10 118

10_117

10_119

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10 118 Quick Notes

10 118 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_18,6,19,5 X_20,13,1,14 X_12,19,13,20 X_14,7,15,8 X₈₃₉₄ X_2,16,3,15 X_10,18,11,17 X_16,10,17,9 X_4,11,5,12
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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8-a^2 z^6-z^6 a^{-2} +5 z^6-3 a^2 z^4-3 z^4 a^{-2} +8 z^4-2 a^2 z^2-2 z^2 a^{-2} +4 z^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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-13 q^{70}+2 q^{68}+22 q^{66}-48 q^{64}+77 q^{62}-93 q^{60}+75 q^{58}-27 q^{56}-60 q^{54}+162 q^{52}-237 q^{50}+259 q^{48}-188 q^{46}+27 q^{44}+173 q^{42}-342 q^{40}+401 q^{38}-319 q^{36}+115 q^{34}+129 q^{32}-313 q^{30}+362 q^{28}-237 q^{26}+12 q^{24}+212 q^{22}-326 q^{20}+260 q^{18}-55 q^{16}-209 q^{14}+412 q^{12}-458 q^{10}+343 q^8-79 q^6-234 q^4+477 q^2-571+477 q^{-2} -234 q^{-4} -79 q^{-6} +343 q^{-8} -458 q^{-10} +412 q^{-12} -209 q^{-14} -55 q^{-16} +260 q^{-18} -326 q^{-20} +212 q^{-22} +12 q^{-24} -237 q^{-26} +362 q^{-28} -313 q^{-30} +129 q^{-32} +115 q^{-34} -319 q^{-36} +401 q^{-38} -342 q^{-40} +173 q^{-42} +27 q^{-44} -188 q^{-46} +259 q^{-48} -237 q^{-50} +162 q^{-52} -60 q^{-54} -27 q^{-56} +75 q^{-58} -93 q^{-60} +77 q^{-62} -48 q^{-64} +22 q^{-66} +2 q^{-68} -13 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{63}+3 q^{61}-6 q^{57}-q^{55}+13 q^{53}+7 q^{51}-35 q^{49}-18 q^{47}+63 q^{45}+58 q^{43}-88 q^{41}-135 q^{39}+89 q^{37}+232 q^{35}-40 q^{33}-321 q^{31}-62 q^{29}+371 q^{27}+187 q^{25}-357 q^{23}-300 q^{21}+286 q^{19}+371 q^{17}-180 q^{15}-389 q^{13}+66 q^{11}+357 q^9+45 q^7-301 q^5-135 q^3+223 q+223 q^{-1} -135 q^{-3} -301 q^{-5} +45 q^{-7} +357 q^{-9} +66 q^{-11} -389 q^{-13} -180 q^{-15} +371 q^{-17} +286 q^{-19} -300 q^{-21} -357 q^{-23} +187 q^{-25} +371 q^{-27} -62 q^{-29} -321 q^{-31} -40 q^{-33} +232 q^{-35} +89 q^{-37} -135 q^{-39} -88 q^{-41} +58 q^{-43} +63 q^{-45} -18 q^{-47} -35 q^{-49} +7 q^{-51} +13 q^{-53} - q^{-55} -6 q^{-57} +3 q^{-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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{216}-3 q^{214}+6 q^{210}-3 q^{208}-3 q^{206}-2 q^{204}+16 q^{202}-q^{200}-21 q^{198}+5 q^{196}-31 q^{194}-23 q^{192}+26 q^{190}+135 q^{188}+115 q^{186}-35 q^{184}-129 q^{182}-370 q^{180}-394 q^{178}-57 q^{176}+715 q^{174}+1160 q^{172}+903 q^{170}+84 q^{168}-1724 q^{166}-3109 q^{164}-2846 q^{162}+192 q^{160}+4246 q^{158}+7066 q^{156}+6743 q^{154}+532 q^{152}-8833 q^{150}-16186 q^{148}-14323 q^{146}-2320 q^{144}+16041 q^{142}+31650 q^{140}+30246 q^{138}+7750 q^{136}-28523 q^{134}-56380 q^{132}-56893 q^{130}-19585 q^{128}+43960 q^{126}+95825 q^{124}+99536 q^{122}+37744 q^{120}-62744 q^{118}-149946 q^{116}-160730 q^{114}-67673 q^{112}+88593 q^{110}+221448 q^{108}+237152 q^{106}+106692 q^{104}-119529 q^{102}-308474 q^{100}-329701 q^{98}-144970 q^{96}+159969 q^{94}+403519 q^{92}+426370 q^{90}+177213 q^{88}-211477 q^{86}-505167 q^{84}-507961 q^{82}-189595 q^{80}+271458 q^{78}+598132 q^{76}+563336 q^{74}+173244 q^{72}-342966 q^{70}-662971 q^{68}-575952 q^{66}-127186 q^{64}+412211 q^{62}+691710 q^{60}+539184 q^{58}+50324 q^{56}-461414 q^{54}-672910 q^{52}-458530 q^{50}+38248 q^{48}+484300 q^{46}+606625 q^{44}+342381 q^{42}-117159 q^{40}-471582 q^{38}-504579 q^{36}-216335 q^{34}+178004 q^{32}+425404 q^{30}+380058 q^{28}+101732 q^{26}-212475 q^{24}-357505 q^{22}-256363 q^{20}-4192 q^{18}+224252 q^{16}+279709 q^{14}+146025 q^{12}-74126 q^{10}-223914 q^8-207137 q^6-46900 q^4+141254 q^2+221433+141254 q^{-2} -46900 q^{-4} -207137 q^{-6} -223914 q^{-8} -74126 q^{-10} +146025 q^{-12} +279709 q^{-14} +224252 q^{-16} -4192 q^{-18} -256363 q^{-20} -357505 q^{-22} -212475 q^{-24} +101732 q^{-26} +380058 q^{-28} +425404 