10 118

10_117

10_119

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10 118 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X18,6,19,5 X20,13,1,14 X12,19,13,20 X14,7,15,8 X8394 X2,16,3,15 X10,18,11,17 X16,10,17,9 X4,11,5,12
Gauss code	1, -7, 6, -10, 2, -1, 5, -6, 9, -8, 10, -4, 3, -5, 7, -9, 8, -2, 4, -3
Dowker-Thistlethwaite code	6 8 18 14 16 4 20 2 10 12
Conway Notation	[8*2:.2]

Three dimensional invariants

Symmetry type Negative amphicheiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 15.5452 A-Polynomial See Data:10 118/A-polynomial

[edit Notes for 10 118'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 10 118's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+12t^{2}-19t+23-19t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 97, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+12q^{2}-15q+17-15q^{-1}+12q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-3z^{4}a^{-2}+8z^{4}-2a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}+1$
Kauffman polynomial (db, data sources)	$3az^{9}+3z^{9}a^{-1}+7a^{2}z^{8}+7z^{8}a^{-2}+14z^{8}+7a^{3}z^{7}+6az^{7}+6z^{7}a^{-1}+7z^{7}a^{-3}+4a^{4}z^{6}-11a^{2}z^{6}-11z^{6}a^{-2}+4z^{6}a^{-4}-30z^{6}+a^{5}z^{5}-12a^{3}z^{5}-20az^{5}-20z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+6a^{2}z^{4}+6z^{4}a^{-2}-6z^{4}a^{-4}+24z^{4}-a^{5}z^{3}+5a^{3}z^{3}+15az^{3}+15z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+a^{4}z^{2}-2a^{2}z^{2}-2z^{2}a^{-2}+z^{2}a^{-4}-6z^{2}-a^{3}z-3az-3za^{-1}-za^{-3}+1$
The A2 invariant	$-q^{14}+2q^{12}-2q^{10}+2q^{8}-2q^{4}+4q^{2}-3+4q^{-2}-2q^{-4}+2q^{-8}-2q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+15q^{72}-13q^{70}+2q^{68}+22q^{66}-48q^{64}+77q^{62}-93q^{60}+75q^{58}-27q^{56}-60q^{54}+162q^{52}-237q^{50}+259q^{48}-188q^{46}+27q^{44}+173q^{42}-342q^{40}+401q^{38}-319q^{36}+115q^{34}+129q^{32}-313q^{30}+362q^{28}-237q^{26}+12q^{24}+212q^{22}-326q^{20}+260q^{18}-55q^{16}-209q^{14}+412q^{12}-458q^{10}+343q^{8}-79q^{6}-234q^{4}+477q^{2}-571+477q^{-2}-234q^{-4}-79q^{-6}+343q^{-8}-458q^{-10}+412q^{-12}-209q^{-14}-55q^{-16}+260q^{-18}-326q^{-20}+212q^{-22}+12q^{-24}-237q^{-26}+362q^{-28}-313q^{-30}+129q^{-32}+115q^{-34}-319q^{-36}+401q^{-38}-342q^{-40}+173q^{-42}+27q^{-44}-188q^{-46}+259q^{-48}-237q^{-50}+162q^{-52}-60q^{-54}-27q^{-56}+75q^{-58}-93q^{-60}+77q^{-62}-48q^{-64}+22q^{-66}+2q^{-68}-13q^{-70}+15q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_118.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+3q^{9}-4q^{7}+4q^{5}-3q^{3}+2q+2q^{-1}-3q^{-3}+4q^{-5}-4q^{-7}+3q^{-9}-q^{-11}$
