10 56

Polynomial invariants

Alexander polynomial	$-2t^{3}+8t^{2}-14t+17-14t^{-1}+8t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-4z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 65, 4 }
Jones polynomial	$q^{10}-3q^{9}+6q^{8}-9q^{7}+10q^{6}-11q^{5}+10q^{4}-7q^{3}+5q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-2z^{2}a^{-4}-3z^{2}a^{-6}+2z^{2}a^{-8}+2a^{-2}-2a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+6z^{8}a^{-6}+4z^{8}a^{-8}+2z^{7}a^{-3}+3z^{7}a^{-5}+7z^{7}a^{-7}+6z^{7}a^{-9}+z^{6}a^{-2}-3z^{6}a^{-4}-14z^{6}a^{-6}-5z^{6}a^{-8}+5z^{6}a^{-10}-6z^{5}a^{-3}-13z^{5}a^{-5}-21z^{5}a^{-7}-11z^{5}a^{-9}+3z^{5}a^{-11}-4z^{4}a^{-2}-3z^{4}a^{-4}+12z^{4}a^{-6}+4z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+4z^{3}a^{-3}+11z^{3}a^{-5}+21z^{3}a^{-7}+11z^{3}a^{-9}-3z^{3}a^{-11}+5z^{2}a^{-2}+3z^{2}a^{-4}-7z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}a^{-10}-z^{2}a^{-12}-4za^{-5}-8za^{-7}-4za^{-9}-2a^{-2}+2a^{-6}+a^{-8}$
The A2 invariant	$1+q^{-4}+2q^{-6}-q^{-8}+3q^{-10}-q^{-12}-3q^{-18}+q^{-20}-2q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
The G2 invariant	$q^{-2}-q^{-4}+4q^{-6}-5q^{-8}+6q^{-10}-4q^{-12}-q^{-14}+12q^{-16}-20q^{-18}+30q^{-20}-30q^{-22}+20q^{-24}+2q^{-26}-32q^{-28}+65q^{-30}-79q^{-32}+73q^{-34}-37q^{-36}-17q^{-38}+73q^{-40}-108q^{-42}+110q^{-44}-70q^{-46}+6q^{-48}+57q^{-50}-93q^{-52}+86q^{-54}-39q^{-56}-18q^{-58}+67q^{-60}-80q^{-62}+48q^{-64}+9q^{-66}-79q^{-68}+121q^{-70}-120q^{-72}+71q^{-74}+7q^{-76}-92q^{-78}+146q^{-80}-158q^{-82}+116q^{-84}-44q^{-86}-46q^{-88}+109q^{-90}-130q^{-92}+103q^{-94}-38q^{-96}-28q^{-98}+71q^{-100}-73q^{-102}+35q^{-104}+21q^{-106}-68q^{-108}+89q^{-110}-64q^{-112}+13q^{-114}+50q^{-116}-94q^{-118}+107q^{-120}-83q^{-122}+37q^{-124}+12q^{-126}-54q^{-128}+72q^{-130}-68q^{-132}+50q^{-134}-20q^{-136}-4q^{-138}+19q^{-140}-29q^{-142}+26q^{-144}-19q^{-146}+11q^{-148}-2q^{-150}-3q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}$

