10 56

10_55

10_57

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10 56 Quick Notes

10 56 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X12,6,13,5 X18,14,19,13 X16,7,17,8 X6,17,7,18 X20,16,1,15 X14,20,15,19 X8,12,9,11 X2,10,3,9
Gauss code	1, -10, 2, -1, 3, -6, 5, -9, 10, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7
Dowker-Thistlethwaite code	4 10 12 16 2 8 18 20 6 14
Conway Notation	[221,3,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [1][-13] Hyperbolic Volume 12.3988 A-Polynomial See Data:10 56/A-polynomial

[edit Notes for 10 56'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 2} 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 2} 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 -4

[edit Notes for 10 56'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 -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	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- q^{-1} +3 q^{-3} -2 q^{-5} +3 q^{-7} - q^{-9} - q^{-11} + q^{-13} -3 q^{-15} +3 q^{-17} -2 q^{-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	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^{33}-2 q^{31}-4 q^{29}+4 q^{27}+6 q^{25}+q^{23}-q^{21}-13 q^{19}-18 q^{17}+4 q^{15}+34 q^{13}+37 q^{11}+10 q^9-47 q^7-91 q^5-55 q^3+55 q+158 q^{-1} +156 q^{-3} -4 q^{-5} -225 q^{-7} -314 q^{-9} -149 q^{-11} +214 q^{-13} +519 q^{-15} +435 q^{-17} -62 q^{-19} -637 q^{-21} -827 q^{-23} -347 q^{-25} +578 q^{-27} +1225 q^{-29} +958 q^{-31} -169 q^{-33} -1401 q^{-35} -1708 q^{-37} -617 q^{-39} +1206 q^{-41} +2328 q^{-43} +1674 q^{-45} -490 q^{-47} -2576 q^{-49} -2787 q^{-51} -672 q^{-53} +2279 q^{-55} +3643 q^{-57} +2059 q^{-59} -1415 q^{-61} -3997 q^{-63} -3386 q^{-65} +148 q^{-67} +3776 q^{-69} +4349 q^{-71} +1221 q^{-73} -3045 q^{-75} -4789 q^{-77} -2428 q^{-79} +2023 q^{-81} +4709 q^{-83} +3267 q^{-85} -997 q^{-87} -4217 q^{-89} -3627 q^{-91} +95 q^{-93} +3529 q^{-95} +3648 q^{-97} +510 q^{-99} -2830 q^{-101} -3367 q^{-103} -897 q^{-105} +2183 q^{-107} +3071 q^{-109} +1118 q^{-111} -1684 q^{-113} -2771 q^{-115} -1338 q^{-117} +1199 q^{-119} +2594 q^{-121} +1672 q^{-123} -685 q^{-125} -2452 q^{-127} -2175 q^{-129} -16 q^{-131} +2274 q^{-133} +2790 q^{-135} +959 q^{-137} -1893 q^{-139} -3410 q^{-141} -2108 q^{-143} +1204 q^{-145} +3818 q^{-147} +3325 q^{-149} -161 q^{-151} -3825 q^{-153} -4394 q^{-155} -1116 q^{-157} +3324 q^{-159} +5039 q^{-161} +2407 q^{-163} -2327 q^{-165} -5079 q^{-167} -3463 q^{-169} +1071 q^{-171} +4497 q^{-173} +3972 q^{-175} +208 q^{-177} -3418 q^{-179} -3921 q^{-181} -1180 q^{-183} +2171 q^{-185} +3312 q^{-187} +1679 q^{-189} -998 q^{-191} -2423 q^{-193} -1712 q^{-195} +161 q^{-197} +1511 q^{-199} +1397 q^{-201} +280 q^{-203} -752 q^{-205} -956 q^{-207} -412 q^{-209} +269 q^{-211} +557 q^{-213} +340 q^{-215} -36 q^{-217} -252 q^{-219} -220 q^{-221} -52 q^{-223} +94 q^{-225} +121 q^{-227} +44 q^{-229} -23 q^{-231} -45 q^{-233} -31 q^{-235} -3 q^{-237} +19 q^{-239} +19 q^{-241} -2 q^{-243} -7 q^{-245} -2 q^{-247} -4 q^{-249} +5 q^{-253} + q^{-255} -3 q^{-257} +2 q^{-259} -2 q^{-263} + q^{-265} }