q^{-30} +178004 q^{-32} -216335 q^{-34} -504579 q^{-36} -471582 q^{-38} -117159 q^{-40} +342381 q^{-42} +606625 q^{-44} +484300 q^{-46} +38248 q^{-48} -458530 q^{-50} -672910 q^{-52} -461414 q^{-54} +50324 q^{-56} +539184 q^{-58} +691710 q^{-60} +412211 q^{-62} -127186 q^{-64} -575952 q^{-66} -662971 q^{-68} -342966 q^{-70} +173244 q^{-72} +563336 q^{-74} +598132 q^{-76} +271458 q^{-78} -189595 q^{-80} -507961 q^{-82} -505167 q^{-84} -211477 q^{-86} +177213 q^{-88} +426370 q^{-90} +403519 q^{-92} +159969 q^{-94} -144970 q^{-96} -329701 q^{-98} -308474 q^{-100} -119529 q^{-102} +106692 q^{-104} +237152 q^{-106} +221448 q^{-108} +88593 q^{-110} -67673 q^{-112} -160730 q^{-114} -149946 q^{-116} -62744 q^{-118} +37744 q^{-120} +99536 q^{-122} +95825 q^{-124} +43960 q^{-126} -19585 q^{-128} -56893 q^{-130} -56380 q^{-132} -28523 q^{-134} +7750 q^{-136} +30246 q^{-138} +31650 q^{-140} +16041 q^{-142} -2320 q^{-144} -14323 q^{-146} -16186 q^{-148} -8833 q^{-150} +532 q^{-152} +6743 q^{-154} +7066 q^{-156} +4246 q^{-158} +192 q^{-160} -2846 q^{-162} -3109 q^{-164} -1724 q^{-166} +84 q^{-168} +903 q^{-170} +1160 q^{-172} +715 q^{-174} -57 q^{-176} -394 q^{-178} -370 q^{-180} -129 q^{-182} -35 q^{-184} +115 q^{-186} +135 q^{-188} +26 q^{-190} -23 q^{-192} -31 q^{-194} +5 q^{-196} -21 q^{-198} - q^{-200} +16 q^{-202} -2 q^{-204} -3 q^{-206} -3 q^{-208} +6 q^{-210} -3 q^{-214} + q^{-216} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{14}+2 q^{12}-2 q^{10}+2 q^8-2 q^4+4 q^2-3+4 q^{-2} -2 q^{-4} +2 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{44}-6 q^{42}+20 q^{40}-50 q^{38}+105 q^{36}-196 q^{34}+336 q^{32}-542 q^{30}+804 q^{28}-1110 q^{26}+1446 q^{24}-1736 q^{22}+1906 q^{20}-1892 q^{18}+1646 q^{16}-1140 q^{14}+379 q^{12}+542 q^{10}-1518 q^8+2440 q^6-3194 q^4+3692 q^2-3858+3692 q^{-2} -3194 q^{-4} +2440 q^{-6} -1518 q^{-8} +542 q^{-10} +379 q^{-12} -1140 q^{-14} +1646 q^{-16} -1892 q^{-18} +1906 q^{-20} -1736 q^{-22} +1446 q^{-24} -1110 q^{-26} +804 q^{-28} -542 q^{-30} +336 q^{-32} -196 q^{-34} +105 q^{-36} -50 q^{-38} +20 q^{-40} -6 q^{-42} + q^{-44} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{38}-2 q^{36}-q^{34}+5 q^{32}-4 q^{30}-6 q^{28}+8 q^{26}+6 q^{24}-8 q^{22}-6 q^{20}+13 q^{18}+7 q^{16}-21 q^{14}+2 q^{12}+16 q^{10}-12 q^8-9 q^6+13 q^4+6 q^2-10+6 q^{-2} +13 q^{-4} -9 q^{-6} -12 q^{-8} +16 q^{-10} +2 q^{-12} -21 q^{-14} +7 q^{-16} +13 q^{-18} -6 q^{-20} -8 q^{-22} +6 q^{-24} +8 q^{-26} -6 q^{-28} -4 q^{-30} +5 q^{-32} - q^{-34} -2 q^{-36} + q^{-38} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{34}-3 q^{32}+q^{30}+7 q^{28}-12 q^{26}+3 q^{24}+18 q^{22}-27 q^{20}+4 q^{18}+30 q^{16}-35 q^{14}+2 q^{12}+31 q^{10}-25 q^8-5 q^6+19 q^4-2 q^2-8-2 q^{-2} +19 q^{-4} -5 q^{-6} -25 q^{-8} +31 q^{-10} +2 q^{-12} -35 q^{-14} +30 q^{-16} +4 q^{-18} -27 q^{-20} +18 q^{-22} +3 q^{-24} -12 q^{-26} +7 q^{-28} + q^{-30} -3 q^{-32} + q^{-34} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{17}+2 q^{15}-3 q^{13}+4 q^{11}-3 q^9+3 q^7-2 q^5+2 q^3+2 q^{-3} -2 q^{-5} +3 q^{-7} -3 q^{-9} +4 q^{-11} -3 q^{-13} +2 q^{-15} - q^{-17} }
1,0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{56}-6 q^{54}+17 q^{52}-27 q^{50}+16 q^{48}+36 q^{46}-120 q^{44}+182 q^{42}-140 q^{40}-66 q^{38}+374 q^{36}-615 q^{34}+575 q^{32}-129 q^{30}-594 q^{28}+1268 q^{26}-1479 q^{24}+995 q^{22}+75 q^{20}-1276 q^{18}+2014 q^{16}-1920 q^{14}+1030 q^{12}+162 q^{10}-1046 q^8+1230 q^6-780 q^4+170 q^2+121+170 q^{-2} -780 q^{-4} +1230 q^{-6} -1046 q^{-8} +162 q^{-10} +1030 q^{-12} -1920 q^{-14} +2014 q^{-16} -1276 q^{-18} +75 q^{-20} +995 q^{-22} -1479 q^{-24} +1268 q^{-26} -594 q^{-28} -129 q^{-30} +575 q^{-32} -615 q^{-34} +374 q^{-36} -66 q^{-38} -140 q^{-40} +182 q^{-42} -120 q^{-44} +36 q^{-46} +16 q^{-48} -27 q^{-50} +17 q^{-52} -6 q^{-54} + q^{-56} }

