10 81

10_80

10_82

Visit 10 81's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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Visit 10 81's page at the original Knot Atlas!

10 81 Quick Notes

10 81 Further Notes and Views

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,6,13,5 X_16,9,17,10 X_20,17,1,18 X_18,13,19,14 X_14,19,15,20 X_10,15,11,16 X_6,12,7,11 X₂₈₃₇
Gauss code	1, -10, 2, -1, 3, -9, 10, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5
Dowker-Thistlethwaite code	4 8 12 2 16 6 18 10 20 14
Conway Notation	[(21,2)(21,2)]

Three dimensional invariants

Symmetry type	Negative amphicheiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	14.4927
A-Polynomial	See Data:10 81/A-polynomial

[edit Notes for 10 81'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	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 3}
Rasmussen s-Invariant	0

[edit Notes for 10 81's four dimensional invariants]

Polynomial invariants

Alexander 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 -t^3+8 t^2-20 t+27-20 t^{-1} +8 t^{-2} - t^{-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 -z^6+2 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	{ 85, 0 }
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+11q^{2}-13q+15-13q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}-a^{4}z^{2}+3a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}-z^{2}-a^{4}+a^{2}+a^{-2}-a^{-4}+1$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+5a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}+3z^{6}a^{-4}-6z^{6}+a^{5}z^{5}-8a^{3}z^{5}-31az^{5}-31z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}-9a^{2}z^{4}-9z^{4}a^{-2}-5z^{4}a^{-4}-8z^{4}-2a^{5}z^{3}+5a^{3}z^{3}+25az^{3}+25z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+6a^{2}z^{2}+6z^{2}a^{-2}+3z^{2}a^{-4}+6z^{2}+a^{5}z-2a^{3}z-8az-8za^{-1}-2za^{-3}+za^{-5}-a^{4}-a^{2}-a^{-2}-a^{-4}+1$
The A2 invariant	$-q^{16}+q^{12}-3q^{10}+2q^{8}-q^{4}+4q^{2}-1+4q^{-2}-q^{-4}+2q^{-8}-3q^{-10}+q^{-12}-q^{-16}$
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}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-9 q^{70}+q^{68}+14 q^{66}-35 q^{64}+56 q^{62}-69 q^{60}+55 q^{58}-16 q^{56}-50 q^{54}+129 q^{52}-183 q^{50}+191 q^{48}-130 q^{46}+4 q^{44}+139 q^{42}-255 q^{40}+293 q^{38}-227 q^{36}+79 q^{34}+91 q^{32}-219 q^{30}+247 q^{28}-166 q^{26}+14 q^{24}+137 q^{22}-214 q^{20}+173 q^{18}-34 q^{16}-147 q^{14}+296 q^{12}-335 q^{10}+249 q^8-55 q^6-172 q^4+360 q^2-427+360 q^{-2} -172 q^{-4} -55 q^{-6} +249 q^{-8} -335 q^{-10} +296 q^{-12} -147 q^{-14} -34 q^{-16} +173 q^{-18} -214 q^{-20} +137 q^{-22} +14 q^{-24} -166 q^{-26} +247 q^{-28} -219 q^{-30} +91 q^{-32} +79 q^{-34} -227 q^{-36} +293 q^{-38} -255 q^{-40} +139 q^{-42} +4 q^{-44} -130 q^{-46} +191 q^{-48} -183 q^{-50} +129 q^{-52} -50 q^{-54} -16 q^{-56} +55 q^{-58} -69 q^{-60} +56 q^{-62} -35 q^{-64} +14 q^{-66} + q^{-68} -9 q^{-70} +9 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} }

Further Quantum Invariants

Further quantum knot invariants for 10_81.

