10 56

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 -2 t^3+8 t^2-14 t+17-14 t^{-1} +8 t^{-2} -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 -2 z^6-4 z^4+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	{ 65, 4 }
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^{10}-3 q^9+6 q^8-9 q^7+10 q^6-11 q^5+10 q^4-7 q^3+5 q^2-2 q+1}
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^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -2 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-8} +2 a^{-2} -2 a^{-6} + a^{-8} }
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^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +6 z^8 a^{-6} +4 z^8 a^{-8} +2 z^7 a^{-3} +3 z^7 a^{-5} +7 z^7 a^{-7} +6 z^7 a^{-9} +z^6 a^{-2} -3 z^6 a^{-4} -14 z^6 a^{-6} -5 z^6 a^{-8} +5 z^6 a^{-10} -6 z^5 a^{-3} -13 z^5 a^{-5} -21 z^5 a^{-7} -11 z^5 a^{-9} +3 z^5 a^{-11} -4 z^4 a^{-2} -3 z^4 a^{-4} +12 z^4 a^{-6} +4 z^4 a^{-8} -6 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-3} +11 z^3 a^{-5} +21 z^3 a^{-7} +11 z^3 a^{-9} -3 z^3 a^{-11} +5 z^2 a^{-2} +3 z^2 a^{-4} -7 z^2 a^{-6} -2 z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} -4 z a^{-5} -8 z a^{-7} -4 z a^{-9} -2 a^{-2} +2 a^{-6} + a^{-8} }
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 1+ q^{-4} +2 q^{-6} - q^{-8} +3 q^{-10} - q^{-12} -3 q^{-18} + q^{-20} -2 q^{-22} + q^{-24} + q^{-26} - q^{-28} + q^{-30} }
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^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +6 q^{-10} -4 q^{-12} - q^{-14} +12 q^{-16} -20 q^{-18} +30 q^{-20} -30 q^{-22} +20 q^{-24} +2 q^{-26} -32 q^{-28} +65 q^{-30} -79 q^{-32} +73 q^{-34} -37 q^{-36} -17 q^{-38} +73 q^{-40} -108 q^{-42} +110 q^{-44} -70 q^{-46} +6 q^{-48} +57 q^{-50} -93 q^{-52} +86 q^{-54} -39 q^{-56} -18 q^{-58} +67 q^{-60} -80 q^{-62} +48 q^{-64} +9 q^{-66} -79 q^{-68} +121 q^{-70} -120 q^{-72} +71 q^{-74} +7 q^{-76} -92 q^{-78} +146 q^{-80} -158 q^{-82} +116 q^{-84} -44 q^{-86} -46 q^{-88} +109 q^{-90} -130 q^{-92} +103 q^{-94} -38 q^{-96} -28 q^{-98} +71 q^{-100} -73 q^{-102} +35 q^{-104} +21 q^{-106} -68 q^{-108} +89 q^{-110} -64 q^{-112} +13 q^{-114} +50 q^{-116} -94 q^{-118} +107 q^{-120} -83 q^{-122} +37 q^{-124} +12 q^{-126} -54 q^{-128} +72 q^{-130} -68 q^{-132} +50 q^{-134} -20 q^{-136} -4 q^{-138} +19 q^{-140} -29 q^{-142} +26 q^{-144} -19 q^{-146} +11 q^{-148} -2 q^{-150} -3 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162} }

Further Quantum Invariants

Further quantum knot invariants for 10_56.

A1 Invariants.

