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]

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

A Morse Link Presentation:

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a257, ...}

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

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)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{15}-4q^{14}+3q^{13}+11q^{12}-27q^{11}+8q^{10}+52q^{9}-76q^{8}-8q^{7}+130q^{6}-122q^{5}-55q^{4}+208q^{3}-134q^{2}-107q+241-107q^{-1}-134q^{-2}+208q^{-3}-55q^{-4}-122q^{-5}+130q^{-6}-8q^{-7}-76q^{-8}+52q^{-9}+8q^{-10}-27q^{-11}+11q^{-12}+3q^{-13}-4q^{-14}+q^{-15}$
3	$-q^{30}+4q^{29}-3q^{28}-6q^{27}+4q^{26}+18q^{25}-9q^{24}-48q^{23}+21q^{22}+99q^{21}-14q^{20}-194q^{19}-26q^{18}+323q^{17}+129q^{16}-466q^{15}-307q^{14}+582q^{13}+562q^{12}-650q^{11}-851q^{10}+639q^{9}+1148q^{8}-565q^{7}-1402q^{6}+430q^{5}+1603q^{4}-274q^{3}-1714q^{2}+84q+1769+84q^{-1}-1714q^{-2}-274q^{-3}+1603q^{-4}+430q^{-5}-1402q^{-6}-565q^{-7}+1148q^{-8}+639q^{-9}-851q^{-10}-650q^{-11}+562q^{-12}+582q^{-13}-307q^{-14}-466q^{-15}+129q^{-16}+323q^{-17}-26q^{-18}-194q^{-19}-14q^{-20}+99q^{-21}+21q^{-22}-48q^{-23}-9q^{-24}+18q^{-25}+4q^{-26}-6q^{-27}-3q^{-28}+4q^{-29}-q^{-30}$
4	$q^{50}-4q^{49}+3q^{48}+6q^{47}-9q^{46}+5q^{45}-17q^{44}+22q^{43}+31q^{42}-60q^{41}-4q^{40}-61q^{39}+131q^{38}+193q^{37}-192q^{36}-199q^{35}-372q^{34}+386q^{33}+907q^{32}-7q^{31}-645q^{30}-1686q^{29}+78q^{28}+2318q^{27}+1584q^{26}-286q^{25}-4171q^{24}-2229q^{23}+2995q^{22}+4722q^{21}+2593q^{20}-6078q^{19}-6548q^{18}+1037q^{17}+7450q^{16}+7776q^{15}-5461q^{14}-10670q^{13}-3245q^{12}+7881q^{11}+12898q^{10}-2626q^{9}-12675q^{8}-7693q^{7}+6233q^{6}+16074q^{5}+739q^{4}-12520q^{3}-10838q^{2}+3699q+17069+3699q^{-1}-10838q^{-2}-12520q^{-3}+739q^{-4}+16074q^{-5}+6233q^{-6}-7693q^{-7}-12675q^{-8}-2626q^{-9}+12898q^{-10}+7881q^{-11}-3245q^{-12}-10670q^{-13}-5461q^{-14}+7776q^{-15}+7450q^{-16}+1037q^{-17}-6548q^{-18}-6078q^{-19}+2593q^{-20}+4722q^{-21}+2995q^{-22}-2229q^{-23}-4171q^{-24}-286q^{-25}+1584q^{-26}+2318q^{-27}+78q^{-28}-1686q^{-29}-645q^{-30}-7q^{-31}+907q^{-32}+386q^{-33}-372q^{-34}-199q^{-35}-192q^{-36}+193q^{-37}+131q^{-38}-61q^{-39}-4q^{-40}-60q^{-41}+31q^{-42}+22q^{-43}-17q^{-44}+5q^{-45}-9q^{-46}+6q^{-47}+3q^{-48}-4q^{-49}+q^{-50}$