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}+2 q^9-4 q^7+4 q^5-2 q^3+2 q+2 q^{-1} -2 q^{-3} +4 q^{-5} -4 q^{-7} +2 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}-2 q^{30}+8 q^{26}-10 q^{24}-8 q^{22}+26 q^{20}-13 q^{18}-27 q^{16}+38 q^{14}-38 q^{10}+26 q^8+15 q^6-26 q^4+q^2+21+ q^{-2} -26 q^{-4} +15 q^{-6} +26 q^{-8} -38 q^{-10} +38 q^{-14} -27 q^{-16} -13 q^{-18} +26 q^{-20} -8 q^{-22} -10 q^{-24} +8 q^{-26} -2 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}+2 q^{61}-4 q^{57}-2 q^{55}+12 q^{53}+9 q^{51}-26 q^{49}-25 q^{47}+41 q^{45}+58 q^{43}-47 q^{41}-113 q^{39}+41 q^{37}+173 q^{35}-3 q^{33}-226 q^{31}-68 q^{29}+260 q^{27}+140 q^{25}-250 q^{23}-215 q^{21}+207 q^{19}+260 q^{17}-137 q^{15}-277 q^{13}+64 q^{11}+255 q^9+21 q^7-215 q^5-91 q^3+159 q+159 q^{-1} -91 q^{-3} -215 q^{-5} +21 q^{-7} +255 q^{-9} +64 q^{-11} -277 q^{-13} -137 q^{-15} +260 q^{-17} +207 q^{-19} -215 q^{-21} -250 q^{-23} +140 q^{-25} +260 q^{-27} -68 q^{-29} -226 q^{-31} -3 q^{-33} +173 q^{-35} +41 q^{-37} -113 q^{-39} -47 q^{-41} +58 q^{-43} +41 q^{-45} -25 q^{-47} -26 q^{-49} +9 q^{-51} +12 q^{-53} -2 q^{-55} -4 q^{-57} +2 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}-2 q^{102}+4 q^{98}-2 q^{96}-13 q^{92}+q^{90}+31 q^{88}+8 q^{86}-9 q^{84}-80 q^{82}-34 q^{80}+117 q^{78}+121 q^{76}+39 q^{74}-273 q^{72}-277 q^{70}+153 q^{68}+456 q^{66}+436 q^{64}-420 q^{62}-907 q^{60}-292 q^{58}+744 q^{56}+1385 q^{54}+71 q^{52}-1490 q^{50}-1436 q^{48}+256 q^{46}+2270 q^{44}+1356 q^{42}-1174 q^{40}-2491 q^{38}-1058 q^{36}+2127 q^{34}+2496 q^{32}+51 q^{30}-2484 q^{28}-2193 q^{26}+1010 q^{24}+2579 q^{22}+1212 q^{20}-1509 q^{18}-2375 q^{16}-229 q^{14}+1776 q^{12}+1719 q^{10}-323 q^8-1855 q^6-1131 q^4+741 q^2+1799+741 q^{-2} -1131 q^{-4} -1855 q^{-6} -323 q^{-8} +1719 q^{-10} +1776 q^{-12} -229 q^{-14} -2375 q^{-16} -1509 q^{-18} +1212 q^{-20} +2579 q^{-22} +1010 q^{-24} -2193 q^{-26} -2484 q^{-28} +51 q^{-30} +2496 q^{-32} +2127 q^{-34} -1058 q^{-36} -2491 q^{-38} -1174 q^{-40} +1356 q^{-42} +2270 q^{-44} +256 q^{-46} -1436 q^{-48} -1490 q^{-50} +71 q^{-52} +1385 q^{-54} +744 q^{-56} -292 q^{-58} -907 q^{-60} -420 q^{-62} +436 q^{-64} +456 q^{-66} +153 q^{-68} -277 q^{-70} -273 q^{-72} +39 q^{-74} +121 q^{-76} +117 q^{-78} -34 q^{-80} -80 q^{-82} -9 q^{-84} +8 q^{-86} +31 q^{-88} + q^{-90} -13 q^{-92} -2 q^{-96} +4 q^{-98} -2 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}+2 q^{153}-4 q^{149}+2 q^{147}+4 q^{145}+q^{143}+3 q^{141}-6 q^{139}-24 q^{137}-7 q^{135}+34 q^{133}+49 q^{131}+30 q^{129}-49 q^{127}-139 q^{125}-122 q^{123}+73 q^{121}+307 q^{119}+323 q^{117}-3 q^{115}-536 q^{113}-776 q^{111}-304 q^{109}+763 q^{107}+1544 q^{105}+1063 q^{103}-721 q^{101}-2541 q^{99}-2552 q^{97}-33 q^{95}+3502 q^{93}+4831 q^{91}+1897 q^{89}-3746 q^{87}-7535 q^{85}-5268 q^{83}+2547 q^{81}+9987 q^{79}+9878 q^{77}+624 q^{75}-11060 q^{73}-14906 q^{71}-5906 q^{69}+9898 q^{67}+19206 q^{65}+12485 q^{63}-6200 q^{61}-21361 q^{59}-19110 q^{57}+243 q^{55}+20810 q^{53}+24323 q^{51}+6647 q^{49}-17440 q^{47}-26957 q^{45}-13219 q^{43}+12096 q^{41}+26728 q^{39}+18078 q^{37}-5902 q^{35}-23973 q^{33}-20683 q^{31}+113 q^{29}+19538 q^{27}+21025 q^{25}+4588 q^{23}-14499 q^{21}-19760 q^{19}-7842 q^{17}+9640 q^{15}+17570 q^{13}+10084 q^{11}-5387 q^9-15327 q^7-11708 q^5+1734 q^3+13340 q+13340 q^{-1} +1734 q^{-3} -11708 q^{-5} -15327 q^{-7} -5387 q^{-9} +10084 q^{-11} +17570 q^{-13} +9640 q^{-15} -7842 q^{-17} -19760 q^{-19} -14499 q^{-21} +4588 q^{-23} +21025 q^{-25} +19538 q^{-27} +113 q^{-29} -20683 q^{-31} -23973 q^{-33} -5902 q^{-35} +18078 q^{-37} +26728 q^{-39} +12096 q^{-41} -13219 q^{-43} -26957 q^{-45} -17440 q^{-47} +6647 q^{-49} +24323 q^{-51} +20810 q^{-53} +243 q^{-55} -19110 q^{-57} -21361 q^{-59} -6200 q^{-61} +12485 q^{-63} +19206 q^{-65} +9898 q^{-67} -5906 q^{-69} -14906 q^{-71} -11060 q^{-73} +624 q^{-75} +9878 q^{-77} +9987 q^{-79} +2547 q^{-81} -5268 q^{-83} -7535 q^{-85} -3746 q^{-87} +1897 q^{-89} +4831 q^{-91} +3502 q^{-93} -33 q^{-95} -2552 q^{-97} -2541 q^{-99} -721 q^{-101} +1063 q^{-103} +1544 q^{-105} +763 q^{-107} -304 q^{-109} -776 q^{-111} -536 q^{-113} -3 q^{-115} +323 q^{-117} +307 q^{-119} +73 q^{-121} -122 q^{-123} -139 q^{-125} -49 q^{-127} +30 q^{-129} +49 q^{-131} +34 q^{-133} -7 q^{-135} -24 q^{-137} -6 q^{-139} +3 q^{-141} + q^{-143} +4 q^{-145} +2 q^{-147} -4 q^{-149} +2 q^{-153} - q^{-155} }

