60}+31753q^{58}+3049q^{56}-27098q^{54}-39538q^{52}-26785q^{50}+3146q^{48}+28958q^{46}+35370q^{44}+18968q^{42}-8079q^{40}-28122q^{38}-28875q^{36}-10823q^{34}+11680q^{32}+24759q^{30}+20638q^{28}+4229q^{26}-13088q^{24}-20018q^{22}-13027q^{20}+972q^{18}+12802q^{16}+14704q^{14}+7084q^{12}-4556q^{10}-11920q^{8}-10508q^{6}-2176q^{4}+7427q^{2}+11325+7427q^{-2}-2176q^{-4}-10508q^{-6}-11920q^{-8}-4556q^{-10}+7084q^{-12}+14704q^{-14}+12802q^{-16}+972q^{-18}-13027q^{-20}-20018q^{-22}-13088q^{-24}+4229q^{-26}+20638q^{-28}+24759q^{-30}+11680q^{-32}-10823q^{-34}-28875q^{-36}-28122q^{-38}-8079q^{-40}+18968q^{-42}+35370q^{-44}+28958q^{-46}+3146q^{-48}-26785q^{-50}-39538q^{-52}-27098q^{-54}+3049q^{-56}+31753q^{-58}+40371q^{-60}+23696q^{-62}-8484q^{-64}-33953q^{-66}-37998q^{-68}-18682q^{-70}+11153q^{-72}+32854q^{-74}+33801q^{-76}+13861q^{-78}-12009q^{-80}-29020q^{-82}-27701q^{-84}-10682q^{-86}+10906q^{-88}+24060q^{-90}+21510q^{-92}+7925q^{-94}-8512q^{-96}-17997q^{-98}-16477q^{-100}-6006q^{-102}+6128q^{-104}+12553q^{-106}+11716q^{-108}+4686q^{-110}-3498q^{-112}-8542q^{-114}-7953q^{-116}-3379q^{-118}+1689q^{-120}+5175q^{-122}+5189q^{-124}+2618q^{-126}-871q^{-128}-2931q^{-130}-3053q^{-132}-1824q^{-134}+215q^{-136}+1590q^{-138}+1846q^{-140}+1001q^{-142}+18q^{-144}-724q^{-146}-1014q^{-148}-612q^{-150}-38q^{-152}+397q^{-154}+424q^{-156}+318q^{-158}+77q^{-160}-192q^{-162}-221q^{-164}-133q^{-166}+41q^{-170}+90q^{-172}+79q^{-174}-5q^{-176}-33q^{-178}-31q^{-180}-8q^{-182}-14q^{-184}+9q^{-186}+23q^{-188}+2q^{-190}-4q^{-192}-5q^{-194}+3q^{-196}-5q^{-198}-2q^{-200}+6q^{-202}-q^{-206}-2q^{-208}+q^{-210}-q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}-3q^{10}+5q^{2}+1+5q^{-2}-3q^{-10}-q^{-14}$
1,1	$q^{44}-2q^{42}+6q^{40}-12q^{38}+25q^{36}-42q^{34}+68q^{32}-106q^{30}+155q^{28}-214q^{26}+276q^{24}-332q^{22}+363q^{20}-368q^{18}+312q^{16}-224q^{14}+67q^{12}+102q^{10}-294q^{8}+476q^{6}-610q^{4}+724q^{2}-734+724q^{-2}-610q^{-4}+476q^{-6}-294q^{-8}+102q^{-10}+67q^{-12}-224q^{-14}+312q^{-16}-368q^{-18}+363q^{-20}-332q^{-22}+276q^{-24}-214q^{-26}+155q^{-28}-106q^{-30}+68q^{-32}-42q^{-34}+25q^{-36}-12q^{-38}+6q^{-40}-2q^{-42}+q^{-44}$
2,0	$q^{38}+q^{34}+3q^{32}-2q^{28}+3q^{26}-7q^{22}-7q^{20}+q^{18}-q^{16}-13q^{14}-2q^{12}+5q^{10}-4q^{8}-q^{6}+12q^{4}+11q^{2}+6+11q^{-2}+12q^{-4}-q^{-6}-4q^{-8}+5q^{-10}-2q^{-12}-13q^{-14}-q^{-16}+q^{-18}-7q^{-20}-7q^{-22}+3q^{-26}-2q^{-28}+3q^{-32}+q^{-34}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-q^{32}+q^{30}+3q^{28}-4q^{26}+q^{24}+7q^{22}-12q^{20}+9q^{16}-20q^{14}-5q^{12}+7q^{10}-13q^{8}-q^{6}+15q^{4}+10q^{2}+10+10q^{-2}+15q^{-4}-q^{-6}-13q^{-8}+7q^{-10}-5q^{-12}-20q^{-14}+9q^{-16}-12q^{-20}+7q^{-22}+q^{-24}-4q^{-26}+3q^{-28}+q^{-30}-q^{-32}+q^{-34}$
1,0,0	$-q^{17}-4q^{13}-4q^{9}+q^{7}+5q^{3}+5q+5q^{-1}+5q^{-3}+q^{-7}-4q^{-9}-4q^{-13}-q^{-17}$
