10 9

10_8

10_10

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10 9 Quick Notes

10 9 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X12,3,13,4 X2,15,3,16 X14,5,15,6 X4,13,5,14 X18,10,19,9 X20,12,1,11 X8,18,9,17 X10,20,11,19
Gauss code	1, -4, 3, -6, 5, -1, 2, -9, 7, -10, 8, -3, 6, -5, 4, -2, 9, -7, 10, -8
Dowker-Thistlethwaite code	6 12 14 16 18 20 4 2 8 10
Conway Notation	[5113]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 8.2941 A-Polynomial See Data:10 9/A-polynomial

[edit Notes for 10 9'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 4} Rasmussen s-Invariant -2

[edit Notes for 10 9'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^4+3 t^3-5 t^2+7 t-7+7 t^{-1} -5 t^{-2} +3 t^{-3} - t^{-4} }
Conway polynomial	$-z^{8}-5z^{6}-7z^{4}-2z^{2}+1$
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	{ 39, 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^7-2 q^6+3 q^5-5 q^4+6 q^3-6 q^2+6 q-4+3 q^{-1} -2 q^{-2} + q^{-3} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8 a^{-2} -7 z^6 a^{-2} +z^6 a^{-4} +z^6-17 z^4 a^{-2} +5 z^4 a^{-4} +5 z^4-16 z^2 a^{-2} +7 z^2 a^{-4} +7 z^2-4 a^{-2} +2 a^{-4} +3}
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 z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+2 a z^7-2 z^7 a^{-1} -2 z^7 a^{-3} +2 z^7 a^{-5} +a^2 z^6-18 z^6 a^{-2} -7 z^6 a^{-4} +2 z^6 a^{-6} -8 z^6-8 a z^5-2 z^5 a^{-1} -4 z^5 a^{-5} +2 z^5 a^{-7} -4 a^2 z^4+31 z^4 a^{-2} +13 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +10 z^4+7 a z^3+4 z^3 a^{-1} +5 z^3 a^{-3} +4 z^3 a^{-5} -4 z^3 a^{-7} +3 a^2 z^2-22 z^2 a^{-2} -8 z^2 a^{-4} +z^2 a^{-6} -2 z^2 a^{-8} -8 z^2-a z-2 z a^{-1} -2 z a^{-3} +z a^{-7} +4 a^{-2} +2 a^{-4} +3}
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^8+q^4+ q^{-2} - q^{-4} +2 q^{-6} - q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }
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^{46}-q^{44}+2 q^{42}-3 q^{40}+2 q^{38}-2 q^{36}-q^{34}+6 q^{32}-8 q^{30}+10 q^{28}-9 q^{26}+5 q^{24}+2 q^{22}-10 q^{20}+16 q^{18}-16 q^{16}+14 q^{14}-4 q^{12}-5 q^{10}+15 q^8-14 q^6+12 q^4-4 q^2-4+10 q^{-2} -9 q^{-4} +3 q^{-6} +6 q^{-8} -10 q^{-10} +14 q^{-12} -9 q^{-14} -2 q^{-16} +8 q^{-18} -16 q^{-20} +18 q^{-22} -17 q^{-24} +7 q^{-26} +3 q^{-28} -13 q^{-30} +18 q^{-32} -18 q^{-34} +11 q^{-36} -3 q^{-38} -6 q^{-40} +10 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-50} +6 q^{-52} -4 q^{-54} -2 q^{-56} +7 q^{-58} -8 q^{-60} +7 q^{-62} -4 q^{-64} - q^{-66} +6 q^{-68} -8 q^{-70} +11 q^{-72} -8 q^{-74} +6 q^{-76} - q^{-78} -2 q^{-80} +5 q^{-82} -8 q^{-84} +10 q^{-86} -7 q^{-88} +4 q^{-90} -4 q^{-94} +6 q^{-96} -6 q^{-98} +5 q^{-100} -3 q^{-102} + q^{-106} -3 q^{-108} +2 q^{-110} - q^{-112} + q^{-114} }

Further Quantum Invariants

Further quantum knot invariants for 10_9.