5	$-q^{75}+4q^{74}-3q^{73}-6q^{72}+9q^{71}-6q^{69}+4q^{68}-5q^{67}-9q^{66}+37q^{65}+29q^{64}-48q^{63}-83q^{62}-68q^{61}+46q^{60}+244q^{59}+301q^{58}-24q^{57}-573q^{56}-787q^{55}-287q^{54}+875q^{53}+1843q^{52}+1364q^{51}-921q^{50}-3428q^{49}-3602q^{48}-259q^{47}+4964q^{46}+7568q^{45}+3700q^{44}-5394q^{43}-12667q^{42}-10365q^{41}+2685q^{40}+17588q^{39}+20463q^{38}+4813q^{37}-19877q^{36}-32656q^{35}-18124q^{34}+16897q^{33}+44345q^{32}+36630q^{31}-6744q^{30}-52443q^{29}-57942q^{28}-10715q^{27}+54076q^{26}+78732q^{25}+34017q^{24}-48179q^{23}-95785q^{22}-59901q^{21}+35233q^{20}+106740q^{19}+85226q^{18}-17417q^{17}-111131q^{16}-107011q^{15}-2543q^{14}+109646q^{13}+123904q^{12}+22010q^{11}-104034q^{10}-135442q^{9}-39509q^{8}+96005q^{7}+142679q^{6}+54100q^{5}-86907q^{4}-146180q^{3}-66504q^{2}+77050q+147507+77050q^{-1}-66504q^{-2}-146180q^{-3}-86907q^{-4}+54100q^{-5}+142679q^{-6}+96005q^{-7}-39509q^{-8}-135442q^{-9}-104034q^{-10}+22010q^{-11}+123904q^{-12}+109646q^{-13}-2543q^{-14}-107011q^{-15}-111131q^{-16}-17417q^{-17}+85226q^{-18}+106740q^{-19}+35233q^{-20}-59901q^{-21}-95785q^{-22}-48179q^{-23}+34017q^{-24}+78732q^{-25}+54076q^{-26}-10715q^{-27}-57942q^{-28}-52443q^{-29}-6744q^{-30}+36630q^{-31}+44345q^{-32}+16897q^{-33}-18124q^{-34}-32656q^{-35}-19877q^{-36}+4813q^{-37}+20463q^{-38}+17588q^{-39}+2685q^{-40}-10365q^{-41}-12667q^{-42}-5394q^{-43}+3700q^{-44}+7568q^{-45}+4964q^{-46}-259q^{-47}-3602q^{-48}-3428q^{-49}-921q^{-50}+1364q^{-51}+1843q^{-52}+875q^{-53}-287q^{-54}-787q^{-55}-573q^{-56}-24q^{-57}+301q^{-58}+244q^{-59}+46q^{-60}-68q^{-61}-83q^{-62}-48q^{-63}+29q^{-64}+37q^{-65}-9q^{-66}-5q^{-67}+4q^{-68}-6q^{-69}+9q^{-71}-6q^{-72}-3q^{-73}+4q^{-74}-q^{-75}$
6	$q^{105}-4q^{104}+3q^{103}+6q^{102}-9q^{101}+q^{99}+19q^{98}-21q^{97}-17q^{96}+32q^{95}-45q^{94}+8q^{93}+50q^{92}+128q^{91}-41q^{90}-167q^{89}-62q^{88}-286q^{87}-16q^{86}+387q^{85}+900q^{84}+404q^{83}-424q^{82}-881q^{81}-2094q^{80}-1401q^{79}+650q^{78}+3938q^{77}+4458q^{76}+2396q^{75}-1204q^{74}-8305q^{73}-10766q^{72}-6703q^{71}+5801q^{70}+16461q^{69}+20757q^{68}+14405q^{67}-9709q^{66}-33262q^{65}-42976q^{64}-22056q^{63}+15948q^{62}+58065q^{61}+77950q^{60}+42156q^{59}-29551q^{58}-104768q^{57}-122544q^{56}-71254q^{55}+47281q^{54}+171007q^{53}+198422q^{52}+103304q^{51}-89064q^{50}-253004q^{49}-297475q^{48}-141664q^{47}+149780q^{46}+383153q^{45}+408243q^{44}+154486q^{43}-230153q^{42}-546632q^{41}-530354q^{40}-143910q^{39}+380359q^{38}+726609q^{37}+615539q^{36}+96521q^{35}-581428q^{34}-920446q^{33}-660120q^{32}+60354q^{31}+813628q^{30}+1064305q^{29}+635918q^{28}-301929q^{27}-1072972q^{26}-1148980q^{25}-451384q^{24}+602132q^{23}+1278685q^{22}+1132696q^{21}+144123