1,0,1	$q^{56}-2q^{54}+5q^{52}-5q^{50}+2q^{48}+10q^{46}-24q^{44}+34q^{42}-24q^{40}-14q^{38}+67q^{36}-113q^{34}+108q^{32}-22q^{30}-107q^{28}+245q^{26}-276q^{24}+187q^{22}+8q^{20}-265q^{18}+360q^{16}-401q^{14}+173q^{12}+9q^{10}-205q^{8}+256q^{6}-120q^{4}+85q^{2}+71+85q^{-2}-120q^{-4}+256q^{-6}-205q^{-8}+9q^{-10}+173q^{-12}-401q^{-14}+360q^{-16}-265q^{-18}+8q^{-20}+187q^{-22}-276q^{-24}+245q^{-26}-107q^{-28}-22q^{-30}+108q^{-32}-113q^{-34}+67q^{-36}-14q^{-38}-24q^{-40}+34q^{-42}-24q^{-44}+10q^{-46}+2q^{-48}-5q^{-50}+5q^{-52}-2q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}+q^{36}+4q^{34}+q^{32}+8q^{28}-q^{26}-8q^{24}+q^{22}-2q^{20}-22q^{18}-19q^{16}-7q^{14}-15q^{12}-21q^{10}+3q^{8}+23q^{6}+9q^{4}+26q^{2}+46+26q^{-2}+9q^{-4}+23q^{-6}+3q^{-8}-21q^{-10}-15q^{-12}-7q^{-14}-19q^{-16}-22q^{-18}-2q^{-20}+q^{-22}-8q^{-24}-q^{-26}+8q^{-28}+q^{-32}+4q^{-34}+q^{-36}+q^{-40}$
1,0,0,0	$-q^{20}-4q^{16}-q^{14}-4q^{12}-3q^{10}+6q^{4}+5q^{2}+9+5q^{-2}+6q^{-4}-3q^{-10}-4q^{-12}-q^{-14}-4q^{-16}-q^{-20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+q^{32}-3q^{30}+5q^{28}-8q^{26}+11q^{24}-15q^{22}+16q^{20}-18q^{18}+15q^{16}-12q^{14}+5q^{12}+3q^{10}-11q^{8}+21q^{6}-25q^{4}+34q^{2}-32+34q^{-2}-25q^{-4}+21q^{-6}-11q^{-8}+3q^{-10}+5q^{-12}-12q^{-14}+15q^{-16}-18q^{-18}+16q^{-20}-15q^{-22}+11q^{-24}-8q^{-26}+5q^{-28}-3q^{-30}+q^{-32}-q^{-34}$
1,0	$q^{56}-q^{52}-q^{50}+2q^{48}+4q^{46}-6q^{42}-4q^{40}+6q^{38}+11q^{36}-3q^{34}-16q^{32}-8q^{30}+13q^{28}+13q^{26}-10q^{24}-21q^{22}-4q^{20}+15q^{18}+6q^{16}-13q^{14}-11q^{12}+9q^{10}+13q^{8}-8q^{4}+7q^{2}+17+7q^{-2}-8q^{-4}+13q^{-8}+9q^{-10}-11q^{-12}-13q^{-14}+6q^{-16}+15q^{-18}-4q^{-20}-21q^{-22}-10q^{-24}+13q^{-26}+13q^{-28}-8q^{-30}-16q^{-32}-3q^{-34}+11q^{-36}+6q^{-38}-4q^{-40}-6q^{-42}+4q^{-46}+2q^{-48}-q^{-50}-q^{-52}+q^{-56}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{46}-q^{44}+2q^{42}-2q^{40}+5q^{38}-6q^{36}+7q^{34}-9q^{32}+12q^{30}-14q^{28}+12q^{26}-14q^{24}+11q^{22}-15q^{20}-q^{18}-10q^{16}-7q^{14}+q^{12}-17q^{10}+16q^{8}-12q^{6}+36q^{4}-11q^{2}+40-11q^{-2}+36q^{-4}-12q^{-6}+16q^{-8}-17q^{-10}+q^{-12}-7q^{-14}-10q^{-16}-q^{-18}-15q^{-20}+11q^{-22}-14q^{-24}+12q^{-26}-14q^{-28}+12q^{-30}-9q^{-32}+7q^{-34}-6q^{-36}+5q^{-38}-2q^{-40}+2q^{-42}-q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-q^{78}+3q^{76}-4q^{74}+4q^{72}-3q^{70}-q^{68}+8q^{66}-15q^{64}+21q^{62}-23q^{60}+16q^{58}-4q^{56}-18q^{54}+44q^{52}-63q^{50}+64q^{48}-47q^{46}+q^{44}+45q^{42}-88q^{40}+101q^{38}-82q^{36}+29q^{34}+29q^{32}-79q^{30}+87q^{28}-58q^{26}+4q^{24}+50q^{22}-75q^{20}+60q^{18}-7q^{16}-53q^{14}+105q^{12}-113q^{10}+85q^{8}-15q^{6}-59q^{4}+126q^{2}-143+126q^{-2}-59q^{-4}-15q^{-6}+85q^{-8}-113q^{-10}+105q^{-12}-53q^{-14}-7q^{-16}+60q^{-18}-75q^{-20}+50q^{-22}+4q^{-24}-58q^{-26}+87q^{-28}-79q^{-30}+29q^{-32}+29q^{-34}-82q^{-36}+101q^{-38}-88q^{-40}+45q^{-42}+q^{-44}-47q^{-46}+64q^{-48}-63q^{-50}+44q^{-52}-18q^{-54}-4q^{-56}+16q^{-58}-23q^{-60}+21q^{-62}-15q^{-64}+8q^{-66}-q^{-68}-3q^{-70}+4q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