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	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-q^4-q^2+5-2 q^{-2} -6 q^{-4} +12 q^{-6} -16 q^{-10} +14 q^{-12} +9 q^{-14} -21 q^{-16} +7 q^{-18} +14 q^{-20} -15 q^{-22} -4 q^{-24} +11 q^{-26} -11 q^{-30} +2 q^{-32} +16 q^{-34} -13 q^{-36} -9 q^{-38} +22 q^{-40} -9 q^{-42} -13 q^{-44} +16 q^{-46} - q^{-48} -8 q^{-50} +5 q^{-52} -2 q^{-56} + q^{-58} }
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^{15}-q^{13}-q^{11}+q^9+4 q^7-2 q^5-7 q^3+2 q+15 q^{-1} -23 q^{-5} -9 q^{-7} +36 q^{-9} +24 q^{-11} -39 q^{-13} -48 q^{-15} +36 q^{-17} +74 q^{-19} -18 q^{-21} -93 q^{-23} -9 q^{-25} +101 q^{-27} +40 q^{-29} -96 q^{-31} -66 q^{-33} +79 q^{-35} +81 q^{-37} -55 q^{-39} -93 q^{-41} +34 q^{-43} +86 q^{-45} -5 q^{-47} -79 q^{-49} -19 q^{-51} +63 q^{-53} +47 q^{-55} -44 q^{-57} -70 q^{-59} +18 q^{-61} +91 q^{-63} +11 q^{-65} -103 q^{-67} -39 q^{-69} +99 q^{-71} +63 q^{-73} -83 q^{-75} -72 q^{-77} +56 q^{-79} +73 q^{-81} -33 q^{-83} -59 q^{-85} +12 q^{-87} +39 q^{-89} -23 q^{-93} -2 q^{-95} +12 q^{-97} -4 q^{-101} +2 q^{-105} -2 q^{-109} + q^{-111} }
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^{28}-q^{26}-q^{24}+q^{22}+4 q^{18}-4 q^{16}-5 q^{14}+4 q^{12}+3 q^{10}+15 q^8-12 q^6-25 q^4-q^2+16+59 q^{-2} -4 q^{-4} -70 q^{-6} -60 q^{-8} -6 q^{-10} +150 q^{-12} +97 q^{-14} -64 q^{-16} -183 q^{-18} -171 q^{-20} +163 q^{-22} +285 q^{-24} +140 q^{-26} -196 q^{-28} -447 q^{-30} -76 q^{-32} +330 q^{-34} +472 q^{-36} +77 q^{-38} -549 q^{-40} -451 q^{-42} +66 q^{-44} +622 q^{-46} +471 q^{-48} -333 q^{-50} -639 q^{-52} -307 q^{-54} +470 q^{-56} +670 q^{-58} -5 q^{-60} -555 q^{-62} -503 q^{-64} +215 q^{-66} +613 q^{-68} +210 q^{-70} -357 q^{-72} -501 q^{-74} + q^{-76} +442 q^{-78} +336 q^{-80} -148 q^{-82} -439 q^{-84} -211 q^{-86} +226 q^{-88} +456 q^{-90} +128 q^{-92} -320 q^{-94} -472 q^{-96} -95 q^{-98} +515 q^{-100} +467 q^{-102} -58 q^{-104} -630 q^{-106} -481 q^{-108} +349 q^{-110} +659 q^{-112} +306 q^{-114} -489 q^{-116} -683 q^{-118} +15 q^{-120} +508 q^{-122} +499 q^{-124} -142 q^{-126} -532 q^{-128} -199 q^{-130} +172 q^{-132} +379 q^{-134} +80 q^{-136} -222 q^{-138} -156 q^{-140} -33 q^{-142} +150 q^{-144} +84 q^{-146} -40 q^{-148} -42 q^{-150} -47 q^{-152} +31 q^{-154} +23 q^{-156} -5 q^{-158} +4 q^{-160} -16 q^{-162} +5 q^{-164} +3 q^{-166} -3 q^{-168} +4 q^{-170} -3 q^{-172} +2 q^{-174} -2 q^{-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	$q^{-2}+q^{-6}+3q^{-8}+2q^{-10}+q^{-12}+4q^{-14}+4q^{-16}+8q^{-22}+3q^{-24}-6q^{-26}+4q^{-28}+11q^{-30}-10q^{-32}-13q^{-34}+4q^{-36}-4q^{-38}-21q^{-40}-9q^{-42}+8q^{-44}-4q^{-46}-5q^{-48}+17q^{-50}+14q^{-52}-5q^{-54}+8q^{-56}+13q^{-58}-9q^{-60}-10q^{-62}+7q^{-64}+q^{-66}-12q^{-68}-2q^{-70}+9q^{-72}-q^{-74}-7q^{-76}+3q^{-78}+4q^{-80}-2q^{-82}-q^{-84}+q^{-86}$
1,0,0,0	$q^{-2}+2q^{-6}+q^{-8}+q^{-10}+3q^{-12}-q^{-14}+3q^{-16}-q^{-18}+q^{-20}-q^{-24}-q^{-26}-2q^{-28}-q^{-30}-3q^{-32}+q^{-34}-2q^{-36}+2q^{-38}+2q^{-44}-q^{-46}+q^{-48}$

B2 Invariants.

Weight	Invariant
0,1	$1-q^{-2}+4q^{-4}-5q^{-6}+9q^{-8}-11q^{-10}+16q^{-12}-17q^{-14}+19q^{-16}-17q^{-18}+13q^{-20}-7q^{-22}-q^{-24}+10q^{-26}-19q^{-28}+28q^{-30}-34q^{-32}+36q^{-34}-36q^{-36}+31q^{-38}-26q^{-40}+15q^{-42}-6q^{-44}-3q^{-46}+10q^{-48}-15q^{-50}+19q^{-52}-19q^{-54}+18q^{-56}-14q^{-58}+10q^{-60}-7q^{-62}+4q^{-64}-2q^{-66}+q^{-68}$
1,0	$q^{2}-q^{-2}-q^{-4}+3q^{-6}+4q^{-8}-q^{-10}-6q^{-12}-q^{-14}+9q^{-16}+10q^{-18}-6q^{-20}-14q^{-22}+19q^{-26}+11q^{-28}-14q^{-30}-18q^{-32}+5q^{-34}+20q^{-36}+4q^{-38}-18q^{-40}-11q^{-42}+9q^{-44}+10q^{-46}-8q^{-48}-13q^{-50}+3q^{-52}+11q^{-54}-2q^{-56}-13q^{-58}+14q^{-62}+7q^{-64}-12q^{-66}-9q^{-68}+12q^{-70}+16q^{-72}-6q^{-74}-19q^{-76}-2q^{-78}+19q^{-80}+11q^{-82}-13q^{-84}-17q^{-86}+2q^{-88}+15q^{-90}+5q^{-92}-8q^{-94}-8q^{-96}+q^{-98}+6q^{-100}+2q^{-102}-2q^{-104}-2q^{-106}+q^{-110}$

D4 Invariants.

Weight

Invariant

1,0,0,0

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

G2 Invariants.

Weight

Invariant

1,0

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

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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]=

$-2t^{3}+8t^{2}-14t+17-14t^{-1}+8t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-4z^{4}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 65, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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