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^{16}+q^{12}-3 q^{10}+2 q^8-q^4+4 q^2-1+4 q^{-2} - q^{-4} +2 q^{-8} -3 q^{-10} + q^{-12} - q^{-16} }

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^{42}-2 q^{38}+5 q^{34}+3 q^{32}-8 q^{30}-5 q^{28}+11 q^{26}+q^{24}-17 q^{22}-6 q^{20}+17 q^{18}+5 q^{16}-20 q^{14}+3 q^{12}+18 q^{10}-5 q^8-10 q^6+9 q^4+6 q^2-6+6 q^{-2} +9 q^{-4} -10 q^{-6} -5 q^{-8} +18 q^{-10} +3 q^{-12} -20 q^{-14} +5 q^{-16} +17 q^{-18} -6 q^{-20} -17 q^{-22} + q^{-24} +11 q^{-26} -5 q^{-28} -8 q^{-30} +3 q^{-32} +5 q^{-34} -2 q^{-38} + q^{-42} }

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}-2 q^{32}+q^{30}+5 q^{28}-10 q^{26}+2 q^{24}+15 q^{22}-23 q^{20}+3 q^{18}+24 q^{16}-31 q^{14}+24 q^{10}-22 q^8-4 q^6+19 q^4+2 q^2-2+2 q^{-2} +19 q^{-4} -4 q^{-6} -22 q^{-8} +24 q^{-10} -31 q^{-14} +24 q^{-16} +3 q^{-18} -23 q^{-20} +15 q^{-22} +2 q^{-24} -10 q^{-26} +5 q^{-28} + q^{-30} -2 q^{-32} + q^{-34} }