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^7-q^5+q^3-q+2 q^{-1} + q^{-7} -2 q^{-9} + q^{-11} - q^{-13} + q^{-15} }
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^{22}-q^{20}-q^{18}+3 q^{16}-q^{14}-3 q^{12}+4 q^{10}+q^8-5 q^6+2 q^4+3 q^2-4+ q^{-2} +4 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} - q^{-16} +4 q^{-18} -3 q^{-20} -3 q^{-22} +5 q^{-24} -2 q^{-26} -2 q^{-28} +3 q^{-30} - q^{-32} - q^{-34} +2 q^{-36} - q^{-38} - q^{-40} + q^{-42} }
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^{45}-q^{43}-q^{41}+q^{39}+2 q^{37}-q^{35}-4 q^{33}+q^{31}+6 q^{29}-7 q^{25}-3 q^{23}+8 q^{21}+7 q^{19}-6 q^{17}-9 q^{15}+2 q^{13}+9 q^{11}+2 q^9-9 q^7-6 q^5+5 q^3+10 q- q^{-1} -9 q^{-3} - q^{-5} +11 q^{-7} +4 q^{-9} -9 q^{-11} -3 q^{-13} +8 q^{-15} +5 q^{-17} -6 q^{-19} -5 q^{-21} +2 q^{-23} +5 q^{-25} -6 q^{-29} -6 q^{-31} +7 q^{-33} +6 q^{-35} -4 q^{-37} -9 q^{-39} +2 q^{-41} +9 q^{-43} + q^{-45} -4 q^{-47} -3 q^{-49} +2 q^{-51} +3 q^{-53} +2 q^{-55} -3 q^{-57} -3 q^{-59} +3 q^{-63} + q^{-65} -2 q^{-67} -2 q^{-69} + q^{-71} +2 q^{-73} - q^{-77} - q^{-79} + q^{-81} }
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^{76}-q^{74}-q^{72}+q^{70}+2 q^{66}-3 q^{64}-2 q^{62}+3 q^{60}+q^{58}+6 q^{56}-6 q^{54}-9 q^{52}+2 q^{50}+5 q^{48}+16 q^{46}-3 q^{44}-17 q^{42}-10 q^{40}-3 q^{38}+25 q^{36}+14 q^{34}-8 q^{32}-17 q^{30}-21 q^{28}+13 q^{26}+19 q^{24}+13 q^{22}+q^{20}-24 q^{18}-11 q^{16}-5 q^{14}+14 q^{12}+27 q^{10}-16 q^6-32 q^4-6 q^2+36+28 q^{-2} -39 q^{-6} -27 q^{-8} +25 q^{-10} +36 q^{-12} +11 q^{-14} -30 q^{-16} -29 q^{-18} +11 q^{-20} +26 q^{-22} +11 q^{-24} -17 q^{-26} -20 q^{-28} +2 q^{-30} +18 q^{-32} +9 q^{-34} -7 q^{-36} -15 q^{-38} -10 q^{-40} +9 q^{-42} +16 q^{-44} +14 q^{-46} -11 q^{-48} -32 q^{-50} -7 q^{-52} +20 q^{-54} +41 q^{-56} +9 q^{-58} -44 q^{-60} -34 q^{-62} - q^{-64} +50 q^{-66} +36 q^{-68} -25 q^{-70} -39 q^{-72} -28 q^{-74} +27 q^{-76} +43 q^{-78} +4 q^{-80} -20 q^{-82} -35 q^{-84} +26 q^{-88} +15 q^{-90} +2 q^{-92} -24 q^{-94} -10 q^{-96} +9 q^{-98} +11 q^{-100} +10 q^{-102} -11 q^{-104} -8 q^{-106} - q^{-108} +3 q^{-110} +9 q^{-112} -3 q^{-114} -3 q^{-116} -2 q^{-118} - q^{-120} +4 q^{-122} - q^{-128} - q^{-130} + q^{-132} }
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^{115}-q^{113}-q^{111}+q^{109}-q^{101}-q^{99}+3 q^{97}+4 q^{95}-q^{93}-4 q^{91}-6 q^{89}-4 q^{87}+4 q^{85}+14 q^{83}+11 q^{81}-4 q^{79}-17 q^{77}-22 q^{75}-8 q^{73}+18 q^{71}+35 q^{69}+24 q^{67}-9 q^{65}-37 q^{63}-42 q^{61}-14 q^{59}+29 q^{57}+54 q^{55}+37 q^{53}-9 q^{51}-46 q^{49}-52 q^{47}-21 q^{45}+24 q^{43}+48 q^{41}+38 q^{39}+10 q^{37}-20 q^{35}-37 q^{33}-36 q^{31}-22 q^{29}+12 q^{27}+47 q^{25}+60 q^{23}+35 q^{21}-28 q^{19}-87 q^{17}-87 q^{15}-9 q^{13}+87 q^{11}+122 q^9+61 q^7-63 q^5-146 q^3-107 q+29 q^{-1} +142 q^{-3} +140 q^{-5} +18 q^{-7} -124 q^{-9} -155 q^{-11} -47 q^{-13} +97 q^{-15} +148 q^{-17} +71 q^{-19} -63 q^{-21} -133 q^{-23} -75 q^{-25} +40 q^{-27} +105 q^{-29} +69 q^{-31} -20 q^{-33} -78 q^{-35} -59 q^{-37} +12 q^{-39} +58 q^{-41} +40 q^{-43} -8 q^{-45} -42 q^{-47} -35 q^{-49} +5 q^{-51} +34 q^{-53} +38 q^{-55} +8 q^{-57} -33 q^{-59} -52 q^{-61} -34 q^{-63} +18 q^{-65} +74 q^{-67} +79 q^{-69} +5 q^{-71} -91 