10 17: Difference between revisions

Revision as of 21:04, 29 August 2005

10_16

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10 17 Quick Notes

10 17 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_12,4,13,3 X_20,15,1,16 X_16,7,17,8 X_18,9,19,10 X_8,17,9,18 X_10,19,11,20 X_14,6,15,5 X_2,12,3,11 X_4,14,5,13
Gauss code	1, -9, 2, -10, 8, -1, 4, -6, 5, -7, 9, -2, 10, -8, 3, -4, 6, -5, 7, -3
Dowker-Thistlethwaite code	6 12 14 16 18 2 4 20 8 10
Conway Notation	[4114]

Minimum Braid Representative:

Length is 10, width is 3.

Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Fully amphicheiral
Unknotting number	1
3-genus	4
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	8.53676
A-Polynomial	See Data:10 17/A-polynomial

[edit Notes for 10 17'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	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}
Concordance genus	$4$
Rasmussen s-Invariant	0

[edit Notes for 10 17's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-3t^{3}+5t^{2}-7t+9-7t^{-1}+5t^{-2}-3t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+5z^{6}+7z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, 0 }
Jones polynomial	$-q^{5}+2q^{4}-3q^{3}+5q^{2}-6q+7-6q^{-1}+5q^{-2}-3q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+7z^{6}-5a^{2}z^{4}-5z^{4}a^{-2}+17z^{4}-7a^{2}z^{2}-7z^{2}a^{-2}+16z^{2}-2a^{2}-2a^{-2}+5$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+2a^{3}z^{7}-2az^{7}-2z^{7}a^{-1}+2z^{7}a^{-3}+2a^{4}z^{6}-7a^{2}z^{6}-7z^{6}a^{-2}+2z^{6}a^{-4}-18z^{6}+a^{5}z^{5}-5a^{3}z^{5}-5z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+11a^{2}z^{4}+11z^{4}a^{-2}-6z^{4}a^{-4}+34z^{4}-3a^{5}z^{3}+2a^{3}z^{3}+6az^{3}+6z^{3}a^{-1}+2z^{3}a^{-3}-3z^{3}a^{-5}+3a^{4}z^{2}-8a^{2}z^{2}-8z^{2}a^{-2}+3z^{2}a^{-4}-22z^{2}+a^{5}z-3az-3za^{-1}+za^{-5}+2a^{2}+2a^{-2}+5$
The A2 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^{14}-q^{10}+q^8+q^6+2 q^2-1+2 q^{-2} + q^{-6} + q^{-8} - q^{-10} - 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}-q^{78}+2 q^{76}-3 q^{74}+2 q^{72}-q^{70}-2 q^{68}+6 q^{66}-7 q^{64}+8 q^{62}-7 q^{60}+2 q^{58}+2 q^{56}-8 q^{54}+11 q^{52}-15 q^{50}+12 q^{48}-7 q^{46}-2 q^{44}+9 q^{42}-16 q^{40}+19 q^{38}-14 q^{36}+4 q^{34}+5 q^{32}-14 q^{30}+15 q^{28}-7 q^{26}-q^{24}+11 q^{22}-12 q^{20}+9 q^{18}+2 q^{16}-12 q^{14}+21 q^{12}-20 q^{10}+14 q^8-q^6-12 q^4+23 q^2-25+23 q^{-2} -12 q^{-4} - q^{-6} +14 q^{-8} -20 q^{-10} +21 q^{-12} -12 q^{-14} +2 q^{-16} +9 q^{-18} -12 q^{-20} +11 q^{-22} - q^{-24} -7 q^{-26} +15 q^{-28} -14 q^{-30} +5 q^{-32} +4 q^{-34} -14 q^{-36} +19 q^{-38} -16 q^{-40} +9 q^{-42} -2 q^{-44} -7 q^{-46} +12 q^{-48} -15 q^{-50} +11 q^{-52} -8 q^{-54} +2 q^{-56} +2 q^{-58} -7 q^{-60} +8 q^{-62} -7 q^{-64} +6 q^{-66} -2 q^{-68} - q^{-70} +2 q^{-72} -3 q^{-74} +2 q^{-76} - q^{-78} + q^{-80} }

Further Quantum Invariants

Further quantum knot invariants for 10_17.

A1 Invariants.