A4 Invariants.

Weight

Invariant

0,1,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{40}-2 q^{38}-q^{36}+6 q^{34}-5 q^{32}-6 q^{30}+15 q^{28}-3 q^{26}-18 q^{24}+14 q^{22}+16 q^{20}-23 q^{18}-8 q^{16}+25 q^{14}-30 q^{10}+11 q^8+30 q^6-29 q^4-8 q^2+40-8 q^{-2} -29 q^{-4} +30 q^{-6} +11 q^{-8} -30 q^{-10} +25 q^{-14} -8 q^{-16} -23 q^{-18} +16 q^{-20} +14 q^{-22} -18 q^{-24} -3 q^{-26} +15 q^{-28} -6 q^{-30} -5 q^{-32} +6 q^{-34} - q^{-36} -2 q^{-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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{34}+3 q^{32}-7 q^{30}+13 q^{28}-20 q^{26}+29 q^{24}-38 q^{22}+43 q^{20}-44 q^{18}+40 q^{16}-29 q^{14}+14 q^{12}+7 q^{10}-29 q^8+51 q^6-69 q^4+82 q^2-86+82 q^{-2} -69 q^{-4} +51 q^{-6} -29 q^{-8} +7 q^{-10} +14 q^{-12} -29 q^{-14} +40 q^{-16} -44 q^{-18} +43 q^{-20} -38 q^{-22} +29 q^{-24} -20 q^{-26} +13 q^{-28} -7 q^{-30} +3 q^{-32} - q^{-34} }

