10 15: Difference between revisions

Revision as of 21:04, 29 August 2005

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Visit 10 15's page at the original Knot Atlas!

10 15 Quick Notes

10 15 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_3,12,4,13 X_9,14,10,15 X_13,10,14,11 X_15,1,16,20 X_5,17,6,16 X_7,19,8,18 X_17,7,18,6 X_19,9,20,8 X_11,2,12,3
Gauss code	-1, 10, -2, 1, -6, 8, -7, 9, -3, 4, -10, 2, -4, 3, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 12 16 18 14 2 10 20 6 8
Conway Notation	[4132]

Minimum Braid Representative:

Length is 11, width is 4.

Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-4][-8]
Hyperbolic Volume	8.97345
A-Polynomial	See Data:10 15/A-polynomial

[edit Notes for 10 15'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	$3$
Rasmussen s-Invariant	2

[edit Notes for 10 15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-6t^{2}+9t-9+9t^{-1}-6t^{-2}+2t^{-3}$
Conway polynomial	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 2 z^6+6 z^4+3 z^2+1}
2nd Alexander ideal (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 \{1\}}
Determinant and Signature	{ 43, 2 }
Jones polynomial	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^6+2 q^5-4 q^4+6 q^3-6 q^2+7 q-6+5 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} }
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^6 a^{-2} +z^6-a^2 z^4+4 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} +4 z^2-a^2+3 a^{-2} -2 a^{-4} +1}
Kauffman 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 a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+a^3 z^7-2 a z^7+3 z^7 a^{-3} -10 a^2 z^6-z^6 a^{-2} +4 z^6 a^{-4} -15 z^6-5 a^3 z^5-4 a z^5-5 z^5 a^{-1} -3 z^5 a^{-3} +3 z^5 a^{-5} +15 a^2 z^4-8 z^4 a^{-2} -7 z^4 a^{-4} +2 z^4 a^{-6} +16 z^4+7 a^3 z^3+8 a z^3+z^3 a^{-1} -3 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+8 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} -7 z^2-2 a^3 z-3 a z+3 z a^{-3} +z a^{-5} -z a^{-7} +a^2-3 a^{-2} -2 a^{-4} +1}
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^{12}+q^4-q^2+1+ q^{-2} + q^{-4} +3 q^{-6} + q^{-10} - q^{-12} - q^{-14} - q^{-18} }
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^{60}-q^{58}+3 q^{56}-5 q^{54}+4 q^{52}-3 q^{50}-2 q^{48}+9 q^{46}-15 q^{44}+17 q^{42}-14 q^{40}+q^{38}+10 q^{36}-22 q^{34}+25 q^{32}-20 q^{30}+8 q^{28}+7 q^{26}-17 q^{24}+21 q^{22}-16 q^{20}+5 q^{18}+6 q^{16}-11 q^{14}+10 q^{12}-5 q^{10}-2 q^8+12 q^6-14 q^4+15 q^2-8-7 q^{-2} +18 q^{-4} -26 q^{-6} +27 q^{-8} -16 q^{-10} +3 q^{-12} +15 q^{-14} -23 q^{-16} +29 q^{-18} -19 q^{-20} +6 q^{-22} +8 q^{-24} -13 q^{-26} +15 q^{-28} -6 q^{-30} + q^{-32} +7 q^{-34} -6 q^{-36} +5 q^{-38} -7 q^{-42} +9 q^{-44} -9 q^{-46} +7 q^{-48} -2 q^{-50} -4 q^{-52} +7 q^{-54} -12 q^{-56} +14 q^{-58} -13 q^{-60} +5 q^{-62} -9 q^{-66} +11 q^{-68} -13 q^{-70} +11 q^{-72} -6 q^{-74} + q^{-76} +3 q^{-78} -7 q^{-80} +6 q^{-82} -5 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }

Further Quantum Invariants

Further quantum knot invariants for 10_15.