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

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

Vassiliev invariants

V2 and V3:

(0, -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 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 -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 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 -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 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 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 \frac{224}{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}} 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 112} 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 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 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 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 752} 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 96} 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{1984}{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{304}{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 16}

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=} 4 is the signature of 10 56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-2	-1	0	1	2	3	4	5	6	7	8	χ
21											1	1
19										2		-2
17									4	1		3
15								5	2			-3
13							5	4				1
11						6	5					-1
9					4	5						-1
7				3	6							3
5			2	4								-2
3		1	4									3
1		1										-1
-1	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, 56]]
Out[2]=	10
In[3]:=	PD[Knot[10, 56]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 6, 13, 5], X[18, 14, 19, 13], X[16, 7, 17, 8], X[6, 17, 7, 18], X[20, 16, 1, 15], X[14, 20, 15, 19], X[8, 12, 9, 11], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 56]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -6, 5, -9, 10, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7]
In[5]:=	BR[Knot[10, 56]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, 2, -3, 2, 2, 2, -3}]
In[6]:=	alex = Alexander[Knot[10, 56]][t]
Out[6]=	2 8 14 2 3 17 - -- + -- - -- - 14 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 56]][z]
Out[7]=	4 6 1 - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 25], Knot[10, 56], Knot[11, Alternating, 140]}
In[9]:=	{KnotDet[Knot[10, 56]], KnotSignature[Knot[10, 56]]}
Out[9]=	{65, 4}
In[10]:=	J=Jones[Knot[10, 56]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 7 q + 10 q - 11 q + 10 q - 9 q + 6 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 25], Knot[10, 56]}
In[12]:=	A2Invariant[Knot[10, 56]][q]
Out[12]=	4 6 8 10 12 18 20 22 24 26 1 + q + 2 q - q + 3 q - q - 3 q + q - 2 q + q + q - 28 30 q + q
In[13]:=	Kauffman[Knot[10, 56]][a, z]
Out[13]=	2 2 2 2 2 -8 2 2 4 z 8 z 4 z z 2 z 2 z 7 z 3 z a + -- - -- - --- - --- - --- - --- + ---- - ---- - ---- + ---- + 6 2 9 7 5 12 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 4 5 z 3 z 11 z 21 z 11 z 4 z z 6 z 4 z ---- - ---- + ----- + ----- + ----- + ---- + --- - ---- + ---- + 2 11 9 7 5 3 12 10 8 a a a a a a a a a 4 4 4 5 5 5 5 5 6 12 z 3 z 4 z 3 z 11 z 21 z 13 z 6 z 5 z ----- - ---- - ---- + ---- - ----- - ----- - ----- - ---- + ---- - 6 4 2 11 9 7 5 3 10 a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 5 z 14 z 3 z z 6 z 7 z 3 z 2 z 4 z 6 z ---- - ----- - ---- + -- + ---- + ---- + ---- + ---- + ---- + ---- + 8 6 4 2 9 7 5 3 8 6 a a a a a a a a a a 8 9 9 2 z z z ---- + -- + -- 4 7 5 a a a
In[14]:=	{Vassiliev[2][Knot[10, 56]], Vassiliev[3][Knot[10, 56]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[10, 56]][q, t]
Out[15]=	3 3 5 1 q q 5 7 7 2 9 2 4 q + 2 q + ---- + - + -- + 4 q t + 3 q t + 6 q t + 4 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 5 q t + 6 q t + 5 q t + 5 q t + 4 q t + 5 q t + 15 6 17 6 17 7 19 7 21 8 2 q t + 4 q t + q t + 2 q t + q t