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+10 q^{46}-16 q^{42}-12 q^{40}+16 q^{38}+28 q^{36}-5 q^{34}-39 q^{32}-17 q^{30}+36 q^{28}+38 q^{26}-19 q^{24}-47 q^{22}-3 q^{20}+44 q^{18}+20 q^{16}-32 q^{14}-28 q^{12}+20 q^{10}+30 q^8-10 q^6-30 q^4+4 q^2+31+4 q^{-2} -30 q^{-4} -10 q^{-6} +30 q^{-8} +20 q^{-10} -28 q^{-12} -32 q^{-14} +20 q^{-16} +44 q^{-18} -3 q^{-20} -47 q^{-22} -19 q^{-24} +38 q^{-26} +36 q^{-28} -17 q^{-30} -39 q^{-32} -5 q^{-34} +28 q^{-36} +16 q^{-38} -12 q^{-40} -16 q^{-42} +10 q^{-46} +4 q^{-48} -3 q^{-50} -3 q^{-52} + q^{-56} }

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{46}-3 q^{44}+4 q^{42}-6 q^{40}+11 q^{38}-16 q^{36}+19 q^{34}-23 q^{32}+31 q^{30}-35 q^{28}+33 q^{26}-33 q^{24}+34 q^{22}-27 q^{20}+14 q^{18}-8 q^{16}+18 q^{12}-33 q^{10}+39 q^8-48 q^6+64 q^4-64 q^2+64-64 q^{-2} +64 q^{-4} -48 q^{-6} +39 q^{-8} -33 q^{-10} +18 q^{-12} -8 q^{-16} +14 q^{-18} -27 q^{-20} +34 q^{-22} -33 q^{-24} +33 q^{-26} -35 q^{-28} +31 q^{-30} -23 q^{-32} +19 q^{-34} -16 q^{-36} +11 q^{-38} -6 q^{-40} +4 q^{-42} -3 q^{-44} + q^{-46} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-13 q^{70}+2 q^{68}+22 q^{66}-48 q^{64}+77 q^{62}-93 q^{60}+75 q^{58}-27 q^{56}-60 q^{54}+162 q^{52}-237 q^{50}+259 q^{48}-188 q^{46}+27 q^{44}+173 q^{42}-342 q^{40}+401 q^{38}-319 q^{36}+115 q^{34}+129 q^{32}-313 q^{30}+362 q^{28}-237 q^{26}+12 q^{24}+212 q^{22}-326 q^{20}+260 q^{18}-55 q^{16}-209 q^{14}+412 q^{12}-458 q^{10}+343 q^8-79 q^6-234 q^4+477 q^2-571+477 q^{-2} -234 q^{-4} -79 q^{-6} +343 q^{-8} -458 q^{-10} +412 q^{-12} -209 q^{-14} -55 q^{-16} +260 q^{-18} -326 q^{-20} +212 q^{-22} +12 q^{-24} -237 q^{-26} +362 q^{-28} -313 q^{-30} +129 q^{-32} +115 q^{-34} -319 q^{-36} +401 q^{-38} -342 q^{-40} +173 q^{-42} +27 q^{-44} -188 q^{-46} +259 q^{-48} -237 q^{-50} +162 q^{-52} -60 q^{-54} -27 q^{-56} +75 q^{-58} -93 q^{-60} +77 q^{-62} -48 q^{-64} +22 q^{-66} +2 q^{-68} -13 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 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.

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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8-a^2 z^6-z^6 a^{-2} +5 z^6-3 a^2 z^4-3 z^4 a^{-2} +8 z^4-2 a^2 z^2-2 z^2 a^{-2} +4 z^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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {...}

Vassiliev invariants

V₂ and V₃:

(0, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

0

0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -8}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{224}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{448}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{248}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums

\chi

(fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

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

-11

1

-1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1}
$r=-5$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{9}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{8}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{15}-4 q^{14}+3 q^{13}+11 q^{12}-27 q^{11}+8 q^{10}+52 q^9-76 q^8-8 q^7+130 q^6-122 q^5-55 q^4+208 q^3-134 q^2-107 q+241-107 q^{-1} -134 q^{-2} +208 q^{-3} -55 q^{-4} -122 q^{-5} +130 q^{-6} -8 q^{-7} -76 q^{-8} +52 q^{-9} +8 q^{-10} -27 q^{-11} +11 q^{-12} +3 q^{-13} -4 q^{-14} + q^{-15} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{30}+4 q^{29}-3 q^{28}-6 q^{27}+4 q^{26}+18 q^{25}-9 q^{24}-48 q^{23}+21 q^{22}+99 q^{21}-14 q^{20}-194 q^{19}-26 q^{18}+323 q^{17}+129 q^{16}-466 q^{15}-307 q^{14}+582 q^{13}+562 q^{12}-650 q^{11}-851 q^{10}+639 q^9+1148 q^8-565 q^7-1402 q^6+430 q^5+1603 q^4-274 q^3-1714 q^2+84 q+1769+84 q^{-1} -1714 q^{-2} -274 q^{-3} +1603 q^{-4} +430 q^{-5} -1402 q^{-6} -565 q^{-7} +1148 q^{-8} +639 q^{-9} -851 q^{-10} -650 q^{-11} +562 q^{-12} +582 q^{-13} -307 q^{-14} -466 q^{-15} +129 q^{-16} +323 q^{-17} -26 q^{-18} -194 q^{-19} -14 q^{-20} +99 q^{-21} +21 q^{-22} -48 q^{-23} -9 q^{-24} +18 q^{-25} +4 q^{-26} -6 q^{-27} -3 q^{-28} +4 q^{-29} - q^{-30} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{50}-4 q^{49}+3 q^{48}+6 q^{47}-9 q^{46}+5 q^{45}-17 q^{44}+22 q^{43}+31 q^{42}-60 q^{41}-4 q^{40}-61 q^{39}+131 q^{38}+193 q^{37}-192 q^{36}-199 q^{35}-372 q^{34}+386 q^{33}+907 q^{32}-7 q^{31}-645 q^{30}-1686 q^{29}+78 q^{28}+2318 q^{27}+1584 q^{26}-286 q^{25}-4171 q^{24}-2229 q^{23}+2995 q^{22}+4722 q^{21}+2593 q^{20}-6078 q^{19}-6548 q^{18}+1037 q^{17}+7450 q^{16}+7776 q^{15}-5461 q^{14}-10670 q^{13}-3245 q^{12}+7881 q^{11}+12898 q^{10}-2626 q^9-12675 q^8-7693 q^7+6233 q^6+16074 q^5+739 q^4-12520 q^3-10838 q^2+3699 q+17069+3699 q^{-1} -10838 q^{-2} -12520 q^{-3} +739 q^{-4} +16074 q^{-5} +6233 q^{-6} -7693 q^{-7} -12675 q^{-8} -2626 q^{-9} +12898 q^{-10} +7881 q^{-11} -3245 q^{-12} -10670 q^{-13} -5461 q^{-14} +7776 q^{-15} +7450 q^{-16} +1037 q^{-17} -6548 q^{-18} -6078 q^{-19} +2593 q^{-20} +4722 q^{-21} +2995 q^{-22} -2229 q^{-23} -4171 q^{-24} -286 q^{-25} +1584 q^{-26} +2318 q^{-27} +78 q^{-28} -1686 q^{-29} -645 q^{-30} -7 q^{-31} +907 q^{-32} +386 q^{-33} -372 q^{-34} -199 q^{-35} -192 q^{-36} +193 q^{-37} +131 q^{-38} -61 q^{-39} -4 q^{-40} -60 q^{-41} +31 q^{-42} +22 q^{-43} -17 q^{-44} +5 q^{-45} -9 q^{-46} +6 q^{-47} +3 q^{-48} -4 q^{-49} + q^{-50} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{75}+4 q^{74}-3 q^{73}-6 q^{72}+9 q^{71}-6 q^{69}+4 q^{68}-5 q^{67}-9 q^{66}+37 q^{65}+29 q^{64}-48 q^{63}-83 q^{62}-68 q^{61}+46 q^{60}+244 q^{59}+301 q^{58}-24 q^{57}-573 q^{56}-787 q^{55}-287 q^{54}+875 q^{53}+1843 q^{52}+1364 q^{51}-921 q^{50}-3428 q^{49}-3602 q^{48}-259 q^{47}+4964 q^{46}+7568 q^{45}+3700 