1,0,0

-q^{21}-q^{17}+q^{15}-3q^{13}+3q^{11}-2q^{9}+2q^{7}-q^{5}+3q^{3}+q+q^{-1}+3q^{-3}-q^{-5}+2q^{-7}-2q^{-9}+3q^{-11}-3q^{-13}+q^{-15}-q^{-17}-q^{-21}

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}+2 q^{32}-5 q^{30}+9 q^{28}-16 q^{26}+22 q^{24}-29 q^{22}+33 q^{20}-35 q^{18}+32 q^{16}-23 q^{14}+12 q^{12}+6 q^{10}-22 q^8+40 q^6-53 q^4+64 q^2-68+64 q^{-2} -53 q^{-4} +40 q^{-6} -22 q^{-8} +6 q^{-10} +12 q^{-12} -23 q^{-14} +32 q^{-16} -35 q^{-18} +33 q^{-20} -29 q^{-22} +22 q^{-24} -16 q^{-26} +9 q^{-28} -5 q^{-30} +2 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}-2 q^{52}-2 q^{50}+3 q^{48}+7 q^{46}-13 q^{42}-9 q^{40}+13 q^{38}+22 q^{36}-4 q^{34}-31 q^{32}-14 q^{30}+28 q^{28}+29 q^{26}-16 q^{24}-38 q^{22}-4 q^{20}+34 q^{18}+15 q^{16}-25 q^{14}-22 q^{12}+16 q^{10}+24 q^8-6 q^6-22 q^4+5 q^2+27+5 q^{-2} -22 q^{-4} -6 q^{-6} +24 q^{-8} +16 q^{-10} -22 q^{-12} -25 q^{-14} +15 q^{-16} +34 q^{-18} -4 q^{-20} -38 q^{-22} -16 q^{-24} +29 q^{-26} +28 q^{-28} -14 q^{-30} -31 q^{-32} -4 q^{-34} +22 q^{-36} +13 q^{-38} -9 q^{-40} -13 q^{-42} +7 q^{-46} +3 q^{-48} -2 q^{-50} -2 q^{-52} + q^{-56} }

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}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-9 q^{70}+q^{68}+14 q^{66}-35 q^{64}+56 q^{62}-69 q^{60}+55 q^{58}-16 q^{56}-50 q^{54}+129 q^{52}-183 q^{50}+191 q^{48}-130 q^{46}+4 q^{44}+139 q^{42}-255 q^{40}+293 q^{38}-227 q^{36}+79 q^{34}+91 q^{32}-219 q^{30}+247 q^{28}-166 q^{26}+14 q^{24}+137 q^{22}-214 q^{20}+173 q^{18}-34 q^{16}-147 q^{14}+296 q^{12}-335 q^{10}+249 q^8-55 q^6-172 q^4+360 q^2-427+360 q^{-2} -172 q^{-4} -55 q^{-6} +249 q^{-8} -335 q^{-10} +296 q^{-12} -147 q^{-14} -34 q^{-16} +173 q^{-18} -214 q^{-20} +137 q^{-22} +14 q^{-24} -166 q^{-26} +247 q^{-28} -219 q^{-30} +91 q^{-32} +79 q^{-34} -227 q^{-36} +293 q^{-38} -255 q^{-40} +139 q^{-42} +4 q^{-44} -130 q^{-46} +191 q^{-48} -183 q^{-50} +129 q^{-52} -50 q^{-54} -16 q^{-56} +55 q^{-58} -69 q^{-60} +56 q^{-62} -35 q^{-64} +14 q^{-66} + q^{-68} -9 q^{-70} +9 q^{-72} -8 q^{-74} +5 q^{-76} -2 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 81"];

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

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

Out[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 -t^3+8 t^2-20 t+27-20 t^{-1} +8 t^{-2} - t^{-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 -z^6+2 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]= { 85, 0 }

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

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

Out[8]= $-q^{5}+3q^{4}-7q^{3}+11q^{2}-13q+15-13q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$