q^{-73} -119 q^{-75} -52 q^{-77} +86 q^{-79} +170 q^{-81} +104 q^{-83} -65 q^{-85} -189 q^{-87} -156 q^{-89} +16 q^{-91} +184 q^{-93} +191 q^{-95} +34 q^{-97} -150 q^{-99} -194 q^{-101} -73 q^{-103} +95 q^{-105} +175 q^{-107} +99 q^{-109} -50 q^{-111} -134 q^{-113} -99 q^{-115} +9 q^{-117} +91 q^{-119} +85 q^{-121} +11 q^{-123} -55 q^{-125} -65 q^{-127} -21 q^{-129} +32 q^{-131} +47 q^{-133} +21 q^{-135} -14 q^{-137} -33 q^{-139} -23 q^{-141} +8 q^{-143} +24 q^{-145} +18 q^{-147} -16 q^{-151} -17 q^{-153} -5 q^{-155} +11 q^{-157} +14 q^{-159} +5 q^{-161} -4 q^{-163} -9 q^{-165} -7 q^{-167} + q^{-169} +7 q^{-171} +4 q^{-173} -2 q^{-177} -4 q^{-179} - q^{-181} +2 q^{-183} +2 q^{-185} - q^{-191} - q^{-193} + q^{-195} }
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^{162}-q^{160}-q^{158}+q^{156}-2 q^{150}+2 q^{148}-q^{144}+5 q^{142}+2 q^{140}-2 q^{138}-9 q^{136}-2 q^{134}-3 q^{132}+15 q^{128}+14 q^{126}+6 q^{124}-17 q^{122}-15 q^{120}-22 q^{118}-16 q^{116}+21 q^{114}+39 q^{112}+42 q^{110}+4 q^{108}-17 q^{106}-55 q^{104}-69 q^{102}-22 q^{100}+34 q^{98}+85 q^{96}+70 q^{94}+49 q^{92}-31 q^{90}-104 q^{88}-107 q^{86}-58 q^{84}+40 q^{82}+89 q^{80}+134 q^{78}+78 q^{76}-22 q^{74}-99 q^{72}-125 q^{70}-71 q^{68}-23 q^{66}+74 q^{64}+99 q^{62}+79 q^{60}+32 q^{58}-9 q^{56}-25 q^{54}-81 q^{52}-72 q^{50}-76 q^{48}-38 q^{46}+33 q^{44}+131 q^{42}+197 q^{40}+117 q^{38}-7 q^{36}-189 q^{34}-289 q^{32}-238 q^{30}-7 q^{28}+269 q^{26}+379 q^{24}+313 q^{22}+22 q^{20}-311 q^{18}-497 q^{16}-368 q^{14}+q^{12}+358 q^{10}+553 q^8+407 q^6-5 q^4-445 q^2-598-379 q^{-2} +52 q^{-4} +484 q^{-6} +620 q^{-8} +353 q^{-10} -157 q^{-12} -534 q^{-14} -558 q^{-16} -252 q^{-18} +225 q^{-20} +554 q^{-22} +495 q^{-24} +98 q^{-26} -314 q^{-28} -485 q^{-30} -349 q^{-32} +11 q^{-34} +350 q^{-36} +410 q^{-38} +176 q^{-40} -125 q^{-42} -302 q^{-44} -267 q^{-46} -62 q^{-48} +167 q^{-50} +236 q^{-52} +121 q^{-54} -40 q^{-56} -142 q^{-58} -130 q^{-60} -36 q^{-62} +77 q^{-64} +113 q^{-66} +53 q^{-68} -34 q^{-70} -93 q^{-72} -81 q^{-74} -20 q^{-76} +71 q^{-78} +127 q^{-80} +106 q^{-82} +5 q^{-84} -124 q^{-86} -191 q^{-88} -163 q^{-90} +207 q^{-94} +327 q^{-96} +231 q^{-98} -48 q^{-100} -325 q^{-102} -461 q^{-104} -290 q^{-106} +115 q^{-108} +502 q^{-110} +577 q^{-112} +264 q^{-114} -220 q^{-116} -629 q^{-118} -639 q^{-120} -216 q^{-122} +367 q^{-124} +697 q^{-126} +573 q^{-128} +125 q^{-130} -429 q^{-132} -678 q^{-134} -479 q^{-136} +8 q^{-138} +431 q^{-140} +535 q^{-142} +349 q^{-144} -63 q^{-146} -371 q^{-148} -399 q^{-150} -187 q^{-152} +84 q^{-154} +234 q^{-156} +262 q^{-158} +102 q^{-160} -70 q^{-162} -146 q^{-164} -118 q^{-166} -42 q^{-168} +12 q^{-170} +77 q^{-172} +48 q^{-174} +11 q^{-176} -3 q^{-178} -2 q^{-180} -4 q^{-182} -24 q^{-184} +2 q^{-186} -17 q^{-188} -17 q^{-190} + q^{-192} +22 q^{-194} +24 q^{-196} +4 q^{-198} +15 q^{-200} -16 q^{-202} -23 q^{-204} -19 q^{-206} - q^{-208} +9 q^{-210} +7 q^{-212} +26 q^{-214} +5 q^{-216} -4 q^{-218} -14 q^{-220} -10 q^{-222} -8 q^{-224} -4 q^{-226} +15 q^{-228} +7 q^{-230} +7 q^{-232} -3 q^{-234} -3 q^{-236} -7 q^{-238} -7 q^{-240} +5 q^{-242} +2 q^{-244} +5 q^{-246} -3 q^{-252} -3 q^{-254} +2 q^{-256} +2 q^{-260} - q^{-266} - q^{-268} + q^{-270} }