q^{20}-950647q^{19}-1413224q^{18}-910924q^{17}+247709q^{16}+1245688q^{15}+1420940q^{14}+538462q^{13}-703247q^{12}-1458570q^{11}-1189250q^{10}-66498q^{9}+1100658q^{8}+1522082q^{7}+790633q^{6}-474803q^{5}-1403113q^{4}-1322934q^{3}-286649q^{2}+950870q+1538859+950870q^{-1}-286649q^{-2}-1322934q^{-3}-1403113q^{-4}-474803q^{-5}+790633q^{-6}+1522082q^{-7}+1100658q^{-8}-66498q^{-9}-1189250q^{-10}-1458570q^{-11}-703247q^{-12}+538462q^{-13}+1420940q^{-14}+1245688q^{-15}+247709q^{-16}-910924q^{-17}-1413224q^{-18}-950647q^{-19}+144123q^{-20}+1132696q^{-21}+1278685q^{-22}+602132q^{-23}-451384q^{-24}-1148980q^{-25}-1072972q^{-26}-301929q^{-27}+635918q^{-28}+1064305q^{-29}+813628q^{-30}+60354q^{-31}-660120q^{-32}-920446q^{-33}-581428q^{-34}+96521q^{-35}+615539q^{-36}+726609q^{-37}+380359q^{-38}-143910q^{-39}-530354q^{-40}-546632q^{-41}-230153q^{-42}+154486q^{-43}+408243q^{-44}+383153q^{-45}+149780q^{-46}-141664q^{-47}-297475q^{-48}-253004q^{-49}-89064q^{-50}+103304q^{-51}+198422q^{-52}+171007q^{-53}+47281q^{-54}-71254q^{-55}-122544q^{-56}-104768q^{-57}-29551q^{-58}+42156q^{-59}+77950q^{-60}+58065q^{-61}+15948q^{-62}-22056q^{-63}-42976q^{-64}-33262q^{-65}-9709q^{-66}+14405q^{-67}+20757q^{-68}+16461q^{-69}+5801q^{-70}-6703q^{-71}-10766q^{-72}-8305q^{-73}-1204q^{-74}+2396q^{-75}+4458q^{-76}+3938q^{-77}+650q^{-78}-1401q^{-79}-2094q^{-80}-881q^{-81}-424q^{-82}+404q^{-83}+900q^{-84}+387q^{-85}-16q^{-86}-286q^{-87}-62q^{-88}-167q^{-89}-41q^{-90}+128q^{-91}+50q^{-92}+8q^{-93}-45q^{-94}+32q^{-95}-17q^{-96}-21q^{-97}+19q^{-98}+q^{-99}-9q^{-101}+6q^{-102}+3q^{-103}-4q^{-104}+q^{-105}$
7	$-q^{140}+4q^{139}-3q^{138}-6q^{137}+9q^{136}-q^{134}-14q^{133}-2q^{132}+43q^{131}-6q^{130}-24q^{129}+8q^{128}-27q^{127}-23q^{126}-69q^{125}-8q^{124}+242q^{123}+163q^{122}+35q^{121}-72q^{120}-385q^{119}-411q^{118}-518q^{117}-142q^{116}+1078q^{115}+1554q^{114}+1474q^{113}+483q^{112}-1739q^{111}-3318q^{110}-4454q^{109}-3310q^{108}+1813q^{107}+7114q^{106}+11157q^{105}+10245q^{104}+1633q^{103}-9940q^{102}-22433q^{101}-27507q^{100}-16520q^{99}+6677q^{98}+37057q^{97}+58407q^{96}+52468q^{95}+18370q^{94}-40973q^{93}-100446q^{92}-122513q^{91}-89098q^{90}+7781q^{89}+133531q^{88}+223614q^{87}+228198q^{86}+109054q^{85}-106363q^{84}-325147q^{83}-443178q^{82}-356501q^{81}-55497q^{80}+349552q^{79}+689017q^{78}+753321q^{77}+439152q^{76}-173905q^{75}-857942q^{74}-1254288q^{73}-1089251q^{72}-333822q^{71}+773753q^{70}+1714946q^{69}+1964892q^{68}+1265562q^{67}-240851q^{66}-1909001q^{