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^9+q^7-q^5+2 q^3-q+ q^{-1} + q^{-3} +2 q^{-7} -2 q^{-9} + q^{-11} - q^{-13} }
2	$q^{28}-q^{26}-2q^{24}+3q^{22}+q^{20}-5q^{18}+2q^{16}+4q^{14}-6q^{12}+5q^{8}-4q^{6}-q^{4}+6q^{2}-3q^{-2}+3q^{-4}+3q^{-6}-3q^{-8}-2q^{-10}+4q^{-12}+q^{-14}-5q^{-16}+4q^{-18}+q^{-20}-5q^{-22}+4q^{-24}-4q^{-28}+2q^{-30}-q^{-34}+q^{-36}$
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^{57}+q^{55}+2 q^{53}-4 q^{49}-3 q^{47}+5 q^{45}+7 q^{43}-3 q^{41}-10 q^{39}-2 q^{37}+12 q^{35}+8 q^{33}-11 q^{31}-12 q^{29}+5 q^{27}+15 q^{25}-q^{23}-15 q^{21}-6 q^{19}+11 q^{17}+8 q^{15}-6 q^{13}-9 q^{11}+5 q^9+11 q^7-11 q^3+12 q^{-1} +4 q^{-3} -13 q^{-5} -6 q^{-7} +13 q^{-9} +10 q^{-11} -8 q^{-13} -11 q^{-15} + q^{-17} +11 q^{-19} +5 q^{-21} -8 q^{-23} -10 q^{-25} +2 q^{-27} +13 q^{-29} -8 q^{-33} -4 q^{-35} +7 q^{-37} + q^{-39} -2 q^{-41} -3 q^{-47} +3 q^{-51} -3 q^{-55} +2 q^{-59} + q^{-67} - q^{-69} }
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^{96}-q^{94}-2 q^{92}+q^{88}+6 q^{86}+q^{84}-5 q^{82}-7 q^{80}-8 q^{78}+11 q^{76}+14 q^{74}+6 q^{72}-7 q^{70}-27 q^{68}-7 q^{66}+12 q^{64}+28 q^{62}+21 q^{60}-25 q^{58}-31 q^{56}-20 q^{54}+20 q^{52}+48 q^{50}+12 q^{48}-18 q^{46}-49 q^{44}-21 q^{42}+34 q^{40}+36 q^{38}+24 q^{36}-33 q^{34}-48 q^{32}-9 q^{30}+20 q^{28}+53 q^{26}+11 q^{24}-34 q^{22}-37 q^{20}-18 q^{18}+44 q^{16}+40 q^{14}-4 q^{12}-37 q^{10}-37 q^8+23 q^6+43 q^4+11 q^2-31-35 q^{-2} +18 q^{-4} +44 q^{-6} +11 q^{-8} -38 q^{-10} -36 q^{-12} +12 q^{-14} +48 q^{-16} +25 q^{-18} -31 q^{-20} -46 q^{-22} -19 q^{-24} +34 q^{-26} +47 q^{-28} +11 q^{-30} -24 q^{-32} -54 q^{-34} -19 q^{-36} +34 q^{-38} +54 q^{-40} +29 q^{-42} -51 q^{-44} -59 q^{-46} -7 q^{-48} +48 q^{-50} +64 q^{-52} -17 q^{-54} -53 q^{-56} -32 q^{-58} +13 q^{-60} +55 q^{-62} +6 q^{-64} -21 q^{-66} -25 q^{-68} -10 q^{-70} +27 q^{-72} +6 q^{-74} + q^{-76} -7 q^{-78} -12 q^{-80} +7 q^{-82} - q^{-84} +6 q^{-86} +2 q^{-88} -6 q^{-90} +3 q^{-92} -3 q^{-94} +2 q^{-96} +2 q^{-98} -2 q^{-100} +2 q^{-102} -2 q^{-104} - q^{-110} + q^{-112} }
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^{145}+q^{143}+2 q^{141}-q^{137}-3 q^{135}-4 q^{133}-q^{131}+7 q^{129}+9 q^{127}+5 q^{125}-2 q^{123}-14 q^{121}-19 q^{119}-8 q^{117}+14 q^{115}+27 q^{113}+26 q^{111}+6 q^{109}-28 q^{107}-46 q^{105}-34 q^{103}+8 q^{101}+49 q^{99}+64 q^{97}+35 q^{95}-30 q^{93}-80 q^{91}-78 q^{89}-17 q^{87}+62 q^{85}+106 q^{83}+78 q^{81}-13 q^{79}-101 q^{77}-123 q^{75}-52 q^{73}+55 q^{71}+130 q^{69}+119 q^{67}+17 q^{65}-99 q^{63}-146 q^{61}-90 q^{59}+24 q^{57}+129 q^{55}+149 q^{53}+56 q^{51}-76 q^{49}-158 q^{47}-133 q^{45}-11 q^{43}+131 q^{41}+182 q^{39}+93 q^{37}-70 q^{35}-194 q^{33}-166 q^{31}-7 q^{29}+169 q^{27}+215 q^{25}+81 q^{23}-125 q^{21}-224 q^{19}-136 q^{17}+67 q^{15}+215 q^{13}+174 q^{11}-22 q^9-182 q^7-174 q^5-14 q^3+150 q+163 q^{-1} +21 q^{-3} -124 q^{-5} -137 q^{-7} -14 q^{-9} +117 q^{-11} +122 q^{-13} - q^{-15} -129 q^{-17} -122 q^{-19} +11 q^{-21} +144 q^{-23} +143 q^{-25} +6 q^{-27} -146 q^{-29} -172 q^{-31} -52 q^{-33} +114 q^{-35} +190 q^{-37} +115 q^{-39} -41 q^{-41} -172 q^{-43} -187 q^{-45} -62 q^{-47} +116 q^{-49} +226 q^{-51} +168 q^{-53} -21 q^{-55} -213 q^{-57} -260 q^{-59} -88 q^{-61} +173 q^{-63} +297 q^{-65} +175 q^{-67} -88 q^{-69} -288 q^{-71} -245 q^{-73} +20 q^{-75} +249 q^{-77} +253 q^{-79} +45 q^{-81} -188 q^{-83} -244 q^{-85} -79 q^{-87} +133 q^{-89} +201 q^{-91} +92 q^{-93} -85 q^{-95} -157 q^{-97} -81 q^{-99} +43 q^{-101} +111 q^{-103} +70 q^{-105} -20 q^{-107} -71 q^{-109} -50 q^{-111} +2 q^{-113} +38 q^{-115} +37 q^{-117} +7 q^{-119} -19 q^{-121} -23 q^{-123} -7 q^{-125} +6 q^{-127} +12 q^{-129} +8 q^{-131} -7 q^{-135} -4 q^{-137} +2 q^{-143} +3 q^{-145} -2 q^{-147} - q^{-149} -2 q^{-153} +2 q^{-157} + q^{-163} - q^{-165} }