Weight	Invariant
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^{11}+q^9-q^7+2 q^5-q^3+q+ q^{-1} - q^{-3} +2 q^{-5} - q^{-7} + q^{-9} - q^{-11} }
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^{32}-q^{30}-q^{28}+2 q^{26}-2 q^{24}-2 q^{22}+4 q^{20}-q^{18}-4 q^{16}+6 q^{14}+q^{12}-6 q^{10}+4 q^8+2 q^6-4 q^4+q^2+3+ q^{-2} -4 q^{-4} +2 q^{-6} +4 q^{-8} -6 q^{-10} + q^{-12} +6 q^{-14} -4 q^{-16} - q^{-18} +4 q^{-20} -2 q^{-22} -2 q^{-24} +2 q^{-26} - q^{-28} - q^{-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}+q^{61}+q^{59}-q^{55}+q^{51}-q^{49}-q^{47}+2 q^{45}+q^{43}-4 q^{41}-3 q^{39}+3 q^{37}+7 q^{35}-3 q^{33}-9 q^{31}-2 q^{29}+13 q^{27}+7 q^{25}-11 q^{23}-12 q^{21}+9 q^{19}+15 q^{17}-6 q^{15}-14 q^{13}+2 q^{11}+12 q^9+q^7-9 q^5-2 q^3+6 q+6 q^{-1} -2 q^{-3} -9 q^{-5} + q^{-7} +12 q^{-9} +2 q^{-11} -14 q^{-13} -6 q^{-15} +15 q^{-17} +9 q^{-19} -12 q^{-21} -11 q^{-23} +7 q^{-25} +13 q^{-27} -2 q^{-29} -9 q^{-31} -3 q^{-33} +7 q^{-35} +3 q^{-37} -3 q^{-39} -4 q^{-41} + q^{-43} +2 q^{-45} - q^{-47} - q^{-49} + q^{-51} - q^{-55} + q^{-59} + q^{-61} - q^{-63} }
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^{104}-q^{102}-q^{100}-q^{96}+3 q^{94}+q^{90}-6 q^{86}+2 q^{84}+5 q^{80}+7 q^{78}-8 q^{76}-5 q^{74}-9 q^{72}+6 q^{70}+19 q^{68}+q^{66}-5 q^{64}-25 q^{62}-10 q^{60}+19 q^{58}+19 q^{56}+19 q^{54}-22 q^{52}-32 q^{50}-10 q^{48}+13 q^{46}+50 q^{44}+12 q^{42}-29 q^{40}-43 q^{38}-24 q^{36}+49 q^{34}+46 q^{32}+4 q^{30}-44 q^{28}-52 q^{26}+22 q^{24}+45 q^{22}+25 q^{20}-21 q^{18}-45 q^{16}-2 q^{14}+22 q^{12}+23 q^{10}-2 q^8-22 q^6-8 q^4+8 q^2+17+8 q^{-2} -8 q^{-4} -22 q^{-6} -2 q^{-8} +23 q^{-10} +22 q^{-12} -2 q^{-14} -45 q^{-16} -21 q^{-18} +25 q^{-20} +45 q^{-22} +22 q^{-24} -52 q^{-26} -44 q^{-28} +4 q^{-30} +46 q^{-32} +49 q^{-34} -24 q^{-36} -43 q^{-38} -29 q^{-40} +12 q^{-42} +50 q^{-44} +13 q^{-46} -10 q^{-48} -32 q^{-50} -22 q^{-52} +19 q^{-54} +19 q^{-56} +19 q^{-58} -10 q^{-60} -25 q^{-62} -5 q^{-64} + q^{-66} +19 q^{-68} +6 q^{-70} -9 q^{-72} -5 q^{-74} -8 q^{-76} +7 q^{-78} +5 q^{-80} +2 q^{-84} -6 q^{-86} + q^{-90} +3 q^{-94} - q^{-96} - q^{-100} - q^{-102} + q^{-104} }
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^{155}+q^{153}+q^{151}+q^{147}-q^{145}-3 q^{143}-2 q^{141}+q^{139}+2 q^{137}+5 q^{135}+4 q^{133}-4 q^{131}-9 q^{129}-6 q^{127}+q^{125}+9 q^{123}+14 q^{121}+6 q^{119}-12 q^{117}-19 q^{115}-13 q^{113}+6 q^{111}+25 q^{109}+28 q^{107}+4 q^{105}-29 q^{103}-40 q^{101}-21 q^{99}+18 q^{97}+50 q^{95}+47 q^{93}+3 q^{91}-47 q^{89}-69 q^{87}-43 q^{85}+22 q^{83}+77 q^{81}+82 q^{79}+31 q^{77}-60 q^{75}-116 q^{73}-89 q^{71}+9 q^{69}+116 q^{67}+154 q^{65}+66 q^{63}-89 q^{61}-184 q^{59}-142 q^{57}+20 q^{55}+185 q^{53}+206 q^{51}+52 q^{49}-150 q^{47}-232 q^{45}-119 q^{43}+92 q^{41}+223 q^{39}+162 q^{37}-36 q^{35}-187 q^{33}-165 q^{31}-12 q^{29}+131 q^{27}+148 q^{25}+40 q^{23}-82 q^{21}-112 q^{19}-45 q^{17}+42 q^{15}+73 q^{13}+42 q^{11}-15 q^9-49 q^7-31 q^5+5 q^3+33 q+33 q^{-1} +5 q^{-3} -31 q^{-5} -49 q^{-7} -15 q^{-9} +42 q^{-11} +73 q^{-13} +42 q^{-15} -45 q^{-17} -112 q^{-19} -82 q^{-21} +40 q^{-23} +148 q^{-25} +131 q^{-27} -12 q^{-29} -165 q^{-31} -187 q^{-33} -36 q^{-35} +162 q^{-37} +223 q^{-39} +92 q^{-41} -119 q^{-43} -232 q^{-45} -150 q^{-47} +52 q^{-49} +206 q^{-51} +185 q^{-53} +20 q^{-55} -142 q^{-57} -184 q^{-59} -89 q^{-61} +66 q^{-63} +154 q^{-65} +116 q^{-67} +9 q^{-69} -89 q^{-71} -116 q^{-73} -60 q^{-75} +31 q^{-77} +82 q^{-79} +77 q^{-81} +22 q^{-83} -43 q^{-85} -69 q^{-87} -47 q^{-89} +3 q^{-91} +47 q^{-93} +50 q^{-95} +18 q^{-97} -21 q^{-99} -40 q^{-101} -29 q^{-103} +4 q^{-105} +28 q^{-107} +25 q^{-109} +6 q^{-111} -13 q^{-113} -19 q^{-115} -12 q^{-117} +6 q^{-119} +14 q^{-121} +9 q^{-123} + q^{-125} -6 q^{-127} -9 q^{-129} -4 q^{-131} +4 q^{-133} +5 q^{-135} +2 q^{-137} + q^{-139} -2 q^{-141} -3 q^{-143} - q^{-145} + q^{-147} + q^{-151} + q^{-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}-q^{214}-q^{212}-q^{208}+q^{206}+q^{204}+5 q^{202}-3 q^{198}-2 q^{196}-6 q^{194}-3 q^{192}+q^{190}+13 q^{188}+6 q^{186}-2 q^{182}-13 q^{180}-12 q^{178}-6 q^{176}+18 q^{174}+15 q^{172}+7 q^{170}+3 q^{168}-17 q^{166}-24 q^{164}-18 q^{162}+21 q^{160}+28 q^{158}+22 q^{156}+13 q^{154}-22 q^{152}-47 q^{150}-49 q^{148}+10 q^{146}+44 q^{144}+58 q^{142}+57 q^{140}+4 q^{138}-61 q^{136}-105 q^{134}-58 q^{132}-7 q^{130}+55 q^{128}+121 q^{126}+113 q^{124}+39 q^{122}-74 q^{120}-122 q^{118}-160 q^{116}-118 q^{114}+25 q^{112}+168 q^{110}+241 q^{108}+184 q^{106}+62 q^{104}-182 q^{102}-359 q^{100}-331 q^{98}-111 q^{96}+211 q^{94}+447 q^{92}+512 q^{90}+203 q^{88}-264 q^{86}-609 q^{84}-625 q^{82}-264 q^{80}+284 q^{78}+778 q^{76}+756 q^{74}+273 q^{72}-403 q^{70}-858 q^{68}-811 q^{66}-265 q^{64}+536 q^{62}+938 q^{60}+766 q^{58}+111 q^{56}-589 q^{54}-919 q^{52}-667 q^{50}+61 q^{48}+643 q^{46}+793 q^{44}+429 q^{42}-151 q^{40}-597 q^{38}-624 q^{36}-207 q^{34}+234 q^{32}+473 q^{30}+373 q^{28}+83 q^{26}-224 q^{24}-337 q^{22}-188 q^{20}+14 q^{18}+170 q^{16}+175 q^{14}+92 q^{12}-42 q^{10}-121 q^8-92 q^6-18 q^4+62 q^2+81+62 q^{-2} -18 q^{-4} -92 q^{-6} -121 q^{-8} -42 q^{-10} +92 q^{-12} +175 q^{-14} +170 q^{-16} +14 q^{-18} -188 q^{-20} -337 q^{-22} -224 q^{-24} +83 q^{-26} +373 q^{-28} +473 q^{-30} +234 q^{-32} -207 q^{-34} -624 q^{-36} -597 q^{-38} -151 q^{-40} +429 q^{-42} +793 q^{-44} +643 q^{-46} +61 q^{-48} -667 q^{-50} -919 q^{-52} -589 q^{-54} +111 q^{-56} +766 q^{-58} +938 q^{-60} +536 q^{-62} -265 q^{-64} -811 q^{-66} -858 q^{-68} -403 q^{-70} +273 q^{-72} +756 q^{-74} +778 q^{-76} +284 q^{-78} -264 q^{-80} -625 q^{-82} -609 q^{-84} -264 q^{-86} +203 q^{-88} +512 q^{-90} +447 q^{-92} +211 q^{-94} -111 q^{-96} -331 q^{-98} -359 q^{-100} -182 q^{-102} +62 q^{-104} +184 q^{-106} +241 q^{-108} +168 q^{-110} +25 q^{-112} -118 q^{-114} -160 q^{-116} -122 q^{-118} -74 q^{-120} +39 q^{-122} +113 q^{-124} +121 q^{-126} +55 q^{-128} -7 q^{-130} -58 q^{-132} -105 q^{-134} -61 q^{-136} +4 q^{-138} +57 q^{-140} +58 q^{-142} +44 q^{-144} +10 q^{-146} -49 q^{-148} -47 q^{-150} -22 q^{-152} +13 q^{-154} +22 q^{-156} +28 q^{-158} +21 q^{-160} -18 q^{-162} -24 q^{-164} -17 q^{-166} +3 q^{-168} +7 q^{-170} +15 q^{-172} +18 q^{-174} -6 q^{-176} -12 q^{-178} -13 q^{-180} -2 q^{-182} +6 q^{-186} +13 q^{-188} + q^{-190} -3 q^{-192} -6 q^{-194} -2 q^{-196} -3 q^{-198} +5 q^{-202} + q^{-204} + q^{-206} - q^{-208} - q^{-212} - q^{-214} + q^{-216} }