65}-2902350q^{64}-2597945q^{63}-884016q^{62}+1573715q^{61}+3625400q^{60}+4160778q^{59}+2617311q^{58}-502842q^{57}-3814742q^{56}-5650023q^{55}-4799031q^{54}-1364682q^{53}+3202185q^{52}+6697142q^{51}+7115131q^{50}+3902842q^{49}-1670439q^{48}-6984275q^{47}-9173314q^{46}-6807311q^{45}-696117q^{44}+6335550q^{43}+10610793q^{42}+9681809q^{41}+3632710q^{40}-4775784q^{39}-11201786q^{38}-12142587q^{37}-6760636q^{36}+2515244q^{35}+10900762q^{34}+13915945q^{33}+9700418q^{32}+121154q^{31}-9843831q^{30}-14895676q^{29}-12155512q^{28}-2782605q^{27}+8283604q^{26}+15135646q^{25}+13970837q^{24}+5181766q^{23}-6511733q^{22}-14808343q^{21}-15135897q^{20}-7142924q^{19}+4783786q^{18}+14138213q^{17}+15752305q^{16}+8612048q^{15}-3272853q^{14}-13335658q^{13}-15980950q^{12}-9641991q^{11}+2050425q^{10}+12561147q^{9}+15997522q^{8}+10346608q^{7}-1103802q^{6}-11896387q^{5}-15942378q^{4}-10871239q^{3}+344203q^{2}+11352146q+15915633+11352146q^{-1}+344203q^{-2}-10871239q^{-3}-15942378q^{-4}-11896387q^{-5}-1103802q^{-6}+10346608q^{-7}+15997522q^{-8}+12561147q^{-9}+2050425q^{-10}-9641991q^{-11}-15980950q^{-12}-13335658q^{-13}-3272853q^{-14}+8612048q^{-15}+15752305q^{-16}+14138213q^{-17}+4783786q^{-18}-7142924q^{-19}-15135897q^{-20}-14808343q^{-21}-6511733q^{-22}+5181766q^{-23}+13970837q^{-24}+15135646q^{-25}+8283604q^{-26}-2782605q^{-27}-12155512q^{-28}-14895676q^{-29}-9843831q^{-30}+121154q^{-31}+9700418q^{-32}+13915945q^{-33}+10900762q^{-34}+2515244q^{-35}-6760636q^{-36}-12142587q^{-37}-11201786q^{-38}-4775784q^{-39}+3632710q^{-40}+9681809q^{-41}+10610793q^{-42}+6335550q^{-43}-696117q^{-44}-6807311q^{-45}-9173314q^{-46}-6984275q^{-47}-1670439q^{-48}+3902842q^{-49}+7115131q^{-50}+6697142q^{-51}+3202185q^{-52}-1364682q^{-53}-4799031q^{-54}-5650023q^{-55}-3814742q^{-56}-502842q^{-57}+2617311q^{-58}+4160778q^{-59}+3625400q^{-60}+1573715q^{-61}-884016q^{-62}-2597945q^{-63}-2902350q^{-64}-1909001q^{-65}-240851q^{-66}+1265562q^{-67}+1964892q^{-68}+1714946q^{-69}+773753q^{-70}-333822q^{-71}-1089251q^{-72}-1254288q^{-73}-857942q^{-74}-173905q^{-75}+439152q^{-76}+753321q^{-77}+689017q^{-78}+349552q^{-79}-55497q^{-80}-356501q^{-81}-443178q^{-82}-325147q^{-83}-106363q^{-84}+109054q^{-85}+228198q^{-86}+223614q^{-87}+133531q^{-88}+7781q^{-89}-89098q^{-90}-122513q^{-91}-100446q^{-92}-40973q^{-93}+18370q^{-94}+52468q^{-95}+58407q^{-96}+37057q^{-97}+6677q^{-98}-16520q^{-99}-27507q^{-100}-22433q^{-101}-9940q^{-102}+1633q^{-103}+10245q^{-104}+11157q^{-105}+7114q^{-106}+1813q