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^{12}+q^4-q^2+1+ q^{-2} + q^{-4} +3 q^{-6} + q^{-10} - q^{-12} - q^{-14} - q^{-18} }
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^{36}-2 q^{34}+6 q^{32}-14 q^{30}+23 q^{28}-36 q^{26}+52 q^{24}-64 q^{22}+69 q^{20}-70 q^{18}+62 q^{16}-46 q^{14}+20 q^{12}+4 q^{10}-32 q^8+54 q^6-79 q^4+98 q^2-102+112 q^{-2} -97 q^{-4} +92 q^{-6} -68 q^{-8} +50 q^{-10} -27 q^{-12} +10 q^{-14} +2 q^{-16} -12 q^{-18} +20 q^{-20} -28 q^{-22} +28 q^{-24} -34 q^{-26} +36 q^{-28} -38 q^{-30} +34 q^{-32} -30 q^{-34} +26 q^{-36} -22 q^{-38} +16 q^{-40} -12 q^{-42} +9 q^{-44} -6 q^{-46} +4 q^{-48} -2 q^{-50} + q^{-52} }
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^{34}-q^{30}-q^{28}+q^{26}+q^{24}-2 q^{22}-q^{20}+2 q^{18}+2 q^{16}-3 q^{14}-3 q^{12}-2 q^6+4 q^2+1+2 q^{-2} +4 q^{-4} + q^{-6} +2 q^{-10} +4 q^{-12} -2 q^{-14} +4 q^{-18} +2 q^{-20} -5 q^{-22} -2 q^{-24} + q^{-26} -2 q^{-28} -2 q^{-30} - q^{-32} - q^{-36} + q^{-40} + q^{-46} }

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^{26}-q^{24}+q^{22}-3 q^{18}+2 q^{16}-q^{14}-3 q^{12}+2 q^{10}-q^8-4 q^6+4 q^4-3+6 q^{-2} +3 q^{-4} +4 q^{-8} +5 q^{-10} +3 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} -6 q^{-20} -2 q^{-22} +3 q^{-24} -6 q^{-26} -2 q^{-28} +5 q^{-30} -3 q^{-32} -2 q^{-34} +3 q^{-36} - q^{-40} + q^{-42} }

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^{15}-q^{11}+q^9-q^7+q^5-q^3+q+2 q^{-3} +2 q^{-5} +2 q^{-7} +3 q^{-9} + q^{-13} -2 q^{-15} -2 q^{-19} - q^{-23} }