Weight	Invariant
1	$q-q^{-1}+3q^{-3}-2q^{-5}+3q^{-7}-q^{-9}-q^{-11}+q^{-13}-3q^{-15}+3q^{-17}-2q^{-19}+q^{-21}$
2	$q^{6}-q^{4}-q^{2}+5-2q^{-2}-6q^{-4}+12q^{-6}-16q^{-10}+14q^{-12}+9q^{-14}-21q^{-16}+7q^{-18}+14q^{-20}-15q^{-22}-4q^{-24}+11q^{-26}-11q^{-30}+2q^{-32}+16q^{-34}-13q^{-36}-9q^{-38}+22q^{-40}-9q^{-42}-13q^{-44}+16q^{-46}-q^{-48}-8q^{-50}+5q^{-52}-2q^{-56}+q^{-58}$
3	$q^{15}-q^{13}-q^{11}+q^{9}+4q^{7}-2q^{5}-7q^{3}+2q+15q^{-1}-23q^{-5}-9q^{-7}+36q^{-9}+24q^{-11}-39q^{-13}-48q^{-15}+36q^{-17}+74q^{-19}-18q^{-21}-93q^{-23}-9q^{-25}+101q^{-27}+40q^{-29}-96q^{-31}-66q^{-33}+79q^{-35}+81q^{-37}-55q^{-39}-93q^{-41}+34q^{-43}+86q^{-45}-5q^{-47}-79q^{-49}-19q^{-51}+63q^{-53}+47q^{-55}-44q^{-57}-70q^{-59}+18q^{-61}+91q^{-63}+11q^{-65}-103q^{-67}-39q^{-69}+99q^{-71}+63q^{-73}-83q^{-75}-72q^{-77}+56q^{-79}+73q^{-81}-33q^{-83}-59q^{-85}+12q^{-87}+39q^{-89}-23q^{-93}-2q^{-95}+12q^{-97}-4q^{-101}+2q^{-105}-2q^{-109}+q^{-111}$
4	$q^{28}-q^{26}-q^{24}+q^{22}+4q^{18}-4q^{16}-5q^{14}+4q^{12}+3q^{10}+15q^{8}-12q^{6}-25q^{4}-q^{2}+16+59q^{-2}-4q^{-4}-70q^{-6}-60q^{-8}-6q^{-10}+150q^{-12}+97q^{-14}-64q^{-16}-183q^{-18}-171q^{-20}+163q^{-22}+285q^{-24}+140q^{-26}-196q^{-28}-447q^{-30}-76q^{-32}+330q^{-34}+472q^{-36}+77q^{-38}-549q^{-40}-451q^{-42}+66q^{-44}+622q^{-46}+471q^{-48}-333q^{-50}-639q^{-52}-307q^{-54}+470q^{-56}+670q^{-58}-5q^{-60}-555q^{-62}-503q^{-64}+215q^{-66}+613q^{-68}+210q^{-70}-357q^{-72}-501q^{-74}+q^{-76}+442q^{-78}+336q^{-80}-148q^{-82}-439q^{-84}-211q^{-86}+226q^{-88}+456q^{-90}+128q^{-92}-320q^{-94}-472q^{-96}-95q^{-98}+515q^{-100}+467q^{-102}-58q^{-104}-630q^{-106}-481q^{-108}+349q^{-110}+659q^{-112}+306q^{-114}-489q^{-116}-683q^{-118}+15q^{-120}+508q^{-122}+499q^{-124}-142q^{-126}-532q^{-128}-199q^{-130}+172q^{-132}+379q^{-134}+80q^{-136}-222q^{-138}-156q^{-140}-33q^{-142}+150q^{-144}+84q^{-146}-40q^{-148}-42q^{-150}-47q^{-152}+31q^{-154}+23q^{-156}-5q^{-158}+4q^{-160}-16q^{-162}+5q^{-164}+3q^{-166}-3q^{-168}+4q^{-170}-3q^{-172}+2q^{-174}-2q^{-178}+q^{-180}$
5	$q^{45}-q^{43}-q^{41}+q^{39}+2q^{33}-2q^{31}-4q^{29}+4q^{27}+6q^{25}+q^{23}-q^{21}-13q^{19}-18q^{17}+4q^{15}+34q^{13}+37q^{11}+10q^{9}-47q^{7}-91q^{5}-55q^{3}+55q+158q^{-1}+156q^{-3}-4q^{-5}-225q^{-7}-314q^{-9}-149q^{-11}+214q^{-13}+519q^{-15}+435q^{-17}-62q^{-19}-637q^{-21}-827q^{-23}-347q^{-25}+578q^{-27}+1225q^{-29}+958q^{-31}-169q^{-33}-1401q^{-35}-1708q^{-37}-617q^{-39}+1206q^{-41}+2328q^{-43}+1674q^{-45}-490q^{-47}-2576q^{-49}-2787q^{-51}-672q^{-53}+2279q^{-55}+3643q^{-57}+2059q^{-59}-1415q^{-61}-3997q^{-63}-3386q^{-65}+148q^{-67}+3776q^{-69}+4349q^{-71}+1221q^{-73}-3045q^{-75}-4789q^{-77}-2428q^{-79}+2023q^{-81}+4709q^{-83}+3267q^{-85}-997q^{-87}-4217q^{-89}-3627q^{-91}+95q^{-93}+3529q^{-95}+3648q^{-97}+510q^{-99}-2830q^{-101}-3367q^{-103}-897q^{-105}+2183q^{-107}+3071q^{-109}+1118q^{-111}-1684q^{-113}-2771q^{-115}-1338q^{-117}+1199q^{-119}+2594q^{-121}+1672q^{-123}-685q^{-125}-2452q^{-127}-2175q^{-129}-16q^{-131}+2274q^{-133}+2790q^{-135}+959q^{-137}-1893q^{-139}-3410q^{-141}-2108q^{-143}+1204q^{-145}+3818q^{-147}+3325q^{-149}-161q^{-151}-3825q^{-153}-4394q^{-155}-1116q^{-157}+3324q^{-159}+5039q^{-161}+2407q^{-163}-2327q^{-165}-5079q^{-167}-3463q^{-169}+1071q^{-171}+4497q^{-173}+3972q^{-175}+208q^{-177}-3418q^{-179}-3921q^{-181}-1180q^{-183}+2171q^{-185}+3312q^{-187}+1679q^{-189}-998q^{-191}-2423q^{-193}-1712q^{-195}+161q^{-197}+1511q^{-199}+1397q^{-201}+280q^{-203}-752q^{-205}-956q^{-207}-412q^{-209}+269q^{-211}+557q^{-213}+340q^{-215}-36q^{-217}-252q^{-219}-220q^{-221}-52q^{-223}+94q^{-225}+121q^{-227}+44q^{-229}-23q^{-231}-45q^{-233}-31q^{-235}-3q^{-237}+19q^{-239}+19q^{-241}-2q^{-243}-7q^{-245}-2q^{-247}-4q^{-249}+5q^{-253}+q^{-255}-3q^{-257}+2q^{-259}-2q^{-263}+q^{-265}$