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}-q^{10}+q^{8}+q^{6}+2q^{2}-1+2q^{-2}+q^{-6}+q^{-8}-q^{-10}-q^{-14}$
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}-2 q^{42}+4 q^{40}-8 q^{38}+13 q^{36}-18 q^{34}+22 q^{32}-28 q^{30}+33 q^{28}-38 q^{26}+40 q^{24}-44 q^{22}+44 q^{20}-40 q^{18}+32 q^{16}-14 q^{14}-5 q^{12}+26 q^{10}-50 q^8+70 q^6-85 q^4+98 q^2-94+98 q^{-2} -85 q^{-4} +70 q^{-6} -50 q^{-8} +26 q^{-10} -5 q^{-12} -14 q^{-14} +32 q^{-16} -40 q^{-18} +44 q^{-20} -44 q^{-22} +40 q^{-24} -38 q^{-26} +33 q^{-28} -28 q^{-30} +22 q^{-32} -18 q^{-34} +13 q^{-36} -8 q^{-38} +4 q^{-40} -2 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}-q^{30}-2 q^{28}-q^{26}-2 q^{22}+3 q^{18}+3 q^{16}-2 q^{14}+q^{12}+3 q^{10}-q^8-q^6+2 q^4+q^2-2+ q^{-2} +2 q^{-4} - q^{-6} - q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} +3 q^{-18} -2 q^{-22} - q^{-26} -2 q^{-28} - q^{-30} + 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}-q^{32}+q^{28}-2 q^{26}+q^{24}+2 q^{22}-4 q^{20}+3 q^{16}-7 q^{14}-q^{12}+4 q^{10}-3 q^8+5 q^4+3 q^2+2+3 q^{-2} +5 q^{-4} -3 q^{-8} +4 q^{-10} - q^{-12} -7 q^{-14} +3 q^{-16} -4 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-26} + q^{-28} - 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^{13}+q^{11}-q^9+2 q^7+2 q^3+q+ q^{-1} +2 q^{-3} +2 q^{-7} - q^{-9} + q^{-11} -2 q^{-13} - 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}-2 q^{54}+3 q^{52}-3 q^{50}+q^{48}+4 q^{46}-9 q^{44}+9 q^{42}-7 q^{40}+q^{38}+6 q^{36}-10 q^{34}+10 q^{32}-4 q^{30}-7 q^{28}+17 q^{26}-22 q^{24}+9 q^{22}+9 q^{20}-30 q^{18}+39 q^{16}-36 q^{14}+13 q^{12}+3 q^{10}-24 q^8+28 q^6-13 q^4+18 q^2+7+18 q^{-2} -13 q^{-4} +28 q^{-6} -24 q^{-8} +3 q^{-10} +13 q^{-12} -36 q^{-14} +39 q^{-16} -30 q^{-18} +9 q^{-20} +9 q^{-22} -22 q^{-24} +17 q^{-26} -7 q^{-28} -4 q^{-30} +10 q^{-32} -10 q^{-34} +6 q^{-36} + q^{-38} -7 q^{-40} +9 q^{-42} -9 q^{-44} +4 q^{-46} + q^{-48} -3 q^{-50} +3 q^{-52} -2 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}+q^{34}-q^{30}+q^{28}-q^{26}-3 q^{24}-5 q^{18}-4 q^{16}-2 q^{12}-5 q^{10}+q^8+7 q^6+2 q^4+7 q^2+12+7 q^{-2} +2 q^{-4} +7 q^{-6} + q^{-8} -5 q^{-10} -2 q^{-12} -4 q^{-16} -5 q^{-18} -3 q^{-24} - q^{-26} + q^{-28} - q^{-30} + q^{-34} + q^{-40} }