.

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 79"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-3t^{3}+7t^{2}-12t+15-12t^{-1}+7t^{-2}-3t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+5z^{6}+9z^{4}+5z^{2}+1$

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]=

{ 61, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+7z^{6}-5a^{2}z^{4}-5z^{4}a^{-2}+19z^{4}-9a^{2}z^{2}-9z^{2}a^{-2}+23z^{2}-5a^{2}-5a^{-2}+11$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+3a^{3}z^{7}+4az^{7}+4z^{7}a^{-1}+3z^{7}a^{-3}+2a^{4}z^{6}-7a^{2}z^{6}-7z^{6}a^{-2}+2z^{6}a^{-4}-18z^{6}+a^{5}z^{5}-6a^{3}z^{5}-15az^{5}-15z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}-4a^{4}z^{4}+12a^{2}z^{4}+12z^{4}a^{-2}-4z^{4}a^{-4}+32z^{4}-3a^{5}z^{3}+4a^{3}z^{3}+22az^{3}+22z^{3}a^{-1}+4z^{3}a^{-3}-3z^{3}a^{-5}+a^{4}z^{2}-13a^{2}z^{2}-13z^{2}a^{-2}+z^{2}a^{-4}-28z^{2}+2a^{5}z-2a^{3}z-11az-11za^{-1}-2za^{-3}+2za^{-5}+5a^{2}+5a^{-2}+11$