A2 Invariants.

Weight	Invariant
1,0	$1+q^{-4}+2q^{-6}-q^{-8}+3q^{-10}-q^{-12}-3q^{-18}+q^{-20}-2q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
1,1	$q^{4}-2q^{2}+8-16q^{-2}+35q^{-4}-58q^{-6}+102q^{-8}-152q^{-10}+223q^{-12}-294q^{-14}+366q^{-16}-418q^{-18}+434q^{-20}-408q^{-22}+318q^{-24}-180q^{-26}-5q^{-28}+210q^{-30}-420q^{-32}+610q^{-34}-751q^{-36}+832q^{-38}-844q^{-40}+784q^{-42}-661q^{-44}+488q^{-46}-284q^{-48}+80q^{-50}+107q^{-52}-252q^{-54}+348q^{-56}-396q^{-58}+395q^{-60}-360q^{-62}+306q^{-64}-246q^{-66}+187q^{-68}-134q^{-70}+92q^{-72}-60q^{-74}+36q^{-76}-20q^{-78}+10q^{-80}-4q^{-82}+q^{-84}$
2,0	$q^{4}-1+q^{-2}+4q^{-4}+q^{-6}-3q^{-8}+2q^{-10}+9q^{-12}-q^{-14}-8q^{-16}+3q^{-18}+6q^{-20}-7q^{-22}-6q^{-24}+5q^{-26}+3q^{-28}-6q^{-30}+q^{-32}+6q^{-34}-6q^{-36}-3q^{-38}+7q^{-40}-2q^{-42}-7q^{-44}+5q^{-46}+8q^{-48}-5q^{-50}-6q^{-52}+6q^{-54}+4q^{-56}-6q^{-58}-2q^{-60}+5q^{-62}-2q^{-66}-q^{-68}+q^{-70}-q^{-74}+q^{-76}$

A3 Invariants.