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^8+q^4+ q^{-2} - q^{-4} +2 q^{-6} - q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }
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^{28}-2 q^{26}+4 q^{24}-8 q^{22}+15 q^{20}-20 q^{18}+30 q^{16}-38 q^{14}+45 q^{12}-50 q^{10}+50 q^8-46 q^6+31 q^4-14 q^2-2+30 q^{-2} -50 q^{-4} +68 q^{-6} -76 q^{-8} +82 q^{-10} -84 q^{-12} +74 q^{-14} -62 q^{-16} +46 q^{-18} -32 q^{-20} +18 q^{-22} -2 q^{-24} -4 q^{-26} +17 q^{-28} -18 q^{-30} +18 q^{-32} -22 q^{-34} +22 q^{-36} -26 q^{-38} +20 q^{-40} -22 q^{-42} +25 q^{-44} -20 q^{-46} +18 q^{-48} -14 q^{-50} +11 q^{-52} -8 q^{-54} +4 q^{-56} -2 q^{-58} + q^{-60} }
2,0	$q^{24}+q^{18}+q^{16}-q^{14}-q^{12}+2q^{10}+q^{8}-2q^{6}+2q^{2}-1-3q^{-2}+q^{-4}-q^{-8}+q^{-10}+3q^{-12}+q^{-14}+q^{-16}+4q^{-18}+q^{-20}-3q^{-22}-q^{-24}+q^{-26}-2q^{-28}-q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-52}$