Vassiliev invariants

V2 and V3:

(5, 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 $20$ $0$ $200$ ${\frac {694}{3}}$ ${\frac {74}{3}}$ $0$ $0$ $0$ $0$ ${\frac {4000}{3}}$ $0$ ${\frac {13880}{3}}$ ${\frac {1480}{3}}$ ${\frac {27247}{6}}$ ${\frac {1154}{3}}$ ${\frac {11678}{9}}$ ${\frac {1045}{18}}$ ${\frac {559}{6}}$

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 79. 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                   1   1 7                 4 1   -3 5               4 1     3 3             5 4       -1 1           6 4         2 -1         4 6           2 -3       4 5             -1 -5     1 4               3 -7   1 4                 -3 -9   1                   1 -11 1                     -1

Integral Khovanov Homology (db, data source)

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

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, 79]]
Out[2]=	10
In[3]:=	PD[Knot[10, 79]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[18, 13, 19, 14], X[16, 9, 17, 10], X[10, 17, 11, 18], X[20, 15, 1, 16], X[14, 19, 15, 20], X[2, 8, 3, 7], X[4, 12, 5, 11]]
In[4]:=	GaussCode[Knot[10, 79]]
Out[4]=	GaussCode[1, -9, 2, -10, 3, -1, 9, -2, 5, -6, 10, -3, 4, -8, 7, -5, 6, -4, 8, -7]
In[5]:=	BR[Knot[10, 79]]
Out[5]=	BR[3, {-1, -1, -1, 2, 2, -1, -1, 2, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 79]][t]
Out[6]=	-4 3 7 12 2 3 4 15 + t - -- + -- - -- - 12 t + 7 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 79]][z]
Out[7]=	2 4 6 8 1 + 5 z + 9 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 79]}
In[9]:=	{KnotDet[Knot[10, 79]], KnotSignature[Knot[10, 79]]}
Out[9]=	{61, 0}
In[10]:=	J=Jones[Knot[10, 79]][q]
Out[10]=	-5 2 5 8 9 2 3 4 5 11 - q + -- - -- + -- - - - 9 q + 8 q - 5 q + 2 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, 79]}
In[12]:=	A2Invariant[Knot[10, 79]][q]
Out[12]=	-14 3 5 2 10 14 1 - q - --- + -- + 5 q - 3 q - q 10 2 q q
In[13]:=	Kauffman[Knot[10, 79]][a, z]
Out[13]=	5 2 2 z 2 z 11 z 3 5 2 11 + -- + 5 a + --- - --- - ---- - 11 a z - 2 a z + 2 a z - 28 z + 2 5 3 a a a a 2 2 3 3 3 z 13 z 2 2 4 2 3 z 4 z 22 z 3 -- - ----- - 13 a z + a z - ---- + ---- + ----- + 22 a z + 4 2 5 3 a a a a a 4 4 5 3 3 5 3 4 4 z 12 z 2 4 4 4 z 4 a z - 3 a z + 32 z - ---- + ----- + 12 a z - 4 a z + -- - 4 2 5 a a a 5 5 6 6 6 z 15 z 5 3 5 5 5 6 2 z 7 z ---- - ----- - 15 a z - 6 a z + a z - 18 z + ---- - ---- - 3 a 4 2 a a a 7 7 8 2 6 4 6 3 z 4 z 7 3 7 8 3 z 7 a z + 2 a z + ---- + ---- + 4 a z + 3 a z + 6 z + ---- + 3 a 2 a a 9 2 8 z 9 3 a z + -- + a z a
In[14]:=	{Vassiliev[2][Knot[10, 79]], Vassiliev[3][Knot[10, 79]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 79]][q, t]
Out[15]=	6 1 1 1 4 1 4 4 - + 6 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 5 4 3 3 2 5 2 5 3 7 3 ---- + --- + 4 q t + 5 q t + 4 q t + 4 q t + q t + 4 q t + 3 q t q t 7 4 9 4 11 5 q t + q t + q t