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^{20}-2 q^{16}-q^{12}+q^8+3 q^4+q^2+3+ q^{-2} +3 q^{-4} + q^{-8} - q^{-12} -2 q^{-16} - 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}+q^{32}-2 q^{30}+3 q^{28}-4 q^{26}+5 q^{24}-6 q^{22}+6 q^{20}-6 q^{18}+5 q^{16}-3 q^{14}+q^{12}+2 q^{10}-5 q^8+8 q^6-9 q^4+13 q^2-12+13 q^{-2} -9 q^{-4} +8 q^{-6} -5 q^{-8} +2 q^{-10} + q^{-12} -3 q^{-14} +5 q^{-16} -6 q^{-18} +6 q^{-20} -6 q^{-22} +5 q^{-24} -4 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} - q^{-34} }

1,0

q^{56}-q^{52}-q^{50}+q^{48}+2q^{46}-q^{44}-3q^{42}-q^{40}+3q^{38}+4q^{36}-2q^{34}-6q^{32}-2q^{30}+5q^{28}+5q^{26}-4q^{24}-7q^{22}-q^{20}+6q^{18}+3q^{16}-4q^{14}-3q^{12}+3q^{10}+4q^{8}-3q^{4}+2q^{2}+5+2q^{-2}-3q^{-4}+4q^{-8}+3q^{-10}-3q^{-12}-4q^{-14}+3q^{-16}+6q^{-18}-q^{-20}-7q^{-22}-4q^{-24}+5q^{-26}+5q^{-28}-2q^{-30}-6q^{-32}-2q^{-34}+4q^{-36}+3q^{-38}-q^{-40}-3q^{-42}-q^{-44}+2q^{-46}+q^{-48}-q^{-50}-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}-q^{44}+q^{42}-2 q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-4 q^{32}+5 q^{30}-5 q^{28}+4 q^{26}-5 q^{24}+3 q^{22}-5 q^{20}-q^{18}-2 q^{16}-2 q^{14}+2 q^{12}-6 q^{10}+7 q^8-4 q^6+13 q^4-5 q^2+14-5 q^{-2} +13 q^{-4} -4 q^{-6} +7 q^{-8} -6 q^{-10} +2 q^{-12} -2 q^{-14} -2 q^{-16} - q^{-18} -5 q^{-20} +3 q^{-22} -5 q^{-24} +4 q^{-26} -5 q^{-28} +5 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} +3 q^{-38} -2 q^{-40} + q^{-42} - 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}-q^{78}+2 q^{76}-3 q^{74}+2 q^{72}-q^{70}-2 q^{68}+6 q^{66}-7 q^{64}+8 q^{62}-7 q^{60}+2 q^{58}+2 q^{56}-8 q^{54}+11 q^{52}-15 q^{50}+12 q^{48}-7 q^{46}-2 q^{44}+9 q^{42}-16 q^{40}+19 q^{38}-14 q^{36}+4 q^{34}+5 q^{32}-14 q^{30}+15 q^{28}-7 q^{26}-q^{24}+11 q^{22}-12 q^{20}+9 q^{18}+2 q^{16}-12 q^{14}+21 q^{12}-20 q^{10}+14 q^8-q^6-12 q^4+23 q^2-25+23 q^{-2} -12 q^{-4} - q^{-6} +14 q^{-8} -20 q^{-10} +21 q^{-12} -12 q^{-14} +2 q^{-16} +9 q^{-18} -12 q^{-20} +11 q^{-22} - q^{-24} -7 q^{-26} +15 q^{-28} -14 q^{-30} +5 q^{-32} +4 q^{-34} -14 q^{-36} +19 q^{-38} -16 q^{-40} +9 q^{-42} -2 q^{-44} -7 q^{-46} +12 q^{-48} -15 q^{-50} +11 q^{-52} -8 q^{-54} +2 q^{-56} +2 q^{-58} -7 q^{-60} +8 q^{-62} -7 q^{-64} +6 q^{-66} -2 q^{-68} - q^{-70} +2 q^{-72} -3 q^{-74} +2 q^{-76} - q^{-78} + q^{-80} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 17"];

In[4]:= Alexander[K][t]