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^{20}-q^{18}+q^{14}-2 q^{12}+q^{10}+2 q^8-q^6+4 q^4+3 q^2-2+3 q^{-2} +2 q^{-4} -5 q^{-6} + q^{-10} -3 q^{-12} -2 q^{-14} -2 q^{-20} +5 q^{-24} - q^{-26} - q^{-28} +6 q^{-30} - q^{-32} -3 q^{-34} +3 q^{-36} -3 q^{-40} + q^{-42} - q^{-46} + q^{-48} }
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^9+2 q^5+2 q- q^{-1} + q^{-3} - q^{-5} - q^{-11} -2 q^{-15} + q^{-17} - q^{-19} +2 q^{-21} + q^{-25} }
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^{34}-2 q^{32}+3 q^{30}-3 q^{28}+q^{26}+4 q^{24}-7 q^{22}+13 q^{20}-6 q^{18}+2 q^{16}+9 q^{14}-17 q^{12}+16 q^{10}-14 q^8+5 q^6+4 q^4-11 q^2+22-17 q^{-2} +19 q^{-4} -13 q^{-6} +3 q^{-8} -2 q^{-10} -7 q^{-12} +8 q^{-14} -12 q^{-16} +16 q^{-18} -18 q^{-20} +28 q^{-22} -22 q^{-24} +16 q^{-26} +2 q^{-28} -17 q^{-30} +26 q^{-32} -28 q^{-34} +22 q^{-36} -6 q^{-38} -4 q^{-40} +10 q^{-42} -6 q^{-44} -5 q^{-46} +6 q^{-48} -7 q^{-50} -7 q^{-52} +12 q^{-54} -12 q^{-56} +12 q^{-58} -2 q^{-60} -4 q^{-62} +9 q^{-64} -9 q^{-66} +6 q^{-68} - q^{-70} -3 q^{-72} +3 q^{-74} -2 q^{-76} + q^{-78} }