q^{44}-5394 q^{43}-12667 q^{42}-10365 q^{41}+2685 q^{40}+17588 q^{39}+20463 q^{38}+4813 q^{37}-19877 q^{36}-32656 q^{35}-18124 q^{34}+16897 q^{33}+44345 q^{32}+36630 q^{31}-6744 q^{30}-52443 q^{29}-57942 q^{28}-10715 q^{27}+54076 q^{26}+78732 q^{25}+34017 q^{24}-48179 q^{23}-95785 q^{22}-59901 q^{21}+35233 q^{20}+106740 q^{19}+85226 q^{18}-17417 q^{17}-111131 q^{16}-107011 q^{15}-2543 q^{14}+109646 q^{13}+123904 q^{12}+22010 q^{11}-104034 q^{10}-135442 q^9-39509 q^8+96005 q^7+142679 q^6+54100 q^5-86907 q^4-146180 q^3-66504 q^2+77050 q+147507+77050 q^{-1} -66504 q^{-2} -146180 q^{-3} -86907 q^{-4} +54100 q^{-5} +142679 q^{-6} +96005 q^{-7} -39509 q^{-8} -135442 q^{-9} -104034 q^{-10} +22010 q^{-11} +123904 q^{-12} +109646 q^{-13} -2543 q^{-14} -107011 q^{-15} -111131 q^{-16} -17417 q^{-17} +85226 q^{-18} +106740 q^{-19} +35233 q^{-20} -59901 q^{-21} -95785 q^{-22} -48179 q^{-23} +34017 q^{-24} +78732 q^{-25} +54076 q^{-26} -10715 q^{-27} -57942 q^{-28} -52443 q^{-29} -6744 q^{-30} +36630 q^{-31} +44345 q^{-32} +16897 q^{-33} -18124 q^{-34} -32656 q^{-35} -19877 q^{-36} +4813 q^{-37} +20463 q^{-38} +17588 q^{-39} +2685 q^{-40} -10365 q^{-41} -12667 q^{-42} -5394 q^{-43} +3700 q^{-44} +7568 q^{-45} +4964 q^{-46} -259 q^{-47} -3602 q^{-48} -3428 q^{-49} -921 q^{-50} +1364 q^{-51} +1843 q^{-52} +875 q^{-53} -287 q^{-54} -787 q^{-55} -573 q^{-56} -24 q^{-57} +301 q^{-58} +244 q^{-59} +46 q^{-60} -68 q^{-61} -83 q^{-62} -48 q^{-63} +29 q^{-64} +37 q^{-65} -9 q^{-66} -5 q^{-67} +4 q^{-68} -6 q^{-69} +9 q^{-71} -6 q^{-72} -3 q^{-73} +4 q^{-74} - q^{-75} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{105}-4 q^{104}+3 q^{103}+6 q^{102}-9 q^{101}+q^{99}+19 q^{98}-21 q^{97}-17 q^{96}+32 q^{95}-45 q^{94}+8 q^{93}+50 q^{92}+128 q^{91}-41 q^{90}-167 q^{89}-62 q^{88}-286 q^{87}-16 q^{86}+387 q^{85}+900 q^{84}+404 q^{83}-424 q^{82}-881 q^{81}-2094 q^{80}-1401 q^{79}+650 q^{78}+3938 q^{77}+4458 q^{76}+2396 q^{75}-1204 q^{74}-8305 q^{73}-10766 q^{72}-6703 q^{71}+5801 q^{70}+16461 q^{69}+20757 q^{68}+14405 q^{67}-9709 q^{66}-33262 q^{65}-42976 q^{64}-22056 q^{63}+15948 q^{62}+58065 q^{61}+77950 q^{60}+42156 q^{59}-29551 q^{58}-104768 q^{57}-122544 q^{56}-71254 q^{55}+47281 q^{54}+171007 q^{53}+198422 q^{52}+103304 q^{51}-89064 q^{50}-253004 q^{49}-297475 q^{48}-141664 q^{47}+149780 q^{46}+383153 q^{45}+408243 q^{44}+154486 q^{43}-230153 q^{42}-546632 q^{41}-530354 q^{40}-143910 q^{39}+380359 q^{38}+726609 q^{37}+615539 q^{36}+96521 q^{35}-581428 q^{34}-920446 q^{33}-660120 q^{32}+60354 q^{31}+813628 q^{30}+1064305 q^{29}+635918 q^{28}-301929 q^{27}-1072972 q^{26}-1148980 q^{25}-451384 q^{24}+602132 q^{23}+1278685 q^{22}+1132696 q^{21}+144123 q^{20}-950647 q^{19}-1413224 q^{18}-910924 q^{17}+247709 q^{16}+1245688 q^{15}+1420940 q^{14}+538462 q^{13}-703247 q^{12}-1458570 q^{11}-1189250 q^{10}-66498 q^9+1100658 q^8+1522082 q^7+790633 q^6-474803 q^5-1403113 q^4-1322934 q^3-286649 q^2+950870 q+1538859+950870 q^{-1} -286649 