2	$q^{32}-3q^{30}+10q^{26}-13q^{24}-8q^{22}+33q^{20}-16q^{18}-32q^{16}+46q^{14}-47q^{10}+31q^{8}+19q^{6}-33q^{4}+27-33q^{-4}+19q^{-6}+31q^{-8}-47q^{-10}+46q^{-14}-32q^{-16}-16q^{-18}+33q^{-20}-8q^{-22}-13q^{-24}+10q^{-26}-3q^{-30}+q^{-32}$
3	$-q^{63}+3q^{61}-6q^{57}-q^{55}+13q^{53}+7q^{51}-35q^{49}-18q^{47}+63q^{45}+58q^{43}-88q^{41}-135q^{39}+89q^{37}+232q^{35}-40q^{33}-321q^{31}-62q^{29}+371q^{27}+187q^{25}-357q^{23}-300q^{21}+286q^{19}+371q^{17}-180q^{15}-389q^{13}+66q^{11}+357q^{9}+45q^{7}-301q^{5}-135q^{3}+223q+223q^{-1}-135q^{-3}-301q^{-5}+45q^{-7}+357q^{-9}+66q^{-11}-389q^{-13}-180q^{-15}+371q^{-17}+286q^{-19}-300q^{-21}-357q^{-23}+187q^{-25}+371q^{-27}-62q^{-29}-321q^{-31}-40q^{-33}+232q^{-35}+89q^{-37}-135q^{-39}-88q^{-41}+58q^{-43}+63q^{-45}-18q^{-47}-35q^{-49}+7q^{-51}+13q^{-53}-q^{-55}-6q^{-57}+3q^{-61}-q^{-63}$
5	$-q^{155}+3q^{153}-6q^{149}+3q^{147}+3q^{145}-2q^{143}-2q^{141}-4q^{139}-7q^{137}+21q^{135}+50q^{133}+8q^{131}-79q^{129}-142q^{127}-87q^{125}+120q^{123}+392q^{121}+416q^{119}-74q^{117}-793q^{115}-1126q^{113}-495q^{111}+1047q^{109}+2435q^{107}+2087q^{105}-554q^{103}-3869q^{101}-5003q^{99}-1882q^{97}+4322q^{95}+8943q^{93}+6977q^{91}-2088q^{89}-12194q^{87}-14473q^{85}-4453q^{83}+12310q^{81}+22517q^{79}+15307q^{77}-6984q^{75}-27793q^{73}-28484q^{71}-4602q^{69}+27215q^{67}+40348q^{65}+20561q^{63}-19257q^{61}-46869q^{59}-37138q^{57}+4964q^{55}+45725q^{53}+49989q^{51}+12146q^{49}-37040q^{47}-55883q^{45}-27875q^{43}+23334q^{41}+54096q^{39}+38750q^{37}-8360q^{35}-46136q^{33}-43230q^{31}-4552q^{29}+34875q^{27}+41972q^{25}+13541q^{23}-23425q^{21}-37066q^{19}-18291q^{17}+13799q^{15}+30926q^{13}+20188q^{11}-6807q^{9}-25762q^{7}-20934q^{5}+2016q^{3}+22419q+22419q^{-1}+2016q^{-3}-20934q^{-5}-25762q^{-7}-6807q^{-9}+20188q^{-11}+30926q^{-13}+13799q^{-15}-18291q^{-17}-37066q^{-19}-23425q^{-21}+13541q^{-23}+41972q^{-25}+34875q^{-27}-4552q^{-29}-43230q^{-31}-46136q^{-33}-8360q^{-35}+38750q^{-37}+54096q^{-39}+23334q^{-41}-27875q^{-43}-55883q^{-45}-37040q^{-47}+12146q^{-49}+49989q^{-51}+45725q^{-53}+4964q^{-55}-37138q^{-57}-46869q^{-59}-19257q^{-61}+20561q^{-63}+40348q^{-65}+27215q^{-67}-4602q^{-69}-28484q^{-71}-27793q^{-73}-6984q^{-75}+15307q^{-77}+22517q^{-79}+12310q^{-81}-4453q^{-83}-14473q^{-85}-12194q^{-87}-2088q^{-89}+6977q^{-91}+8943q^{-93}+4322q^{-95}-1882q^{-97}-5003q^{-99}-3869q^{-101}-554q^{-103}+2087q^{-105}+2435q^{-107}+1047q^{-109}-495q^{-111}-1126q^{-113}-793q^{-115}-74q^{-117}+416q^{-119}+392q^{-121}+120q^{-123}-87q^{-125}-142q^{-127}-79q^{-129}+8q^{-131}+50q^{-133}+21q^{-135}-7q^{-137}-4q^{-139}-2q^{-141}-2q^{-143}+3q^{-145}+3q^{-147}-6q^{-149}+3q^{-153}-q^{-155}$