^{-107}-3310q^{-108}-4454q^{-109}-3318q^{-110}-1739q^{-111}+483q^{-112}+1474q^{-113}+1554q^{-114}+1078q^{-115}-142q^{-116}-518q^{-117}-411q^{-118}-385q^{-119}-72q^{-120}+35q^{-121}+163q^{-122}+242q^{-123}-8q^{-124}-69q^{-125}-23q^{-126}-27q^{-127}+8q^{-128}-24q^{-129}-6q^{-130}+43q^{-131}-2q^{-132}-14q^{-133}-q^{-134}+9q^{-136}-6q^{-137}-3q^{-138}+4q^{-139}-q^{-140}$

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, 118]]
Out[2]=	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[3]:=	GaussCode[Knot[10, 118]]
Out[3]=	GaussCode[1, -7, 6, -10, 2, -1, 5, -6, 9, -8, 10, -4, 3, -5, 7, -9, 8, -2, 4, -3]
In[4]:=	DTCode[Knot[10, 118]]
Out[4]=	DTCode[6, 8, 18, 14, 16, 4, 20, 2, 10, 12]
In[5]:=	br = BR[Knot[10, 118]]
Out[5]=	BR[3, {1, 1, -2, 1, -2, 1, -2, -2, 1, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 118]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 118]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 118]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{NegativeAmphicheiral, 1, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 118]][t]
Out[10]=	-4 5 12 19 2 3 4 23 + t - -- + -- - -- - 19 t + 12 t - 5 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 118]][z]
Out[11]=	4 6 8 1 + 2 z + 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 118], Knot[11, Alternating, 257]}
In[13]:=	{KnotDet[Knot[10, 118]], KnotSignature[Knot[10, 118]]}
Out[13]=	{97, 0}
In[14]:=	Jones[Knot[10, 118]][q]
Out[14]=	-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[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 118]}
In[16]:=	A2Invariant[Knot[10, 118]][q]
Out[16]=	-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[17]:=	HOMFLYPT[Knot[10, 118]][a, z]
Out[17]=	2 4 6 2 2 z 2 2 4 3 z 2 4 6 z 2 6 1 + 4 z - ---- - 2 a z + 8 z - ---- - 3 a z + 5 z - -- - a z + 2 2 2 a a a 8 z
In[18]:=	Kauffman[Knot[10, 118]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[10, 118]], Vassiliev[3][Knot[10, 118]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 118]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[10, 118], 2][q]
Out[21]=	-15 4 3 11 27 8 52 76 8 130 122 241 + q - --- + --- + --- - --- + --- + -- - -- - -- + --- - --- - 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q 55 208 134 107 2 3 4 5 -- + --- - --- - --- - 107 q - 134 q + 208 q - 55 q - 122 q + 4 3 2 q q q q 6 7 8 9 10 11 12 13 130 q - 8 q - 76 q + 52 q + 8 q - 27 q + 11 q + 3 q - 14 15 4 q + q

See/edit the Rolfsen_Splice_Template.