Weight	Invariant
0,1,0	$1-q^{-2}+2q^{-4}+3q^{-6}-3q^{-8}+5q^{-10}+6q^{-12}-9q^{-14}+9q^{-16}+7q^{-18}-15q^{-20}+9q^{-22}+7q^{-24}-18q^{-26}+q^{-28}+4q^{-30}-10q^{-32}-4q^{-34}+4q^{-36}+7q^{-38}-4q^{-40}-q^{-42}+16q^{-44}-7q^{-46}-10q^{-48}+17q^{-50}-7q^{-52}-11q^{-54}+12q^{-56}-2q^{-58}-6q^{-60}+5q^{-62}-2q^{-66}+q^{-68}$
1,0,0	$q^{-1}+2q^{-5}+3q^{-9}-q^{-11}+3q^{-13}-q^{-15}+q^{-17}-q^{-19}-q^{-21}-q^{-23}-3q^{-25}+q^{-27}-2q^{-29}+2q^{-31}-q^{-33}+2q^{-35}-q^{-37}+q^{-39}$

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^{-2} + q^{-6} +3 q^{-8} +2 q^{-10} + q^{-12} +4 q^{-14} +4 q^{-16} +8 q^{-22} +3 q^{-24} -6 q^{-26} +4 q^{-28} +11 q^{-30} -10 q^{-32} -13 q^{-34} +4 q^{-36} -4 q^{-38} -21 q^{-40} -9 q^{-42} +8 q^{-44} -4 q^{-46} -5 q^{-48} +17 q^{-50} +14 q^{-52} -5 q^{-54} +8 q^{-56} +13 q^{-58} -9 q^{-60} -10 q^{-62} +7 q^{-64} + q^{-66} -12 q^{-68} -2 q^{-70} +9 q^{-72} - q^{-74} -7 q^{-76} +3 q^{-78} +4 q^{-80} -2 q^{-82} - q^{-84} + q^{-86} }
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^{-2} +2 q^{-6} + q^{-8} + q^{-10} +3 q^{-12} - q^{-14} +3 q^{-16} - q^{-18} + q^{-20} - q^{-24} - q^{-26} -2 q^{-28} - q^{-30} -3 q^{-32} + q^{-34} -2 q^{-36} +2 q^{-38} +2 q^{-44} - q^{-46} + q^{-48} }

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 1- q^{-2} +4 q^{-4} -5 q^{-6} +9 q^{-8} -11 q^{-10} +16 q^{-12} -17 q^{-14} +19 q^{-16} -17 q^{-18} +13 q^{-20} -7 q^{-22} - q^{-24} +10 q^{-26} -19 q^{-28} +28 q^{-30} -34 q^{-32} +36 q^{-34} -36 q^{-36} +31 q^{-38} -26 q^{-40} +15 q^{-42} -6 q^{-44} -3 q^{-46} +10 q^{-48} -15 q^{-50} +19 q^{-52} -19 q^{-54} +18 q^{-56} -14 q^{-58} +10 q^{-60} -7 q^{-62} +4 q^{-64} -2 q^{-66} + q^{-68} }
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^2- q^{-2} - q^{-4} +3 q^{-6} +4 q^{-8} - q^{-10} -6 q^{-12} - q^{-14} +9 q^{-16} +10 q^{-18} -6 q^{-20} -14 q^{-22} +19 q^{-26} +11 q^{-28} -14 q^{-30} -18 q^{-32} +5 q^{-34} +20 q^{-36} +4 q^{-38} -18 q^{-40} -11 q^{-42} +9 q^{-44} +10 q^{-46} -8 q^{-48} -13 q^{-50} +3 q^{-52} +11 q^{-54} -2 q^{-56} -13 q^{-58} +14 q^{-62} +7 q^{-64} -12 q^{-66} -9 q^{-68} +12 q^{-70} +16 q^{-72} -6 q^{-74} -19 q^{-76} -2 q^{-78} +19 q^{-80} +11 q^{-82} -13 q^{-84} -17 q^{-86} +2 q^{-88} +15 q^{-90} +5 q^{-92} -8 q^{-94} -8 q^{-96} + q^{-98} +6 q^{-100} +2 q^{-102} -2 q^{-104} -2 q^{-106} + q^{-110} }