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

Out[4]= $t^{4}-3t^{3}+5t^{2}-7t+9-7t^{-1}+5t^{-2}-3t^{-3}+t^{-4}$

In[5]:= Conway[K][z]

Out[5]= $z^{8}+5z^{6}+7z^{4}+2z^{2}+1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 41, 0 }

In[8]:= Jones[K][q]

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

Out[8]= $-q^{5}+2q^{4}-3q^{3}+5q^{2}-6q+7-6q^{-1}+5q^{-2}-3q^{-3}+2q^{-4}-q^{-5}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $z^{8}-a^{2}z^{6}-z^{6}a^{-2}+7z^{6}-5a^{2}z^{4}-5z^{4}a^{-2}+17z^{4}-7a^{2}z^{2}-7z^{2}a^{-2}+16z^{2}-2a^{2}-2a^{-2}+5$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+2a^{3}z^{7}-2az^{7}-2z^{7}a^{-1}+2z^{7}a^{-3}+2a^{4}z^{6}-7a^{2}z^{6}-7z^{6}a^{-2}+2z^{6}a^{-4}-18z^{6}+a^{5}z^{5}-5a^{3}z^{5}-5z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+11a^{2}z^{4}+11z^{4}a^{-2}-6z^{4}a^{-4}+34z^{4}-3a^{5}z^{3}+2a^{3}z^{3}+6az^{3}+6z^{3}a^{-1}+2z^{3}a^{-3}-3z^{3}a^{-5}+3a^{4}z^{2}-8a^{2}z^{2}-8z^{2}a^{-2}+3z^{2}a^{-4}-22z^{2}+a^{5}z-3az-3za^{-1}+za^{-5}+2a^{2}+2a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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:

(2, 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 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 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 32}

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{20}{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{124}{3}}

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 \frac{256}{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 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 -\frac{160}{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{992}{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{2009}{15}}

{\frac {3476}{15}}

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{25556}{45}}

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{89}{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 -\frac{2489}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials 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 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 \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 17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