q^{-2} -1322934 q^{-3} -1403113 q^{-4} -474803 q^{-5} +790633 q^{-6} +1522082 q^{-7} +1100658 q^{-8} -66498 q^{-9} -1189250 q^{-10} -1458570 q^{-11} -703247 q^{-12} +538462 q^{-13} +1420940 q^{-14} +1245688 q^{-15} +247709 q^{-16} -910924 q^{-17} -1413224 q^{-18} -950647 q^{-19} +144123 q^{-20} +1132696 q^{-21} +1278685 q^{-22} +602132 q^{-23} -451384 q^{-24} -1148980 q^{-25} -1072972 q^{-26} -301929 q^{-27} +635918 q^{-28} +1064305 q^{-29} +813628 q^{-30} +60354 q^{-31} -660120 q^{-32} -920446 q^{-33} -581428 q^{-34} +96521 q^{-35} +615539 q^{-36} +726609 q^{-37} +380359 q^{-38} -143910 q^{-39} -530354 q^{-40} -546632 q^{-41} -230153 q^{-42} +154486 q^{-43} +408243 q^{-44} +383153 q^{-45} +149780 q^{-46} -141664 q^{-47} -297475 q^{-48} -253004 q^{-49} -89064 q^{-50} +103304 q^{-51} +198422 q^{-52} +171007 q^{-53} +47281 q^{-54} -71254 q^{-55} -122544 q^{-56} -104768 q^{-57} -29551 q^{-58} +42156 q^{-59} +77950 q^{-60} +58065 q^{-61} +15948 q^{-62} -22056 q^{-63} -42976 q^{-64} -33262 q^{-65} -9709 q^{-66} +14405 q^{-67} +20757 q^{-68} +16461 q^{-69} +5801 q^{-70} -6703 q^{-71} -10766 q^{-72} -8305 q^{-73} -1204 q^{-74} +2396 q^{-75} +4458 q^{-76} +3938 q^{-77} +650 q^{-78} -1401 q^{-79} -2094 q^{-80} -881 q^{-81} -424 q^{-82} +404 q^{-83} +900 q^{-84} +387 q^{-85} -16 q^{-86} -286 q^{-87} -62 q^{-88} -167 q^{-89} -41 q^{-90} +128 q^{-91} +50 q^{-92} +8 q^{-93} -45 q^{-94} +32 q^{-95} -17 q^{-96} -21 q^{-97} +19 q^{-98} + q^{-99} -9 q^{-101} +6 q^{-102} +3 q^{-103} -4 q^{-104} + q^{-105} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{140}+4 q^{139}-3 q^{138}-6 q^{137}+9 q^{136}-q^{134}-14 q^{133}-2 q^{132}+43 q^{131}-6 q^{130}-24 q^{129}+8 q^{128}-27 q^{127}-23 q^{126}-69 q^{125}-8 q^{124}+242 q^{123}+163 q^{122}+35 q^{121}-72 q^{120}-385 q^{119}-411 q^{118}-518 q^{117}-142 q^{116}+1078 q^{115}+1554 q^{114}+1474 q^{113}+483 q^{112}-1739 q^{111}-3318 q^{110}-4454 q^{109}-3310 q^{108}+1813 q^{107}+7114 q^{106}+11157 q^{105}+10245 q^{104}+1633 q^{103}-9940 q^{102}-22433 q^{101}-27507 q^{100}-16520 q^{99}+6677 q^{98}+37057 q^{97}+58407 q^{96}+52468 q^{95}+18370 q^{94}-40973 q^{93}-100446 q^{92}-122513 q^{91}-89098 q^{90}+7781 q^{89}+133531 q^{88}+223614 q^{87}+228198 q^{86}+109054 q^{85}-106363 q^{84}-325147 q^{83}-443178 q^{82}-356501 q^{81}-55497 q^{80}+349552 q^{79}+689017 q^{78}+753321 q^{77}+439152 q^{76}-173905 q^{75}-857942 q^{74}-1254288 q^{73}-1089251 q^{72}-333822 q^{71}+773753 q^{70}+1714946 q^{69}+1964892 q^{68}+1265562 q^{67}-240851 q^{66}-1909001 q^{65}-2902350 q^{64}-2597945 q^{63}-884016 q^{62}+1573715 q^{61}+3625400 q^{60}+4160778 q^{59}+2617311 q^{58}-502842 q^{57}-3814742 q^{56}-5650023 q^{55}-4799031 q^{54}-1364682 q^{53}+3202185 q^{52}+6697142 q^{51}+7115131 q^{50}+3902842 q^{49}-1670439 q^{48}-6984275 q^{47}-9173314 q^{46}-6807311 q^{45}-696117 q^{44}+6335550 q^{43}+10610793 q^{42}+9681809 q^{41}+3632710 q^{40}-4775784 q^{39}-11201786 q^{38}-12142587 q^{37}-6760636 