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^{22}+q^{16}-q^{12}+q^8+q^6+4 q^4+3 q^2+3+4 q^{-2} +3 q^{-4} - q^{-6} -4 q^{-8} -3 q^{-12} -6 q^{-14} -4 q^{-16} + q^{-18} -5 q^{-20} -2 q^{-22} +3 q^{-24} +5 q^{-30} +5 q^{-32} +3 q^{-36} +4 q^{-38} -4 q^{-42} -3 q^{-48} - q^{-50} + q^{-52} + q^{-58} }
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^{10}+2 q^6+q^4+2 q^2+1+ q^{-4} -2 q^{-6} -2 q^{-10} -2 q^{-14} - q^{-18} + q^{-22} +2 q^{-26} + q^{-30} }

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

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^{26}-q^{24}+q^{22}-2 q^{20}+3 q^{18}-4 q^{16}+3 q^{14}-4 q^{12}+6 q^{10}-3 q^8+6 q^6-q^4+6 q^2+2+2 q^{-2} +2 q^{-4} -2 q^{-6} +2 q^{-8} -8 q^{-10} +4 q^{-12} -10 q^{-14} +5 q^{-16} -11 q^{-18} +6 q^{-20} -8 q^{-22} +7 q^{-24} -5 q^{-26} +5 q^{-28} -2 q^{-30} +4 q^{-32} +2 q^{-34} +2 q^{-38} -2 q^{-40} +5 q^{-42} -3 q^{-44} +3 q^{-46} -4 q^{-48} +4 q^{-50} -3 q^{-52} +2 q^{-54} -3 q^{-56} +2 q^{-58} -2 q^{-60} + q^{-62} - q^{-64} + q^{-66} }

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^{46}-q^{44}+2 q^{42}-3 q^{40}+2 q^{38}-2 q^{36}-q^{34}+6 q^{32}-8 q^{30}+10 q^{28}-9 q^{26}+5 q^{24}+2 q^{22}-10 q^{20}+16 q^{18}-16 q^{16}+14 q^{14}-4 q^{12}-5 q^{10}+15 q^8-14 q^6+12 q^4-4 q^2-4+10 q^{-2} -9 q^{-4} +3 q^{-6} +6 q^{-8} -10 q^{-10} +14 q^{-12} -9 q^{-14} -2 q^{-16} +8 q^{-18} -16 q^{-20} +18 q^{-22} -17 q^{-24} +7 q^{-26} +3 q^{-28} -13 q^{-30} +18 q^{-32} -18 q^{-34} +11 q^{-36} -3 q^{-38} -6 q^{-40} +10 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-50} +6 q^{-52} -4 q^{-54} -2 q^{-56} +7 q^{-58} -8 q^{-60} +7 q^{-62} -4 q^{-64} - q^{-66} +6 q^{-68} -8 q^{-70} +11 q^{-72} -8 q^{-74} +6 q^{-76} - q^{-78} -2 q^{-80} +5 q^{-82} -8 q^{-84} +10 q^{-86} -7 q^{-88} +4 q^{-90} -4 q^{-94} +6 q^{-96} -6 q^{-98} +5 q^{-100} -3 q^{-102} + q^{-106} -3 q^{-108} +2 q^{-110} - q^{-112} + q^{-114} }

.

Computer Talk

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

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

In[1]:=

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

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 9"];

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^4+3 t^3-5 t^2+7 t-7+7 t^{-1} -5 t^{-2} +3 t^{-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]=

Failed to parse (SVG (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]=

{ 39, 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^7-2 q^6+3 q^5-5 q^4+6 q^3-6 q^2+6 q-4+3 q^{-1} -2 q^{-2} + q^{-3} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8 a^{-2} -7 z^6 a^{-2} +z^6 a^{-4} +z^6-17 z^4 a^{-2} +5 z^4 a^{-4} +5 z^4-16 z^2 a^{-2} +7 z^2 a^{-4} +7 z^2-4 a^{-2} +2 a^{-4} +3}

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 z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8+2 a z^7-2 z^7 a^{-1} -2 z^7 a^{-3} +2 z^7 a^{-5} +a^2 z^6-18 z^6 a^{-2} -7 z^6 a^{-4} +2 z^6 a^{-6} -8 z^6-8 a z^5-2 z^5 a^{-1} -4 z^5 a^{-5} +2 z^5 a^{-7} -4 a^2 z^4+31 z^4 a^{-2} +13 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +10 z^4+7 a z^3+4 z^3 a^{-1} +5 z^3 a^{-3} +4 z^3 a^{-5} -4 z^3 a^{-7} +3 a^2 z^2-22 z^2 a^{-2} -8 z^2 a^{-4} +z^2 a^{-6} -2 z^2 a^{-8} -8 z^2-a z-2 z a^{-1} -2 z a^{-3} +z a^{-7} +4 a^{-2} +2 a^{-4} +3}