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^{-2} - q^{-4} +3 q^{-6} -2 q^{-8} +7 q^{-10} -5 q^{-12} +10 q^{-14} -8 q^{-16} +15 q^{-18} -13 q^{-20} +15 q^{-22} -13 q^{-24} +15 q^{-26} -10 q^{-28} +7 q^{-30} -3 q^{-32} - q^{-34} +4 q^{-36} -17 q^{-38} +13 q^{-40} -24 q^{-42} +21 q^{-44} -32 q^{-46} +26 q^{-48} -26 q^{-50} +29 q^{-52} -20 q^{-54} +20 q^{-56} -11 q^{-58} +13 q^{-60} -3 q^{-64} +4 q^{-66} -10 q^{-68} +14 q^{-70} -15 q^{-72} +12 q^{-74} -16 q^{-76} +15 q^{-78} -10 q^{-80} +8 q^{-82} -8 q^{-84} +6 q^{-86} -3 q^{-88} +2 q^{-90} -2 q^{-92} + q^{-94} }

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^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +6 q^{-10} -4 q^{-12} - q^{-14} +12 q^{-16} -20 q^{-18} +30 q^{-20} -30 q^{-22} +20 q^{-24} +2 q^{-26} -32 q^{-28} +65 q^{-30} -79 q^{-32} +73 q^{-34} -37 q^{-36} -17 q^{-38} +73 q^{-40} -108 q^{-42} +110 q^{-44} -70 q^{-46} +6 q^{-48} +57 q^{-50} -93 q^{-52} +86 q^{-54} -39 q^{-56} -18 q^{-58} +67 q^{-60} -80 q^{-62} +48 q^{-64} +9 q^{-66} -79 q^{-68} +121 q^{-70} -120 q^{-72} +71 q^{-74} +7 q^{-76} -92 q^{-78} +146 q^{-80} -158 q^{-82} +116 q^{-84} -44 q^{-86} -46 q^{-88} +109 q^{-90} -130 q^{-92} +103 q^{-94} -38 q^{-96} -28 q^{-98} +71 q^{-100} -73 q^{-102} +35 q^{-104} +21 q^{-106} -68 q^{-108} +89 q^{-110} -64 q^{-112} +13 q^{-114} +50 q^{-116} -94 q^{-118} +107 q^{-120} -83 q^{-122} +37 q^{-124} +12 q^{-126} -54 q^{-128} +72 q^{-130} -68 q^{-132} +50 q^{-134} -20 q^{-136} -4 q^{-138} +19 q^{-140} -29 q^{-142} +26 q^{-144} -19 q^{-146} +11 q^{-148} -2 q^{-150} -3 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162} }

.

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 56"];

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 -2 t^3+8 t^2-14 t+17-14 t^{-1} +8 t^{-2} -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 -2 z^6-4 z^4+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]=

{ 65, 4 }

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^{10}-3 q^9+6 q^8-9 q^7+10 q^6-11 q^5+10 q^4-7 q^3+5 q^2-2 q+1}

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^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -2 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-8} +2 a^{-2} -2 a^{-6} + a^{-8} }

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^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +6 z^8 a^{-6} +4 z^8 a^{-8} +2 z^7 a^{-3} +3 z^7 a^{-5} +7 z^7 a^{-7} +6 z^7 a^{-9} +z^6 a^{-2} -3 z^6 a^{-4} -14 z^6 a^{-6} -5 z^6 a^{-8} +5 z^6 a^{-10} -6 z^5 a^{-3} -13 z^5 a^{-5} -21 z^5 a^{-7} -11 z^5 a^{-9} +3 z^5 a^{-11} -4 z^4 a^{-2} -3 z^4 a^{-4} +12 z^4 a^{-6} +4 z^4 a^{-8} -6 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-3} +11 z^3 a^{-5} +21 z^3 a^{-7} +11 z^3 a^{-9} -3 z^3 a^{-11} +5 z^2 a^{-2} +3 z^2 a^{-4} -7 z^2 a^{-6} -2 z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} -4 z a^{-5} -8 z a^{-7} -4 z a^{-9} -2 a^{-2} +2 a^{-6} + a^{-8} }