10 9

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)	$\{1\}$
Determinant and Signature	{ 39, 2 }
Jones polynomial	$q^{7}-2q^{6}+3q^{5}-5q^{4}+6q^{3}-6q^{2}+6q-4+3q^{-1}-2q^{-2}+q^{-3}$
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	$q^{162}-q^{160}-q^{158}+q^{156}-2q^{150}+2q^{148}-q^{144}+5q^{142}+2q^{140}-2q^{138}-9q^{136}-2q^{134}-3q^{132}+15q^{128}+14q^{126}+6q^{124}-17q^{122}-15q^{120}-22q^{118}-16q^{116}+21q^{114}+39q^{112}+42q^{110}+4q^{108}-17q^{106}-55q^{104}-69q^{102}-22q^{100}+34q^{98}+85q^{96}+70q^{94}+49q^{92}-31q^{90}-104q^{88}-107q^{86}-58q^{84}+40q^{82}+89q^{80}+134q^{78}+78q^{76}-22q^{74}-99q^{72}-125q^{70}-71q^{68}-23q^{66}+74q^{64}+99q^{62}+79q^{60}+32q^{58}-9q^{56}-25q^{54}-81q^{52}-72q^{50}-76q^{48}-38q^{46}+33q^{44}+131q^{42}+197q^{40}+117q^{38}-7q^{36}-189q^{34}-289q^{32}-238q^{30}-7q^{28}+269q^{26}+379q^{24}+313q^{22}+22q^{20}-311q^{18}-497q^{16}-368q^{14}+q^{12}+358q^{10}+553q^{8}+407q^{6}-5q^{4}-445q^{2}-598-379q^{-2}+52q^{-4}+484q^{-6}+620q^{-8}+353q^{-10}-157q^{-12}-534q^{-14}-558q^{-16}-252q^{-18}+225q^{-20}+554q^{-22}+495q^{-24}+98q^{-26}-314q^{-28}-485q^{-30}-349q^{-32}+11q^{-34}+350q^{-36}+410q^{-38}+176q^{-40}-125q^{-42}-302q^{-44}-267q^{-46}-62q^{-48}+167q^{-50}+236q^{-52}+121q^{-54}-40q^{-56}-142q^{-58}-130q^{-60}-36q^{-62}+77q^{-64}+113q^{-66}+53q^{-68}-34q^{-70}-93q^{-72}-81q^{-74}-20q^{-76}+71q^{-78}+127q^{-80}+106q^{-82}+5q^{-84}-124q^{-86}-191q^{-88}-163q^{-90}+207q^{-94}+327q^{-96}+231q^{-98}-48q^{-100}-325q^{-102}-461q^{-104}-290q^{-106}+115q^{-108}+502q^{-110}+577q^{-112}+264q^{-114}-220q^{-116}-629q^{-118}-639q^{-120}-216q^{-122}+367q^{-124}+697q^{-126}+573q^{-128}+125q^{-130}-429q^{-132}-678q^{-134}-479q^{-136}+8q^{-138}+431q^{-140}+535q^{-142}+349q^{-144}-63q^{-146}-371q^{-148}-399q^{-150}-187q^{-152}+84q^{-154}+234q^{-156}+262q^{-158}+102q^{-160}-70q^{-162}-146q^{-164}-118q^{-166}-42q^{-168}+12q^{-170}+77q^{-172}+48q^{-174}+11q^{-176}-3q^{-178}-2q^{-180}-4q^{-182}-24q^{-184}+2q^{-186}-17q^{-188}-17q^{-190}+q^{-192}+22q^{-194}+24q^{-196}+4q^{-198}+15q^{-200}-16q^{-202}-23q^{-204}-19q^{-206}-q^{-208}+9q^{-210}+7q^{-212}+26q^{-214}+5q^{-216}-4q^{-218}-14q^{-220}-10q^{-222}-8q^{-224}-4q^{-226}+15q^{-228}+7q^{-230}+7q^{-232}-3q^{-234}-3q^{-236}-7q^{-238}-7q^{-240}+5q^{-242}+2q^{-244}+5q^{-246}-3q^{-252}-3q^{-254}+2q^{-256}+2q^{-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	Failed to parse (SVG (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^{24}+q^{18}+q^{16}-q^{14}-q^{12}+2 q^{10}+q^8-2 q^6+2 q^2-1-3 q^{-2} + q^{-4} - q^{-8} + q^{-10} +3 q^{-12} + q^{-14} + q^{-16} +4 q^{-18} + q^{-20} -3 q^{-22} - q^{-24} + q^{-26} -2 q^{-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	$q^{20}-q^{18}+2q^{16}-3q^{14}+4q^{12}-5q^{10}+6q^{8}-5q^{6}+6q^{4}-3q^{2}+2+q^{-2}-2q^{-4}+5q^{-6}-8q^{-8}+9q^{-10}-11q^{-12}+10q^{-14}-10q^{-16}+8q^{-18}-6q^{-20}+4q^{-22}-q^{-24}-q^{-26}+3q^{-28}-4q^{-30}+5q^{-32}-5q^{-34}+5q^{-36}-4q^{-38}+3q^{-40}-3q^{-42}+2q^{-44}-q^{-46}+q^{-48}$
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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 39, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

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}