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

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

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

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

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

Out[10]= $az^{9}+z^{9}a^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+5a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}+3z^{6}a^{-4}-6z^{6}+a^{5}z^{5}-8a^{3}z^{5}-31az^{5}-31z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}-9a^{2}z^{4}-9z^{4}a^{-2}-5z^{4}a^{-4}-8z^{4}-2a^{5}z^{3}+5a^{3}z^{3}+25az^{3}+25z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+6a^{2}z^{2}+6z^{2}a^{-2}+3z^{2}a^{-4}+6z^{2}+a^{5}z-2a^{3}z-8az-8za^{-1}-2za^{-3}+za^{-5}-a^{4}-a^{2}-a^{-2}-a^{-4}+1$

Vassiliev invariants

V₂ and V₃:

(3, 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 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 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 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 110}

18

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 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 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 1320}

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 216}

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{15071}{10}}

-{\frac {1466}{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{10942}{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{289}{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{991}{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 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 $s=$ 0 is the signature of 10 81. 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

2

7

5

1

-4

5

6

2

4

3

7

5

-2

1

8

6

2

-1

6

8

2

-3

5

7

-2

-5

2

6

4

-7

1

5

-4

-9

2

-11

1

-1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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 \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 81]]
Out[2]=	10
In[3]:=	PD[Knot[10, 81]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[16, 9, 17, 10], X[20, 17, 1, 18], X[18, 13, 19, 14], X[14, 19, 15, 20], X[10, 15, 11, 16], X[6, 12, 7, 11], X[2, 8, 3, 7]]
In[4]:=	GaussCode[Knot[10, 81]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -9, 10, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5]
In[5]:=	BR[Knot[10, 81]]
Out[5]=	BR[5, {1, 1, -2, 1, 3, 2, 2, -4, -3, -3, -3, -4}]
In[6]:=	alex = Alexander[Knot[10, 81]][t]
Out[6]=	-3 8 20 2 3 27 - t + -- - -- - 20 t + 8 t - t 2 t t
In[7]:=	Conway[Knot[10, 81]][z]
Out[7]=	2 4 6 1 + 3 z + 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 81]}
In[9]:=	{KnotDet[Knot[10, 81]], KnotSignature[Knot[10, 81]]}
Out[9]=	{85, 0}
In[10]:=	J=Jones[Knot[10, 81]][q]
Out[10]=	-5 3 7 11 13 2 3 4 5 15 - q + -- - -- + -- - -- - 13 q + 11 q - 7 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 81], Knot[10, 109]}
In[12]:=	A2Invariant[Knot[10, 81]][q]
Out[12]=	-16 -12 3 2 -4 4 2 4 8 10 -1 - q + q - --- + -- - q + -- + 4 q - q + 2 q - 3 q + 10 8 2 q q q 12 16 q - q
In[13]:=	Kauffman[Knot[10, 81]][a, z]
Out[13]=	-4 -2 2 4 z 2 z 8 z 3 5 1 - a - a - a - a + -- - --- - --- - 8 a z - 2 a z + a z + 5 3 a a a 2 2 3 3 3 2 3 z 6 z 2 2 4 2 2 z 5 z 25 z 6 z + ---- + ---- + 6 a z + 3 a z - ---- + ---- + ----- + 4 2 5 3 a a a a a 4 4 3 3 3 5 3 4 5 z 9 z 2 4 25 a z + 5 a z - 2 a z - 8 z - ---- - ---- - 9 a z - 4 2 a a 5 5 5 4 4 z 8 z 31 z 5 3 5 5 5 6 5 a z + -- - ---- - ----- - 31 a z - 8 a z + a z - 6 z + 5 3 a a a 6 7 7 8 3 z 4 6 5 z 13 z 7 3 7 8 4 z ---- + 3 a z + ---- + ----- + 13 a z + 5 a z + 8 z + ---- + 4 3 a 2 a a a 9 2 8 z 9 4 a z + -- + a z a
In[14]:=	{Vassiliev[2][Knot[10, 81]], Vassiliev[3][Knot[10, 81]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 81]][q, t]
Out[15]=	8 1 2 1 5 2 6 5 - + 8 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 7 6 3 3 2 5 2 5 3 7 3 ---- + --- + 6 q t + 7 q t + 5 q t + 6 q t + 2 q t + 5 q t + 3 q t q t 7 4 9 4 11 5 q t + 2 q t + q t

10 81

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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