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{32}+q^{26}-q^{22}-q^{18}-3q^{16}-q^{14}-2q^{12}-4q^{10}-3q^{8}+2q^{6}+q^{4}-3q^{2}+3+7q^{-2}+q^{-4}+7q^{-8}+6q^{-10}+3q^{-12}+7q^{-14}+7q^{-16}+4q^{-18}+2q^{-22}-3q^{-24}-8q^{-26}-5q^{-28}-2q^{-30}-6q^{-32}-5q^{-34}+q^{-36}+q^{-38}-q^{-40}-q^{-42}+2q^{-44}+2q^{-46}+q^{-52}$
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^{18}-q^{14}-q^8+q^6-q^4+q^2+ q^{-2} +2 q^{-4} +2 q^{-6} +3 q^{-8} +2 q^{-10} +3 q^{-12} + q^{-16} -2 q^{-18} - q^{-20} - q^{-22} -2 q^{-24} - q^{-28} }

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^{26}+q^{24}-3 q^{22}+4 q^{20}-5 q^{18}+6 q^{16}-7 q^{14}+7 q^{12}-6 q^{10}+5 q^8-2 q^6+4 q^2-7+10 q^{-2} -11 q^{-4} +14 q^{-6} -12 q^{-8} +13 q^{-10} -9 q^{-12} +8 q^{-14} -4 q^{-16} + q^{-18} +2 q^{-20} -4 q^{-22} +5 q^{-24} -6 q^{-26} +6 q^{-28} -7 q^{-30} +5 q^{-32} -4 q^{-34} +3 q^{-36} -2 q^{-38} + q^{-40} - q^{-42} }

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^{44}-q^{40}-q^{38}+2 q^{36}+2 q^{34}-3 q^{32}-4 q^{30}+q^{28}+5 q^{26}+q^{24}-6 q^{22}-4 q^{20}+4 q^{18}+6 q^{16}-q^{14}-8 q^{12}-3 q^{10}+5 q^8+6 q^6-2 q^4-6 q^2+6 q^{-2} +3 q^{-4} -2 q^{-6} - q^{-8} +5 q^{-10} +3 q^{-12} -2 q^{-14} -2 q^{-16} +5 q^{-18} +5 q^{-20} - q^{-22} -5 q^{-24} +5 q^{-28} +2 q^{-30} -5 q^{-32} -5 q^{-34} + q^{-36} +5 q^{-38} -6 q^{-42} -5 q^{-44} +2 q^{-46} +6 q^{-48} -4 q^{-52} -3 q^{-54} + q^{-56} +3 q^{-58} + q^{-60} - q^{-62} - q^{-64} + q^{-68} }

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^{34}-q^{32}+2 q^{30}-3 q^{28}+4 q^{26}-5 q^{24}+4 q^{22}-6 q^{20}+5 q^{18}-7 q^{16}+3 q^{14}-5 q^{12}+3 q^{10}-2 q^8-q^6+2 q^4-2 q^2+7-6 q^{-2} +10 q^{-4} -7 q^{-6} +13 q^{-8} -7 q^{-10} +14 q^{-12} -5 q^{-14} +11 q^{-16} -4 q^{-18} +6 q^{-20} -3 q^{-22} -2 q^{-26} -5 q^{-28} -5 q^{-32} +3 q^{-34} -6 q^{-36} +3 q^{-38} -5 q^{-40} +6 q^{-42} -4 q^{-44} +2 q^{-46} -3 q^{-48} +3 q^{-50} - q^{-52} + q^{-54} - q^{-56} + q^{-58} }

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^{60}-q^{58}+3 q^{56}-5 q^{54}+4 q^{52}-3 q^{50}-2 q^{48}+9 q^{46}-15 q^{44}+17 q^{42}-14 q^{40}+q^{38}+10 q^{36}-22 q^{34}+25 q^{32}-20 q^{30}+8 q^{28}+7 q^{26}-17 q^{24}+21 q^{22}-16 q^{20}+5 q^{18}+6 q^{16}-11 q^{14}+10 q^{12}-5 q^{10}-2 q^8+12 q^6-14 q^4+15 q^2-8-7 q^{-2} +18 q^{-4} -26 q^{-6} +27 q^{-8} -16 q^{-10} +3 q^{-12} +15 q^{-14} -23 q^{-16} +29 q^{-18} -19 q^{-20} +6 q^{-22} +8 q^{-24} -13 q^{-26} +15 q^{-28} -6 q^{-30} + q^{-32} +7 q^{-34} -6 q^{-36} +5 q^{-38} -7 q^{-42} +9 q^{-44} -9 q^{-46} +7 q^{-48} -2 q^{-50} -4 q^{-52} +7 q^{-54} -12 q^{-56} +14 q^{-58} -13 q^{-60} +5 q^{-62} -9 q^{-66} +11 q^{-68} -13 q^{-70} +11 q^{-72} -6 q^{-74} + q^{-76} +3 q^{-78} -7 q^{-80} +6 q^{-82} -5 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }

.