1

7

2

1

-1

5

3

1

2

3

2

-1

1

4

3

1

-1

3

4

1

-3

2

3

-1

-5

1

3

2

-7

1

2

-1

-9

1

-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}
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}	${\mathbb {Z} }$
$r=-4$	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}	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}^{2}\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=-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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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}^{3}\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=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}^{4}\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}^{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 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}^{3}\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}^{2}\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=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}\oplus{\mathbb Z}_2^{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}^{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 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}	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=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}-2 q^{14}+4 q^{12}-6 q^{11}+10 q^9-11 q^8-3 q^7+20 q^6-16 q^5-10 q^4+30 q^3-18 q^2-16 q+35-16 q^{-1} -18 q^{-2} +30 q^{-3} -10 q^{-4} -16 q^{-5} +20 q^{-6} -3 q^{-7} -11 q^{-8} +10 q^{-9} -6 q^{-11} +4 q^{-12} -2 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}+2 q^{29}-q^{27}-2 q^{26}+3 q^{25}+q^{24}-3 q^{23}-2 q^{22}+6 q^{21}-8 q^{19}-q^{18}+12 q^{17}+4 q^{16}-18 q^{15}-7 q^{14}+19 q^{13}+19 q^{12}-24 q^{11}-25 q^{10}+18 q^9+40 q^8-18 q^7-46 q^6+10 q^5+56 q^4-8 q^3-57 q^2+63-57 q^{-2} -8 q^{-3} +56 q^{-4} +10 q^{-5} -46 q^{-6} -18 q^{-7} +40 q^{-8} +18 q^{-9} -25 q^{-10} -24 q^{-11} +19 q^{-12} +19 q^{-13} -7 q^{-14} -18 q^{-15} +4 q^{-16} +12 q^{-17} - q^{-18} -8 q^{-19} +6 q^{-21} -2 q^{-22} -3 q^{-23} + q^{-24} +3 q^{-25} -2 q^{-26} - q^{-27} +2 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}-2 q^{49}+q^{47}-q^{46}+5 q^{45}-5 q^{44}+q^{43}-7 q^{41}+13 q^{40}-7 q^{39}+6 q^{38}+2 q^{37}-22 q^{36}+16 q^{35}-11 q^{34}+21 q^{33}+15 q^{32}-40 q^{31}+10 q^{30}-31 q^{29}+36 q^{28}+44 q^{27}-40 q^{26}+10 q^{25}-72 q^{24}+26 q^{23}+66 q^{22}-17 q^{21}+47 q^{20}-110 q^{19}-15 q^{18}+52 q^{17}+2 q^{16}+120 q^{15}-113 q^{14}-57 q^{13}+4 q^{12}-6 q^{11}+194 q^{10}-90 q^9-77 q^8-42 q^7-30 q^6+237 q^5-66 q^4-76 q^3-67 q^2-50 q+251-50 q^{-1} -67 q^{-2} -76 q^{-3} -66 q^{-4} +237 q^{-5} -30 q^{-6} -42 q^{-7} -77 q^{-8} -90 q^{-9} +194 q^{-10} -6 q^{-11} +4 q^{-12} -57 q^{-13} -113 q^{-14} +120 q^{-15} +2 q^{-16} +52 q^{-17} -15 q^{-18} -110 q^{-19} +47 q^{-20} -17 q^{-21} +66 q^{-22} +26 q^{-23} -72 q^{-24} +10 q^{-25} -40 q^{-26} +44 q^{-27} +36 q^{-28} -31 q^{-29} +10 q^{-30} -40 q^{-31} +15 q^{-32} +21 q^{-33} -11 q^{-34} +16 q^{-35} -22 q^{-36} +2 q^{-37} +6 q^{-38} -7 q^{-39} +13 q^{-40} -7 q^{-41} + q^{-43} -5 q^{-44} +5 q^{-45} - q^{-46} + q^{-47} -2 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}+2 q^{74}-q^{72}+q^{71}-2 q^{70}-3 q^{69}+3 q^{68}+3 q^{67}+4 q^{65}-3 q^{64}-11 q^{63}-2 q^{62}+6 q^{61}+7 q^{60}+12 q^{59}+2 q^{58}-19 q^{57}-20 q^{56}-q^{55}+13 q^{54}+31 q^{53}+21 q^{52}-16 q^{51}-44 q^{50}-34 q^{49}+2 q^{48}+50 q^{47}+60 q^{46}+16 q^{45}-47 q^{44}-78 q^{43}-48 q^{42}+28 q^{41}+86 q^{40}+81 q^{39}+8 q^{38}-73 q^{37}-99 q^{36}-63 q^{35}+30 q^{34}+108 q^{33}+106 q^{32}+34 q^{31}-61 q^{30}-151 q^{29}-125 q^{28}+13 q^{27}+148 q^{26}+196 q^{25}+104 q^{24}-130 q^{23}-279 q^{22}-189 q^{21}+66 q^{20}+309 q^{19}+315 q^{18}+q^{17}-340 q^{16}-387 q^{15}-85 q^{14}+331 q^{13}+468 q^{12}+144 q^{11}-323 q^{10}-495 q^9-207 q^8+301 q^7+535 q^6+231 q^5-292 q^4-526 q^3-264 q^2+267 q+553+267 q^{-1} -264 q^{-2} -526 q^{-3} -292 q^{-4} +231 q^{-5} +535 q^{-6} +301 q^{-7} -207 q^{-8} -495 q^{-9} -323 q^{-10} +144 q^{-11} +468 q^{-12} +331 q^{-13} -85 q^{-14} -387 q^{-15} -340 q^{-16} + q^{-17} +315 q^{-18} +309 q^{-19} +66 q^{-20} -189 q^{-21} -279 q^{-22} -130 q^{-23} +104 q^{-24} +196 q^{-25} +148 q^{-26} +13 q^{-27} -125 q^{-28} -151 q^{-29} -61 q^{-30} +34 q^{-31} +106 q^{-32} +108 q^{-33} +30 q^{-34} -63 q^{-35} -99 q^{-36} -73 q^{-37} +8 q^{-38} +81 q^{-39} +86 q^{-40} +28 q^{-41} -48 q^{-42} -78 q^{-43} -47 q^{-44} +16 q^{-45} +60 q^{-46} +50 q^{-47} +2 q^{-48} -34 q^{-49} -44 q^{-50} -16 q^{-51} +21 q^{-52} +31 q^{-53} +13 q^{-54} - q^{-55} -20 q^{-56} -19 q^{-57} +2 