6	$q^{216}-3q^{214}+6q^{210}-3q^{208}-3q^{206}-2q^{204}+16q^{202}-q^{200}-21q^{198}+5q^{196}-31q^{194}-23q^{192}+26q^{190}+135q^{188}+115q^{186}-35q^{184}-129q^{182}-370q^{180}-394q^{178}-57q^{176}+715q^{174}+1160q^{172}+903q^{170}+84q^{168}-1724q^{166}-3109q^{164}-2846q^{162}+192q^{160}+4246q^{158}+7066q^{156}+6743q^{154}+532q^{152}-8833q^{150}-16186q^{148}-14323q^{146}-2320q^{144}+16041q^{142}+31650q^{140}+30246q^{138}+7750q^{136}-28523q^{134}-56380q^{132}-56893q^{130}-19585q^{128}+43960q^{126}+95825q^{124}+99536q^{122}+37744q^{120}-62744q^{118}-149946q^{116}-160730q^{114}-67673q^{112}+88593q^{110}+221448q^{108}+237152q^{106}+106692q^{104}-119529q^{102}-308474q^{100}-329701q^{98}-144970q^{96}+159969q^{94}+403519q^{92}+426370q^{90}+177213q^{88}-211477q^{86}-505167q^{84}-507961q^{82}-189595q^{80}+271458q^{78}+598132q^{76}+563336q^{74}+173244q^{72}-342966q^{70}-662971q^{68}-575952q^{66}-127186q^{64}+412211q^{62}+691710q^{60}+539184q^{58}+50324q^{56}-461414q^{54}-672910q^{52}-458530q^{50}+38248q^{48}+484300q^{46}+606625q^{44}+342381q^{42}-117159q^{40}-471582q^{38}-504579q^{36}-216335q^{34}+178004q^{32}+425404q^{30}+380058q^{28}+101732q^{26}-212475q^{24}-357505q^{22}-256363q^{20}-4192q^{18}+224252q^{16}+279709q^{14}+146025q^{12}-74126q^{10}-223914q^{8}-207137q^{6}-46900q^{4}+141254q^{2}+221433+141254q^{-2}-46900q^{-4}-207137q^{-6}-223914q^{-8}-74126q^{-10}+146025q^{-12}+279709q^{-14}+224252q^{-16}-4192q^{-18}-256363q^{-20}-357505q^{-22}-212475q^{-24}+101732q^{-26}+380058q^{-28}+425404q^{-30}+178004q^{-32}-216335q^{-34}-504579q^{-36}-471582q^{-38}-117159q^{-40}+342381q^{-42}+606625q^{-44}+484300q^{-46}+38248q^{-48}-458530q^{-50}-672910q^{-52}-461414q^{-54}+50324q^{-56}+539184q^{-58}+691710q^{-60}+412211q^{-62}-127186q^{-64}-575952q^{-66}-662971q^{-68}-342966q^{-70}+173244q^{-72}+563336q^{-74}+598132q^{-76}+271458q^{-78}-189595q^{-80}-507961q^{-82}-505167q^{-84}-211477q^{-86}+177213q^{-88}+426370q^{-90}+403519q^{-92}+159969q^{-94}-144970q^{-96}-329701q^{-98}-308474q^{-100}-119529q^{-102}+106692q^{-104}+237152q^{-106}+221448q^{-108}+88593q^{-110}-67673q^{-112}-160730q^{-114}-149946q^{-116}-62744q^{-118}+37744q^{-120}+99536q^{-122}+95825q^{-124}+43960q^{-126}-19585q^{-128}-56893q^{-130}-56380q^{-132}-28523q^{-134}+7750q^{-136}+30246q^{-138}+31650q^{-140}+16041q^{-142}-2320q^{-144}-14323q^{-146}-16186