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 15"];

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

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

Out[4]= $2t^{3}-6t^{2}+9t-9+9t^{-1}-6t^{-2}+2t^{-3}$

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

Out[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 2 z^6+6 z^4+3 z^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]= 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\}}

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

Out[7]= { 43, 2 }

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

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

Out[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 -q^6+2 q^5-4 q^4+6 q^3-6 q^2+7 q-6+5 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} }

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

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

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

Out[10]= 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 a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+a^3 z^7-2 a z^7+3 z^7 a^{-3} -10 a^2 z^6-z^6 a^{-2} +4 z^6 a^{-4} -15 z^6-5 a^3 z^5-4 a z^5-5 z^5 a^{-1} -3 z^5 a^{-3} +3 z^5 a^{-5} +15 a^2 z^4-8 z^4 a^{-2} -7 z^4 a^{-4} +2 z^4 a^{-6} +16 z^4+7 a^3 z^3+8 a z^3+z^3 a^{-1} -3 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} -7 a^2 z^2+8 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} -7 z^2-2 a^3 z-3 a z+3 z a^{-3} +z a^{-5} -z a^{-7} +a^2-3 a^{-2} -2 a^{-4} +1}

"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:

(3, 2)

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 12}

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}

72

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 78}

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 -14}

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 192}

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{736}{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{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 -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 288}

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 128}

936

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 -168}

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{11711}{10}}

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{7334}{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{4738}{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{257}{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 -\frac{1729}{10}}

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=} 2 is the signature of 10 15. 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