q^{-58} +12 q^{-59} +7 q^{-60} +6 q^{-61} -2 q^{-62} -11 q^{-63} -3 q^{-64} +4 q^{-65} +3 q^{-67} +3 q^{-68} -3 q^{-69} -2 q^{-70} + q^{-71} - q^{-72} +2 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}-2 q^{104}+q^{102}-q^{101}+2 q^{100}+5 q^{98}-7 q^{97}-3 q^{96}+2 q^{95}-5 q^{94}+5 q^{93}+4 q^{92}+17 q^{91}-14 q^{90}-9 q^{89}-16 q^{87}+6 q^{86}+10 q^{85}+41 q^{84}-17 q^{83}-17 q^{82}-4 q^{81}-36 q^{80}-q^{79}+16 q^{78}+80 q^{77}-10 q^{76}-23 q^{75}-13 q^{74}-71 q^{73}-26 q^{72}+14 q^{71}+139 q^{70}+24 q^{69}-9 q^{68}-14 q^{67}-124 q^{66}-91 q^{65}-30 q^{64}+186 q^{63}+75 q^{62}+53 q^{61}+52 q^{60}-132 q^{59}-165 q^{58}-143 q^{57}+138 q^{56}+37 q^{55}+95 q^{54}+195 q^{53}+11 q^{52}-92 q^{51}-200 q^{50}+16 q^{49}-207 q^{48}-82 q^{47}+223 q^{46}+231 q^{45}+230 q^{44}+36 q^{43}+81 q^{42}-516 q^{41}-549 q^{40}-122 q^{39}+215 q^{38}+591 q^{37}+584 q^{36}+575 q^{35}-538 q^{34}-1032 q^{33}-798 q^{32}-240 q^{31}+638 q^{30}+1130 q^{29}+1376 q^{28}-136 q^{27}-1204 q^{26}-1453 q^{25}-940 q^{24}+308 q^{23}+1382 q^{22}+2104 q^{21}+446 q^{20}-1054 q^{19}-1817 q^{18}-1520 q^{17}-138 q^{16}+1355 q^{15}+2521 q^{14}+887 q^{13}-815 q^{12}-1917 q^{11}-1810 q^{10}-445 q^9+1242 q^8+2670 q^7+1089 q^6-659 q^5-1912 q^4-1893 q^3-579 q^2+1163 q+2699+1163 q^{-1} -579 q^{-2} -1893 q^{-3} -1912 q^{-4} -659 q^{-5} +1089 q^{-6} +2670 q^{-7} +1242 q^{-8} -445 q^{-9} -1810 q^{-10} -1917 q^{-11} -815 q^{-12} +887 q^{-13} +2521 q^{-14} +1355 q^{-15} -138 q^{-16} -1520 q^{-17} -1817 q^{-18} -1054 q^{-19} +446 q^{-20} +2104 q^{-21} +1382 q^{-22} +308 q^{-23} -940 q^{-24} -1453 q^{-25} -1204 q^{-26} -136 q^{-27} +1376 q^{-28} +1130 q^{-29} +638 q^{-30} -240 q^{-31} -798 q^{-32} -1032 q^{-33} -538 q^{-34} +575 q^{-35} +584 q^{-36} +591 q^{-37} +215 q^{-38} -122 q^{-39} -549 q^{-40} -516 q^{-41} +81 q^{-42} +36 q^{-43} +230 q^{-44} +231 q^{-45} +223 q^{-46} -82 q^{-47} -207 q^{-48} +16 q^{-49} -200 q^{-50} -92 q^{-51} +11 q^{-52} +195 q^{-53} +95 q^{-54} +37 q^{-55} +138 q^{-56} -143 q^{-57} -165 q^{-58} -132 q^{-59} +52 q^{-60} +53 q^{-61} +75 q^{-62} +186 q^{-63} -30 q^{-64} -91 q^{-65} -124 q^{-66} -14 q^{-67} -9 q^{-68} +24 q^{-69} +139 q^{-70} +14 q^{-71} -26 q^{-72} -71 q^{-73} -13 q^{-74} -23 q^{-75} -10 q^{-76} +80 q^{-77} +16 q^{-78} - q^{-79} -36 q^{-80} -4 q^{-81} -17 q^{-82} -17 q^{-83} +41 q^{-84} +10 q^{-85} +6 q^{-86} -16 q^{-87} -9 q^{-89} -14 q^{-90} +17 q^{-91} +4 q^{-92} +5 q^{-93} -5 q^{-94} +2 q^{-95} -3 q^{-96} -7 q^{-97} +5 q^{-98} +2 q^{-100} - q^{-101} + q^{-102} -2 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}+2 q^{139}-q^{137}+q^{136}-2 q^{135}-2 q^{133}-q^{132}+7 q^{131}+q^{130}-q^{129}+4 q^{128}-7 q^{127}-3 q^{126}-6 q^{125}-7 q^{124}+17 q^{123}+6 q^{122}+q^{121}+10 q^{120}-12 q^{119}-5 q^{118}-13 q^{117}-21 q^{116}+27 q^{115}+14 q^{114}+3 q^{113}+18 q^{112}-25 q^{111}-4 q^{110}-13 q^{109}-36 q^{108}+42 q^{107}+30 q^{106}+15 q^{105}+19 q^{104}-62 q^{103}-27 q^{102}-23 q^{101}-51 q^{100}+77 q^{99}+79 q^{98}+71 q^{97}+55 q^{96}-115 q^{95}-103 q^{94}-105 q^{93}-121 q^{92}+93 q^{91}+150 q^{90}+198 q^{89}+198 q^{88}-76 q^{87}-156 q^{86}-240 q^{85}-305 q^{84}-14 q^{83}+106 q^{82}+267 q^{81}+397 q^{80}+112 q^{79}-8 q^{78}-206 q^{77}-410 q^{76}-176 q^{75}-131 q^{74}+48 q^{73}+314 q^{72}+150 q^{71}+217 q^{70}+142 q^{69}-88 q^{68}+62 q^{67}-160 q^{66}-289 q^{65}-216 q^{64}-460 q^{63}-125 q^{62}+261 q^{61}+477 q^{60}+972 q^{59}+675 q^{58}+78 q^{57}-529 q^{56}-1512 q^{55}-1450 q^{54}-731 q^{53}+277 q^{52}+1846 q^{51}+2262 q^{50}+1710 q^{49}+431 q^{48}-1878 q^{47}-3034 q^{46}-2852 q^{45}-1435 q^{44}+1503 q^{43}+3470 q^{42}+3975 q^{41}+2772 q^{40}-718 q^{39}-3639 q^{38}-4963 q^{37}-4104 q^{36}-306 q^{35}+3382 q^{34}+5632 q^{33}+5408 q^{32}+1485 q^{31}-2907 q^{30}-6028 q^{29}-6446 q^{28}-2598 q^{27}+2259 q^{26}+6138 q^{25}+7228 q^{24}+3551 q^{23}-1610 q^{22}-6059 q^{21}-7728 q^{20}-4287 q^{19}+1035 q^{18}+5906 q^{17}+8021 q^{16}+4764 q^{15}-614 q^{14}-5710 q^{13}-8128 q^{12}-5079 q^{11}+298 q^{10}+5565 q^9+8205 q^8+5231 q^7-162 q^6-5447 q^5-8167 q^4-5321 q^3+13 q^2+5376 q+8221+5376 q^{-1} +13 q^{-2} -5321 q^{-3} -8167 q^{-4} -5447 q^{-5} -162 q^{-6} +5231 q^{-7} +8205 q^{-8} +5565 q^{-9} +298 q^{-10} -5079 q^{-11} -8128 q^{-12} -5710 q^{-13} -614 q^{-14} +4764 q^{-15} +8021 q^{-16} +5906 q^{-17} +1035 q^{-18} -4287 q^{-19} -7728 q^{-20} -6059 q^{-21} -1610 q^{-22} +3551 q^{-23} +7228 q^{-24} +6138 q^{-25} +2259 q^{-26} -2598 q^{-27} -6446 q^{-28} -6028 q^{-29} -2907 q^{-30} +1485 q^{-31} +5408 q^{-32} +5632 q^{-33} +3382 q^{-34} -306 q^{-35} -4104 q^{-36} -4963 q^{-37} -3639 q^{-38} -718 