Vassiliev invariants

V2 and V3:

(-2, -2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (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 -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 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{260}{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{220}{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 128} Failed to parse (SVG (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{1088}{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{320}{3}} $144$ Failed to parse (SVG (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 128} Failed to parse (SVG (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{2080}{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{1760}{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{1231}{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{3588}{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 -\frac{54844}{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{2911}{9}} $-{\frac {6271}{15}}$

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=} 2 is the signature of 10 9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-4	-3	-2	-1	0	1	2	3	4	5	6	χ
15											1	1
13										1		-1
11									2	1		1
9								3	1			-2
7							3	2				1
5						3	3					0
3					3	3						0
1				2	4							2
-1			1	2								-1
-3		1	2									1
-5		1										-1
-7	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, 9]]
Out[2]=	10
In[3]:=	PD[Knot[10, 9]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[12, 3, 13, 4], X[2, 15, 3, 16], X[14, 5, 15, 6], X[4, 13, 5, 14], X[18, 10, 19, 9], X[20, 12, 1, 11], X[8, 18, 9, 17], X[10, 20, 11, 19]]
In[4]:=	GaussCode[Knot[10, 9]]
Out[4]=	GaussCode[1, -4, 3, -6, 5, -1, 2, -9, 7, -10, 8, -3, 6, -5, 4, -2, 9, -7, 10, -8]
In[5]:=	BR[Knot[10, 9]]
Out[5]=	BR[3, {1, 1, 1, 1, 1, -2, 1, -2, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 9]][t]
Out[6]=	-4 3 5 7 2 3 4 -7 - t + -- - -- + - + 7 t - 5 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 9]][z]
Out[7]=	2 4 6 8 1 - 2 z - 7 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 9]}
In[9]:=	{KnotDet[Knot[10, 9]], KnotSignature[Knot[10, 9]]}
Out[9]=	{39, 2}
In[10]:=	J=Jones[Knot[10, 9]][q]
Out[10]=	-3 2 3 2 3 4 5 6 7 -4 + q - -- + - + 6 q - 6 q + 6 q - 5 q + 3 q - 2 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 9]}
In[12]:=	A2Invariant[Knot[10, 9]][q]
Out[12]=	-8 -4 2 4 6 8 12 14 16 20 q + q + q - q + 2 q - q - q - q + q + q
In[13]:=	Kauffman[Knot[10, 9]][a, z]
Out[13]=	2 2 2 2 2 4 z 2 z 2 z 2 2 z z 8 z 22 z 3 + -- + -- + -- - --- - --- - a z - 8 z - ---- + -- - ---- - ----- + 4 2 7 3 a 8 6 4 2 a a a a a a a a 3 3 3 3 4 4 2 2 4 z 4 z 5 z 4 z 3 4 z 3 z 3 a z - ---- + ---- + ---- + ---- + 7 a z + 10 z + -- - ---- + 7 5 3 a 8 6 a a a a a 4 4 5 5 5 6 13 z 31 z 2 4 2 z 4 z 2 z 5 6 2 z ----- + ----- - 4 a z + ---- - ---- - ---- - 8 a z - 8 z + ---- - 4 2 7 5 a 6 a a a a a 6 6 7 7 7 8 7 z 18 z 2 6 2 z 2 z 2 z 7 8 2 z ---- - ----- + a z + ---- - ---- - ---- + 2 a z + 2 z + ---- + 4 2 5 3 a 4 a a a a a 8 9 9 4 z z z ---- + -- + -- 2 3 a a a
In[14]:=	{Vassiliev[2][Knot[10, 9]], Vassiliev[3][Knot[10, 9]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[10, 9]][q, t]
Out[15]=	3 1 1 1 2 1 2 2 q 4 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 3 q t + 3 q t + 3 q t + 3 q t + 2 q t + 3 q t + q t + 11 4 11 5 13 5 15 6 2 q t + q t + q t + q t