13

1

-1

11

1

9

3

1

-2

7

3

1

2

5

3

0

3

4

3

1

3

4

1

-1

2

3

-1

-3

1

3

2

-5

1

2

-1

-7

1

-9

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=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}}
$r=-4$	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}^{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=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^{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=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^{17}-2 q^{16}+q^{15}+3 q^{14}-8 q^{13}+5 q^{12}+7 q^{11}-17 q^{10}+11 q^9+10 q^8-26 q^7+17 q^6+13 q^5-32 q^4+16 q^3+19 q^2-32 q+10+22 q^{-1} -26 q^{-2} +3 q^{-3} +19 q^{-4} -17 q^{-5} -2 q^{-6} +13 q^{-7} -7 q^{-8} -4 q^{-9} +6 q^{-10} - q^{-11} -2 q^{-12} + q^{-13} }
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^{33}+2 q^{32}-q^{31}-q^{29}+4 q^{28}-3 q^{27}-3 q^{26}+2 q^{25}+7 q^{24}-6 q^{23}-6 q^{22}+5 q^{21}+7 q^{20}-8 q^{19}-3 q^{18}+11 q^{17}-4 q^{16}-12 q^{15}+5 q^{14}+24 q^{13}-15 q^{12}-24 q^{11}+7 q^{10}+37 q^9-9 q^8-34 q^7-5 q^6+40 q^5+9 q^4-31 q^3-24 q^2+33 q+26-23 q^{-1} -36 q^{-2} +22 q^{-3} +37 q^{-4} -12 q^{-5} -42 q^{-6} +8 q^{-7} +40 q^{-8} +2 q^{-9} -39 q^{-10} -9 q^{-11} +31 q^{-12} +16 q^{-13} -23 q^{-14} -19 q^{-15} +14 q^{-16} +17 q^{-17} -4 q^{-18} -15 q^{-19} +9 q^{-21} +3 q^{-22} -5 q^{-23} -2 q^{-24} + q^{-25} +2 q^{-26} - q^{-27} }
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^{54}-2 q^{53}+q^{52}-2 q^{50}+5 q^{49}-6 q^{48}+5 q^{47}-7 q^{45}+11 q^{44}-15 q^{43}+13 q^{42}+4 q^{41}-14 q^{40}+19 q^{39}-34 q^{38}+18 q^{37}+12 q^{36}-9 q^{35}+40 q^{34}-71 q^{33}+3 q^{32}+16 q^{31}+18 q^{30}+89 q^{29}-113 q^{28}-42 q^{27}-5 q^{26}+54 q^{25}+170 q^{24}-129 q^{23}-97 q^{22}-57 q^{21}+62 q^{20}+250 q^{19}-104 q^{18}-117 q^{17}-110 q^{16}+27 q^{15}+280 q^{14}-69 q^{13}-81 q^{12}-123 q^{11}-26 q^{10}+253 q^9-54 q^8-25 q^7-100 q^6-62 q^5+205 q^4-56 q^3+24 q^2-67 q-88+152 q^{-1} -52 q^{-2} +66 q^{-3} -35 q^{-4} -108 q^{-5} +92 q^{-6} -52 q^{-7} +99 q^{-8} +9 q^{-9} -104 q^{-10} +30 q^{-11} -71 q^{-12} +102 q^{-13} +54 q^{-14} -62 q^{-15} -3 q^{-16} -100 q^{-17} +63 q^{-18} +69 q^{-19} -5 q^{-20} +9 q^{-21} -102 q^{-22} +8 q^{-23} +41 q^{-24} +26 q^{-25} +39 q^{-26} -66 q^{-27} -20 q^{-28} + q^{-29} +15 q^{-30} +45 q^{-31} -20 q^{-32} -13 q^{-33} -15 q^{-34} -4 q^{-35} +25 q^{-36} -7 q^{-39} -7 q^{-40} +6 q^{-41} + q^{-42} +2 q^{-43} - q^{-44} -2 q^{-45} + q^{-46} }
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^{80}+2 q^{79}-q^{78}+2 q^{76}-2 q^{75}-3 q^{74}+4 q^{73}-2 q^{72}-q^{71}+7 q^{70}-3 q^{69}-5 q^{68}+4 q^{67}-6 q^{66}-4 q^{65}+14 q^{64}+5 q^{63}-q^{62}-2 q^{61}-19 q^{60}-20 q^{59}+18 q^{58}+31 q^{57}+29 q^{56}-q^{55}-55 q^{54}-72 q^{53}-3 q^{52}+82 q^{51}+119 q^{50}+40 q^{49}-123 q^{48}-196 q^{47}-79 q^{46}+154 q^{45}+296 q^{44}+149 q^{43}-191 q^{42}-408 q^{41}-244 q^{40}+210 q^{39}+529 q^{38}+357 q^{37}-195 q^{36}-637 q^{35}-509 q^{34}+167 q^{33}+729 q^{32}+620 q^{31}-73 q^{30}-761 q^{29}-770 q^{28}-5 q^{27}+776 q^{26}+812 q^{25}+116 q^{24}-703 q^{23}-880 q^{22}-183 q^{21}+651 q^{20}+827 q^{19}+247 q^{18}-547 q^{17}-805 q^{16}-259 q^{15}+485 q^{14}+707 q^{13}+273 q^{12}-395 q^{11}-668 q^{10}-258 q^9+352 q^8+574 q^7+266 q^6-267 q^5-545 q^4-263 q^3+221 q^2+451 