q^{36}+2515244 q^{35}+10900762 q^{34}+13915945 q^{33}+9700418 q^{32}+121154 q^{31}-9843831 q^{30}-14895676 q^{29}-12155512 q^{28}-2782605 q^{27}+8283604 q^{26}+15135646 q^{25}+13970837 q^{24}+5181766 q^{23}-6511733 q^{22}-14808343 q^{21}-15135897 q^{20}-7142924 q^{19}+4783786 q^{18}+14138213 q^{17}+15752305 q^{16}+8612048 q^{15}-3272853 q^{14}-13335658 q^{13}-15980950 q^{12}-9641991 q^{11}+2050425 q^{10}+12561147 q^9+15997522 q^8+10346608 q^7-1103802 q^6-11896387 q^5-15942378 q^4-10871239 q^3+344203 q^2+11352146 q+15915633+11352146 q^{-1} +344203 q^{-2} -10871239 q^{-3} -15942378 q^{-4} -11896387 q^{-5} -1103802 q^{-6} +10346608 q^{-7} +15997522 q^{-8} +12561147 q^{-9} +2050425 q^{-10} -9641991 q^{-11} -15980950 q^{-12} -13335658 q^{-13} -3272853 q^{-14} +8612048 q^{-15} +15752305 q^{-16} +14138213 q^{-17} +4783786 q^{-18} -7142924 q^{-19} -15135897 q^{-20} -14808343 q^{-21} -6511733 q^{-22} +5181766 q^{-23} +13970837 q^{-24} +15135646 q^{-25} +8283604 q^{-26} -2782605 q^{-27} -12155512 q^{-28} -14895676 q^{-29} -9843831 q^{-30} +121154 q^{-31} +9700418 q^{-32} +13915945 q^{-33} +10900762 q^{-34} +2515244 q^{-35} -6760636 q^{-36} -12142587 q^{-37} -11201786 q^{-38} -4775784 q^{-39} +3632710 q^{-40} +9681809 q^{-41} +10610793 q^{-42} +6335550 q^{-43} -696117 q^{-44} -6807311 q^{-45} -9173314 q^{-46} -6984275 q^{-47} -1670439 q^{-48} +3902842 q^{-49} +7115131 q^{-50} +6697142 q^{-51} +3202185 q^{-52} -1364682 q^{-53} -4799031 q^{-54} -5650023 q^{-55} -3814742 q^{-56} -502842 q^{-57} +2617311 q^{-58} +4160778 q^{-59} +3625400 q^{-60} +1573715 q^{-61} -884016 q^{-62} -2597945 q^{-63} -2902350 q^{-64} -1909001 q^{-65} -240851 q^{-66} +1265562 q^{-67} +1964892 q^{-68} +1714946 q^{-69} +773753 q^{-70} -333822 q^{-71} -1089251 q^{-72} -1254288 q^{-73} -857942 q^{-74} -173905 q^{-75} +439152 q^{-76} +753321 q^{-77} +689017 q^{-78} +349552 q^{-79} -55497 q^{-80} -356501 q^{-81} -443178 q^{-82} -325147 q^{-83} -106363 q^{-84} +109054 q^{-85} +228198 q^{-86} +223614 q^{-87} +133531 q^{-88} +7781 q^{-89} -89098 q^{-90} -122513 q^{-91} -100446 q^{-92} -40973 q^{-93} +18370 q^{-94} +52468 q^{-95} +58407 q^{-96} +37057 q^{-97} +6677 q^{-98} -16520 q^{-99} -27507 q^{-100} -22433 q^{-101} -9940 q^{-102} +1633 q^{-103} +10245 q^{-104} +11157 q^{-105} +7114 q^{-106} +1813 q^{-107} -3310 q^{-108} -4454 q^{-109} -3318 q^{-110} -1739 q^{-111} +483 q^{-112} +1474 q^{-113} +1554 q^{-114} +1078 q^{-115} -142 q^{-116} -518 q^{-117} -411 q^{-118} -385 q^{-119} -72 q^{-120} +35 q^{-121} +163 q^{-122} +242 q^{-123} -8 q^{-124} -69 q^{-125} -23 q^{-126} -27 q^{-127} +8 q^{-128} -24 q^{-129} -6 q^{-130} +43 q^{-131} -2 q^{-132} -14 q^{-133} - q^{-134} +9 q^{-136} -6 q^{-137} -3 q^{-138} +4 q^{-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.

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10_117

10_119

10 118

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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