q^{-148}-8833q^{-150}+532q^{-152}+6743q^{-154}+7066q^{-156}+4246q^{-158}+192q^{-160}-2846q^{-162}-3109q^{-164}-1724q^{-166}+84q^{-168}+903q^{-170}+1160q^{-172}+715q^{-174}-57q^{-176}-394q^{-178}-370q^{-180}-129q^{-182}-35q^{-184}+115q^{-186}+135q^{-188}+26q^{-190}-23q^{-192}-31q^{-194}+5q^{-196}-21q^{-198}-q^{-200}+16q^{-202}-2q^{-204}-3q^{-206}-3q^{-208}+6q^{-210}-3q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}+2q^{12}-2q^{10}+2q^{8}-2q^{4}+4q^{2}-3+4q^{-2}-2q^{-4}+2q^{-8}-2q^{-10}+2q^{-12}-q^{-14}$
1,1	$q^{44}-6q^{42}+20q^{40}-50q^{38}+105q^{36}-196q^{34}+336q^{32}-542q^{30}+804q^{28}-1110q^{26}+1446q^{24}-1736q^{22}+1906q^{20}-1892q^{18}+1646q^{16}-1140q^{14}+379q^{12}+542q^{10}-1518q^{8}+2440q^{6}-3194q^{4}+3692q^{2}-3858+3692q^{-2}-3194q^{-4}+2440q^{-6}-1518q^{-8}+542q^{-10}+379q^{-12}-1140q^{-14}+1646q^{-16}-1892q^{-18}+1906q^{-20}-1736q^{-22}+1446q^{-24}-1110q^{-26}+804q^{-28}-542q^{-30}+336q^{-32}-196q^{-34}+105q^{-36}-50q^{-38}+20q^{-40}-6q^{-42}+q^{-44}$
2,0	$q^{38}-2q^{36}-q^{34}+5q^{32}-4q^{30}-6q^{28}+8q^{26}+6q^{24}-8q^{22}-6q^{20}+13q^{18}+7q^{16}-21q^{14}+2q^{12}+16q^{10}-12q^{8}-9q^{6}+13q^{4}+6q^{2}-10+6q^{-2}+13q^{-4}-9q^{-6}-12q^{-8}+16q^{-10}+2q^{-12}-21q^{-14}+7q^{-16}+13q^{-18}-6q^{-20}-8q^{-22}+6q^{-24}+8q^{-26}-6q^{-28}-4q^{-30}+5q^{-32}-q^{-34}-2q^{-36}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-3q^{32}+q^{30}+7q^{28}-12q^{26}+3q^{24}+18q^{22}-27q^{20}+4q^{18}+30q^{16}-35q^{14}+2q^{12}+31q^{10}-25q^{8}-5q^{6}+19q^{4}-2q^{2}-8-2q^{-2}+19q^{-4}-5q^{-6}-25q^{-8}+31q^{-10}+2q^{-12}-35q^{-14}+30q^{-16}+4q^{-18}-27q^{-20}+18q^{-22}+3q^{-24}-12q^{-26}+7q^{-28}+q^{-30}-3q^{-32}+q^{-34}$
1,0,0	$-q^{17}+2q^{15}-3q^{13}+4q^{11}-3q^{9}+3q^{7}-2q^{5}+2q^{3}+2q^{-3}-2q^{-5}+3q^{-7}-3q^{-9}+4q^{-11}-3q^{-13}+2q^{-15}-q^{-17}$
1,0,1	$q^{56}-6q^{54}+17q^{52}-27q^{50}+16q^{48}+36q^{46}-120q^{44}+182q^{42}-140q^{40}-66q^{38}+374q^{36}-615q^{34}+575q^{32}-129q^{30}-594q^{28}+1268q^{26}-1479q^{24}+995q^{22}+75q^{20}-1276q^{18}+2014q^{16}-1920q^{14}+1030q^{12}+162q^{10}-1046q^{8}+1230q^{6}-780q^{4}+170q^{2}+121+170q^{-2}-780q^{-4}+1230q^{-6}-1046q^{-8}+162q^{-10}+1030q^{-12}-1920q^{-14}+2014q^{-16}-1276q^{-18}+75q^{-20}+995q^{-22}-1479q^{-24}+1268q^{-26}-594q^{-28}-129q^{-30}+575q^{-32}-615q^{-34}+374q^{-36}-66q^{-38}-140q^{-40}+182q^{-42}-120q^{-44}+36q^{-46}+16q^{-48}-27q^{-50}+17q^{-52}-6q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}-2q^{38}-q^{36}+6q^{34}-5q^{32}-6q^{30}+15q^{28}-3q^{26}-18q^{24}+14q^{22}+16q^{20}-23q^{18}-8q^{16}+25q^{14}-30q^{10}+11q^{8}+30q^{6}-29q^{4}-8q^{2}+40-8q^{-2}-29q^{-4}+30q^{-6}+11q^{-8}-30q^{-10}+25q^{-14}-8q^{-16}-23q^{-18}+16q^{-20}+14q^{-22}-18q^{-24}-3q^{-26}+15q^{-28}-6q^{-30}-5q^{-32}+6q^{-34}-q^{-36}-2q^{-38}+q^{-40}$