q+279-122 q^{-1} -403 q^{-2} -276 q^{-3} +57 q^{-4} +291 q^{-5} +271 q^{-6} +38 q^{-7} -207 q^{-8} -235 q^{-9} -91 q^{-10} +88 q^{-11} +183 q^{-12} +137 q^{-13} - q^{-14} -101 q^{-15} -137 q^{-16} -88 q^{-17} +24 q^{-18} +109 q^{-19} +123 q^{-20} +62 q^{-21} -48 q^{-22} -139 q^{-23} -118 q^{-24} -13 q^{-25} +98 q^{-26} +144 q^{-27} +84 q^{-28} -46 q^{-29} -138 q^{-30} -118 q^{-31} -16 q^{-32} +88 q^{-33} +131 q^{-34} +70 q^{-35} -36 q^{-36} -107 q^{-37} -91 q^{-38} -19 q^{-39} +60 q^{-40} +92 q^{-41} +52 q^{-42} -16 q^{-43} -63 q^{-44} -63 q^{-45} -19 q^{-46} +31 q^{-47} +50 q^{-48} +34 q^{-49} +2 q^{-50} -34 q^{-51} -34 q^{-52} -10 q^{-53} +8 q^{-54} +22 q^{-55} +20 q^{-56} -14 q^{-58} -9 q^{-59} -5 q^{-60} +9 q^{-62} +5 q^{-63} -2 q^{-64} -2 q^{-65} - q^{-66} -2 q^{-67} + q^{-68} +2 q^{-69} - q^{-70} }
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^{111}-2 q^{110}+q^{109}-2 q^{107}+2 q^{106}+5 q^{104}-7 q^{103}+3 q^{102}+q^{101}-9 q^{100}+5 q^{99}+q^{98}+11 q^{97}-12 q^{96}+9 q^{95}+q^{94}-28 q^{93}+7 q^{92}+6 q^{91}+23 q^{90}-10 q^{89}+24 q^{88}-2 q^{87}-72 q^{86}-3 q^{85}+8 q^{84}+52 q^{83}+20 q^{82}+66 q^{81}-8 q^{80}-166 q^{79}-55 q^{78}-10 q^{77}+112 q^{76}+119 q^{75}+173 q^{74}-17 q^{73}-350 q^{72}-207 q^{71}-69 q^{70}+238 q^{69}+355 q^{68}+396 q^{67}-46 q^{66}-694 q^{65}-540 q^{64}-194 q^{63}+478 q^{62}+821 q^{61}+811 q^{60}-81 q^{59}-1249 q^{58}-1151 q^{57}-478 q^{56}+785 q^{55}+1552 q^{54}+1522 q^{53}+33 q^{52}-1889 q^{51}-2047 q^{50}-1078 q^{49}+904 q^{48}+2342 q^{47}+2515 q^{46}+522 q^{45}-2251 q^{44}-2949 q^{43}-1979 q^{42}+569 q^{41}+2759 q^{40}+3437 q^{39}+1334 q^{38}-2062 q^{37}-3380 q^{36}-2770 q^{35}-105 q^{34}+2563 q^{33}+3809 q^{32}+2013 q^{31}-1504 q^{30}-3173 q^{29}-3019 q^{28}-649 q^{27}+2009 q^{26}+3576 q^{25}+2210 q^{24}-1025 q^{23}-2653 q^{22}-2797 q^{21}-828 q^{20}+1513 q^{19}+3127 q^{18}+2098 q^{17}-769 q^{16}-2197 q^{15}-2500 q^{14}-861 q^{13}+1144 q^{12}+2758 q^{11}+2026 q^{10}-507 q^9-1810 q^8-2319 q^7-1011 q^6+700 q^5+2401 q^4+2064 q^3-72 q^2-1302 q-2107-1250 q^{-1} +88 q^{-2} +1862 q^{-3} +2008 q^{-4} +432 q^{-5} -612 q^{-6} -1664 q^{-7} -1344 q^{-8} -554 q^{-9} +1099 q^{-10} +1656 q^{-11} +752 q^{-12} +109 q^{-13} -961 q^{-14} -1097 q^{-15} -958 q^{-16} +291 q^{-17} +990 q^{-18} +681 q^{-19} +577 q^{-20} -198 q^{-21} -510 q^{-22} -914 q^{-23} -239 q^{-24} +248 q^{-25} +229 q^{-26} +566 q^{-27} +271 q^{-28} +142 q^{-29} -461 q^{-30} -264 q^{-31} -182 q^{-32} -282 q^{-33} +154 q^{-34} +233 q^{-35} +449 q^{-36} +25 q^{-37} +90 q^{-38} -113 q^{-39} -439 q^{-40} -242 q^{-41} -117 q^{-42} +281 q^{-43} +143 q^{-44} +368 q^{-45} +196 q^{-46} -185 q^{-47} -264 q^{-48} -320 q^{-49} -40 q^{-50} -78 q^{-51} +283 q^{-52} +307 q^{-53} +113 q^{-54} -24 q^{-55} -188 q^{-56} -125 q^{-57} -260 q^{-58} +28 q^{-59} +138 q^{-60} +149 q^{-61} +118 q^{-62} +24 q^{-63} +9 q^{-64} -197 q^{-65} -82 q^{-66} -34 q^{-67} +25 q^{-68} +59 q^{-69} +74 q^{-70} +97 q^{-71} -52 q^{-72} -31 q^{-73} -53 q^{-74} -33 q^{-75} -22 q^{-76} +17 q^{-77} +67 q^{-78} +4 q^{-79} +15 q^{-80} -10 q^{-81} -14 q^{-82} -27 q^{-83} -12 q^{-84} +19 q^{-85} + q^{-86} +11 q^{-87} +4 q^{-88} +3 q^{-89} -9 q^{-90} -7 q^{-91} +4 q^{-92} -2 q^{-93} +2 q^{-94} + q^{-95} +2 q^{-96} - q^{-97} -2 q^{-98} + q^{-99} }
7	Not Available