q^{-39} +2772 q^{-40} +3975 q^{-41} +3470 q^{-42} +1503 q^{-43} -1435 q^{-44} -2852 q^{-45} -3034 q^{-46} -1878 q^{-47} +431 q^{-48} +1710 q^{-49} +2262 q^{-50} +1846 q^{-51} +277 q^{-52} -731 q^{-53} -1450 q^{-54} -1512 q^{-55} -529 q^{-56} +78 q^{-57} +675 q^{-58} +972 q^{-59} +477 q^{-60} +261 q^{-61} -125 q^{-62} -460 q^{-63} -216 q^{-64} -289 q^{-65} -160 q^{-66} +62 q^{-67} -88 q^{-68} +142 q^{-69} +217 q^{-70} +150 q^{-71} +314 q^{-72} +48 q^{-73} -131 q^{-74} -176 q^{-75} -410 q^{-76} -206 q^{-77} -8 q^{-78} +112 q^{-79} +397 q^{-80} +267 q^{-81} +106 q^{-82} -14 q^{-83} -305 q^{-84} -240 q^{-85} -156 q^{-86} -76 q^{-87} +198 q^{-88} +198 q^{-89} +150 q^{-90} +93 q^{-91} -121 q^{-92} -105 q^{-93} -103 q^{-94} -115 q^{-95} +55 q^{-96} +71 q^{-97} +79 q^{-98} +77 q^{-99} -51 q^{-100} -23 q^{-101} -27 q^{-102} -62 q^{-103} +19 q^{-104} +15 q^{-105} +30 q^{-106} +42 q^{-107} -36 q^{-108} -13 q^{-109} -4 q^{-110} -25 q^{-111} +18 q^{-112} +3 q^{-113} +14 q^{-114} +27 q^{-115} -21 q^{-116} -13 q^{-117} -5 q^{-118} -12 q^{-119} +10 q^{-120} + q^{-121} +6 q^{-122} +17 q^{-123} -7 q^{-124} -6 q^{-125} -3 q^{-126} -7 q^{-127} +4 q^{-128} - q^{-129} + q^{-130} +7 q^{-131} - q^{-132} -2 q^{-133} -2 q^{-135} + q^{-136} - q^{-137} +2 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, 17]]
Out[2]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[20, 15, 1, 16], X[16, 7, 17, 8], X[18, 9, 19, 10], X[8, 17, 9, 18], X[10, 19, 11, 20], X[14, 6, 15, 5], X[2, 12, 3, 11], X[4, 14, 5, 13]]
In[3]:=	GaussCode[Knot[10, 17]]
Out[3]=	GaussCode[1, -9, 2, -10, 8, -1, 4, -6, 5, -7, 9, -2, 10, -8, 3, -4, 6, -5, 7, -3]
In[4]:=	DTCode[Knot[10, 17]]
Out[4]=	DTCode[6, 12, 14, 16, 18, 2, 4, 20, 8, 10]
In[5]:=	br = BR[Knot[10, 17]]
Out[5]=	BR[3, {-1, -1, -1, -1, 2, -1, 2, 2, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 17]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 17]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 17]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 1, 4, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 17]][t]
Out[10]=	-4 3 5 7 2 3 4 9 + t - -- + -- - - - 7 t + 5 t - 3 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 17]][z]
Out[11]=	2 4 6 8 1 + 2 z + 7 z + 5 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 17]}
In[13]:=	{KnotDet[Knot[10, 17]], KnotSignature[Knot[10, 17]]}
Out[13]=	{41, 0}
In[14]:=	Jones[Knot[10, 17]][q]
Out[14]=	-5 2 3 5 6 2 3 4 5 7 - q + -- - -- + -- - - - 6 q + 5 q - 3 q + 2 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, 17]}
In[16]:=	A2Invariant[Knot[10, 17]][q]
Out[16]=	-14 -10 -8 -6 2 2 6 8 10 14 -1 - q - q + q + q + -- + 2 q + q + q - q - q 2 q
In[17]:=	HOMFLYPT[Knot[10, 17]][a, z]
Out[17]=	2 4 2 2 2 7 z 2 2 4 5 z 2 4 5 - -- - 2 a + 16 z - ---- - 7 a z + 17 z - ---- - 5 a z + 2 2 2 a a a 6 6 z 2 6 8 7 z - -- - a z + z 2 a
In[18]:=	Kauffman[Knot[10, 17]][a, z]
Out[18]=	2 2 2 2 z 3 z 5 2 3 z 8 z 5 + -- + 2 a + -- - --- - 3 a z + a z - 22 z + ---- - ---- - 2 5 a 4 2 a a a a 3 3 3 2 2 4 2 3 z 2 z 6 z 3 3 3 5 3 8 a z + 3 a z - ---- + ---- + ---- + 6 a z + 2 a z - 3 a z + 5 3 a a a 4 4 5 5 4 6 z 11 z 2 4 4 4 z 5 z 3 5 34 z - ---- + ----- + 11 a z - 6 a z + -- - ---- - 5 a z + 4 2 5 3 a a a a 6 6 7 7 5 5 6 2 z 7 z 2 6 4 6 2 z 2 z a z - 18 z + ---- - ---- - 7 a z + 2 a z + ---- - ---- - 4 2 3 a a a a 8 9 7 3 7 8 2 z 2 8 z 9 2 a z + 2 a z + 4 z + ---- + 2 a z + -- + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 17]], Vassiliev[3][Knot[10, 17]]}
Out[19]=	{2, 0}
In[20]:=	Kh[Knot[10, 17]][q, t]
Out[20]=	4 1 1 1 2 1 3 2 - + 4 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 3 3 3 3 2 5 2 5 3 7 3 ---- + --- + 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + 3 q t q t 7 4 9 4 11 5 q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 17], 2][q]
Out[21]=	-15 2 4 6 10 11 3 20 16 10 30 18 35 + q - --- + --- - --- + -- - -- - -- + -- - -- - -- + -- - -- - 14 12 11 9 8 7 6 5 4 3 2 q q q q q q q q q q q 16 2 3 4 5 6 7 8 -- - 16 q - 18 q + 30 q - 10 q - 16 q + 20 q - 3 q - 11 q + q 9 11 12 14 15 10 q - 6 q + 4 q - 2 q + q

See/edit the Rolfsen_Splice_Template.

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 </table>
+{| width=100%
-See/edit the [[Rolfsen_Splice_Template]].
+|align=left|See/edit the [[Rolfsen_Splice_Template]].
+Back to the [[#top|top]].
+|align=right|{{Knot Navigation Links|ext=gif}}
+|}
  [[Category:Knot Page]]

10 17: Difference between revisions

Revision as of 21:04, 29 August 2005

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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