1,0,0,0	$-q^{20}+2q^{18}-3q^{16}+3q^{14}-q^{12}+2q^{8}-2q^{6}+3q^{4}-2q^{2}+3-2q^{-2}+3q^{-4}-2q^{-6}+2q^{-8}-q^{-12}+3q^{-14}-3q^{-16}+2q^{-18}-q^{-20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+3q^{32}-7q^{30}+13q^{28}-20q^{26}+29q^{24}-38q^{22}+43q^{20}-44q^{18}+40q^{16}-29q^{14}+14q^{12}+7q^{10}-29q^{8}+51q^{6}-69q^{4}+82q^{2}-86+82q^{-2}-69q^{-4}+51q^{-6}-29q^{-8}+7q^{-10}+14q^{-12}-29q^{-14}+40q^{-16}-44q^{-18}+43q^{-20}-38q^{-22}+29q^{-24}-20q^{-26}+13q^{-28}-7q^{-30}+3q^{-32}-q^{-34}$
1,0	$q^{56}-3q^{52}-3q^{50}+4q^{48}+10q^{46}-16q^{42}-12q^{40}+16q^{38}+28q^{36}-5q^{34}-39q^{32}-17q^{30}+36q^{28}+38q^{26}-19q^{24}-47q^{22}-3q^{20}+44q^{18}+20q^{16}-32q^{14}-28q^{12}+20q^{10}+30q^{8}-10q^{6}-30q^{4}+4q^{2}+31+4q^{-2}-30q^{-4}-10q^{-6}+30q^{-8}+20q^{-10}-28q^{-12}-32q^{-14}+20q^{-16}+44q^{-18}-3q^{-20}-47q^{-22}-19q^{-24}+38q^{-26}+36q^{-28}-17q^{-30}-39q^{-32}-5q^{-34}+28q^{-36}+16q^{-38}-12q^{-40}-16q^{-42}+10q^{-46}+4q^{-48}-3q^{-50}-3q^{-52}+q^{-56}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{46}-3q^{44}+4q^{42}-6q^{40}+11q^{38}-16q^{36}+19q^{34}-23q^{32}+31q^{30}-35q^{28}+33q^{26}-33q^{24}+34q^{22}-27q^{20}+14q^{18}-8q^{16}+18q^{12}-33q^{10}+39q^{8}-48q^{6}+64q^{4}-64q^{2}+64-64q^{-2}+64q^{-4}-48q^{-6}+39q^{-8}-33q^{-10}+18q^{-12}-8q^{-16}+14q^{-18}-27q^{-20}+34q^{-22}-33q^{-24}+33q^{-26}-35q^{-28}+31q^{-30}-23q^{-32}+19q^{-34}-16q^{-36}+11q^{-38}-6q^{-40}+4q^{-42}-3q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-3q^{78}+7q^{76}-13q^{74}+15q^{72}-13q^{70}+2q^{68}+22q^{66}-48q^{64}+77q^{62}-93q^{60}+75q^{58}-27q^{56}-60q^{54}+162q^{52}-237q^{50}+259q^{48}-188q^{46}+27q^{44}+173q^{42}-342q^{40}+401q^{38}-319q^{36}+115q^{34}+129q^{32}-313q^{30}+362q^{28}-237q^{26}+12q^{24}+212q^{22}-326q^{20}+260q^{18}-55q^{16}-209q^{14}+412q^{12}-458q^{10}+343q^{8}-79q^{6}-234q^{4}+477q^{2}-571+477q^{-2}-234q^{-4}-79q^{-6}+343q^{-8}-458q^{-10}+412q^{-12}-209q^{-14}-55q^{-16}+260q^{-18}-326q^{-20}+212q^{-22}+12q^{-24}-237q^{-26}+362q^{-28}-313q^{-30}+129q^{-32}+115q^{-34}-319q^{-36}+401q^{-38}-342q^{-40}+173q^{-42}+27q^{-44}-188q^{-46}+259q^{-48}-237q^{-50}+162q^{-52}-60q^{-54}-27q^{-56}+75q^{-58}-93q^{-60}+77q^{-62}-48q^{-64}+22q^{-66}+2q^{-68}-13q^{-70}+15q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