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, 15]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[9, 14, 10, 15], X[13, 10, 14, 11], X[15, 1, 16, 20], X[5, 17, 6, 16], X[7, 19, 8, 18], X[17, 7, 18, 6], X[19, 9, 20, 8], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[10, 15]]
Out[3]=	GaussCode[-1, 10, -2, 1, -6, 8, -7, 9, -3, 4, -10, 2, -4, 3, -5, 6, -8, 7, -9, 5]
In[4]:=	DTCode[Knot[10, 15]]
Out[4]=	DTCode[4, 12, 16, 18, 14, 2, 10, 20, 6, 8]
In[5]:=	br = BR[Knot[10, 15]]
Out[5]=	BR[4, {1, 1, 1, 1, -2, 1, -2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 15]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 15]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 15]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 15]][t]
Out[10]=	2 6 9 2 3 -9 + -- - -- + - + 9 t - 6 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 15]][z]
Out[11]=	2 4 6 1 + 3 z + 6 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 15]}
In[13]:=	{KnotDet[Knot[10, 15]], KnotSignature[Knot[10, 15]]}
Out[13]=	{43, 2}
In[14]:=	Jones[Knot[10, 15]][q]
Out[14]=	-4 2 3 5 2 3 4 5 6 -6 - q + -- - -- + - + 7 q - 6 q + 6 q - 4 q + 2 q - q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 15]}
In[16]:=	A2Invariant[Knot[10, 15]][q]
Out[16]=	-12 -4 -2 2 4 6 10 12 14 18 1 - q + q - q + q + q + 3 q + q - q - q - q
In[17]:=	HOMFLYPT[Knot[10, 15]][a, z]
Out[17]=	2 2 4 4 2 3 2 2 3 z 5 z 2 2 4 z 4 z 1 - -- + -- - a + 4 z - ---- + ---- - 3 a z + 4 z - -- + ---- - 4 2 4 2 4 2 a a a a a a 6 2 4 6 z a z + z + -- 2 a
In[18]:=	Kauffman[Knot[10, 15]][a, z]
Out[18]=	2 2 2 3 2 z z 3 z 3 2 z 7 z 1 - -- - -- + a - -- + -- + --- - 3 a z - 2 a z - 7 z - -- + ---- + 4 2 7 5 3 6 4 a a a a a a a 2 3 3 3 3 8 z 2 2 z 2 z 3 z z 3 3 3 4 ---- - 7 a z + -- - ---- - ---- + -- + 8 a z + 7 a z + 16 z + 2 7 5 3 a a a a a 4 4 4 5 5 5 2 z 7 z 8 z 2 4 3 z 3 z 5 z 5 ---- - ---- - ---- + 15 a z + ---- - ---- - ---- - 4 a z - 6 4 2 5 3 a a a a a a 6 6 7 3 5 6 4 z z 2 6 3 z 7 3 7 5 a z - 15 z + ---- - -- - 10 a z + ---- - 2 a z + a z + 4 2 3 a a a 8 9 8 2 z 2 8 z 9 4 z + ---- + 2 a z + -- + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 15]], Vassiliev[3][Knot[10, 15]]}
Out[19]=	{3, 2}
In[20]:=	Kh[Knot[10, 15]][q, t]
Out[20]=	3 1 1 1 2 1 3 2 4 q + 4 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t 3 3 q 3 5 5 2 7 2 7 3 9 3 --- + --- + 3 q t + 3 q t + 3 q t + 3 q t + q t + 3 q t + q t t 9 4 11 4 13 5 q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 15], 2][q]
Out[21]=	-13 2 -11 6 4 7 13 2 17 19 3 26 10 + q - --- - q + --- - -- - -- + -- - -- - -- + -- + -- - -- + 12 10 9 8 7 6 5 4 3 2 q q q q q q q q q q 22 2 3 4 5 6 7 8 -- - 32 q + 19 q + 16 q - 32 q + 13 q + 17 q - 26 q + 10 q + q 9 10 11 12 13 14 15 16 17 11 q - 17 q + 7 q + 5 q - 8 q + 3 q + q - 2 q + q

See/edit the Rolfsen_Splice_Template.

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10 15: 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

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