.

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["10 118"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+12t^{2}-19t+23-19t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$

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]=

{ 97, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+12q^{2}-15q+17-15q^{-1}+12q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3az^{9}+3z^{9}a^{-1}+7a^{2}z^{8}+7z^{8}a^{-2}+14z^{8}+7a^{3}z^{7}+6az^{7}+6z^{7}a^{-1}+7z^{7}a^{-3}+4a^{4}z^{6}-11a^{2}z^{6}-11z^{6}a^{-2}+4z^{6}a^{-4}-30z^{6}+a^{5}z^{5}-12a^{3}z^{5}-20az^{5}-20z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+6a^{2}z^{4}+6z^{4}a^{-2}-6z^{4}a^{-4}+24z^{4}-a^{5}z^{3}+5a^{3}z^{3}+15az^{3}+15z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+a^{4}z^{2}-2a^{2}z^{2}-2z^{2}a^{-2}+z^{2}a^{-4}-6z^{2}-a^{3}z-3az-3za^{-1}-za^{-3}+1$

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 $0$ $0$ $0$ $-16$ $-16$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $-8$ $-{\frac {224}{3}}$ ${\frac {448}{3}}$ $-{\frac {248}{3}}$ $24$

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 10 118. 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                   3   3 7                 5 1   -4 5               7 3     4 3             8 5       -3 1           9 7         2 -1         7 9           2 -3       5 8             -3 -5     3 7               4 -7   1 5                 -4 -9   3                   3 -11 1                     -1

Integral Khovanov Homology (db, data source)

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

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, 118]]
Out[2]=	10
In[3]:=	PD[Knot[10, 118]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 6, 19, 5], X[20, 13, 1, 14], X[12, 19, 13, 20], X[14, 7, 15, 8], X[8, 3, 9, 4], X[2, 16, 3, 15], X[10, 18, 11, 17], X[16, 10, 17, 9], X[4, 11, 5, 12]]
In[4]:=	GaussCode[Knot[10, 118]]
Out[4]=	GaussCode[1, -7, 6, -10, 2, -1, 5, -6, 9, -8, 10, -4, 3, -5, 7, -9, 8, -2, 4, -3]
In[5]:=	BR[Knot[10, 118]]
Out[5]=	BR[3, {1, 1, -2, 1, -2, 1, -2, -2, 1, -2}]
In[6]:=	alex = Alexander[Knot[10, 118]][t]
Out[6]=	-4 5 12 19 2 3 4 23 + t - -- + -- - -- - 19 t + 12 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 118]][z]
Out[7]=	4 6 8 1 + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 118], Knot[11, Alternating, 257]}
In[9]:=	{KnotDet[Knot[10, 118]], KnotSignature[Knot[10, 118]]}
Out[9]=	{97, 0}
In[10]:=	J=Jones[Knot[10, 118]][q]
Out[10]=	-5 4 8 12 15 2 3 4 5 17 - q + -- - -- + -- - -- - 15 q + 12 q - 8 q + 4 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, 118]}
In[12]:=	A2Invariant[Knot[10, 118]][q]
Out[12]=	-14 2 2 2 2 4 2 4 8 10 -3 - q + --- - --- + -- - -- + -- + 4 q - 2 q + 2 q - 2 q + 12 10 8 4 2 q q q q q 12 14 2 q - q
In[13]:=	Kauffman[Knot[10, 118]][a, z]
Out[13]=	2 2 3 z 3 z 3 2 z 2 z 2 2 4 2 z 1 - -- - --- - 3 a z - a z - 6 z + -- - ---- - 2 a z + a z - -- + 3 a 4 2 5 a a a a 3 3 4 4 5 z 15 z 3 3 3 5 3 4 6 z 6 z ---- + ----- + 15 a z + 5 a z - a z + 24 z - ---- + ---- + 3 a 4 2 a a a 5 5 5 2 4 4 4 z 12 z 20 z 5 3 5 5 5 6 a z - 6 a z + -- - ----- - ----- - 20 a z - 12 a z + a z - 5 3 a a a 6 6 7 7 6 4 z 11 z 2 6 4 6 7 z 6 z 7 30 z + ---- - ----- - 11 a z + 4 a z + ---- + ---- + 6 a z + 4 2 3 a a a a 8 9 3 7 8 7 z 2 8 3 z 9 7 a z + 14 z + ---- + 7 a z + ---- + 3 a z 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 118]], Vassiliev[3][Knot[10, 118]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 118]][q, t]
Out[15]=	9 1 3 1 5 3 7 5 - + 9 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 8 7 3 3 2 5 2 5 3 7 3 ---- + --- + 7 q t + 8 q t + 5 q t + 7 q t + 3 q t + 5 q t + 3 q t q t 7 4 9 4 11 5 q t + 3 q t + q t