10 92

10_91

10_93

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10 92 Quick Notes

10 92 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X18,7,19,8 X12,18,13,17 X6,19,7,20 X8,14,9,13 X2,10,3,9
Gauss code	1, -10, 2, -1, 3, -8, 6, -9, 10, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, -4
Dowker-Thistlethwaite code	4 10 14 18 2 16 8 20 12 6
Conway Notation	[.21.2.20]

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [1][-13] Hyperbolic Volume 14.8554 A-Polynomial See Data:10 92/A-polynomial

[edit Notes for 10 92'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 $3$ Rasmussen s-Invariant -4

[edit Notes for 10 92'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+10 t^2-20 t+25-20 t^{-1} +10 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^6-2 z^4+2 z^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	{ 89, 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}-4 q^9+8 q^8-12 q^7+14 q^6-15 q^5+14 q^4-10 q^3+7 q^2-3 q+1}
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-2}-z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}+a^{-4}-a^{-6}$
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 2 z^9 a^{-5} +2 z^9 a^{-7} +4 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8 a^{-8} +3 z^7 a^{-3} +5 z^7 a^{-5} +12 z^7 a^{-7} +10 z^7 a^{-9} +z^6 a^{-2} -8 z^6 a^{-4} -22 z^6 a^{-6} -5 z^6 a^{-8} +8 z^6 a^{-10} -8 z^5 a^{-3} -22 z^5 a^{-5} -32 z^5 a^{-7} -14 z^5 a^{-9} +4 z^5 a^{-11} -3 z^4 a^{-2} +2 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} +6 z^3 a^{-3} +18 z^3 a^{-5} +21 z^3 a^{-7} +7 z^3 a^{-9} -2 z^3 a^{-11} +3 z^2 a^{-2} +z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-3} -5 z a^{-5} -5 z a^{-7} -z a^{-9} - a^{-2} + a^{-4} + a^{-6} }
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^{-2} + q^{-4} +2 q^{-6} -2 q^{-8} +4 q^{-10} - q^{-12} + q^{-14} + q^{-16} -3 q^{-18} +2 q^{-20} -3 q^{-22} + q^{-24} + q^{-26} -2 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} -2 q^{-4} +6 q^{-6} -10 q^{-8} +13 q^{-10} -12 q^{-12} +3 q^{-14} +19 q^{-16} -45 q^{-18} +75 q^{-20} -88 q^{-22} +65 q^{-24} -7 q^{-26} -85 q^{-28} +181 q^{-30} -229 q^{-32} +207 q^{-34} -97 q^{-36} -66 q^{-38} +227 q^{-40} -319 q^{-42} +300 q^{-44} -165 q^{-46} -29 q^{-48} +202 q^{-50} -276 q^{-52} +226 q^{-54} -72 q^{-56} -100 q^{-58} +223 q^{-60} -234 q^{-62} +120 q^{-64} +62 q^{-66} -250 q^{-68} +358 q^{-70} -329 q^{-72} +175 q^{-74} +56 q^{-76} -283 q^{-78} +422 q^{-80} -428 q^{-82} +287 q^{-84} -63 q^{-86} -176 q^{-88} +333 q^{-90} -352 q^{-92} +236 q^{-94} -38 q^{-96} -143 q^{-98} +230 q^{-100} -197 q^{-102} +55 q^{-104} +115 q^{-106} -235 q^{-108} +256 q^{-110} -158 q^{-112} -6 q^{-114} +173 q^{-116} -275 q^{-118} +278 q^{-120} -193 q^{-122} +61 q^{-124} +66 q^{-126} -157 q^{-128} +184 q^{-130} -156 q^{-132} +99 q^{-134} -28 q^{-136} -26 q^{-138} +54 q^{-140} -65 q^{-142} +54 q^{-144} -34 q^{-146} +16 q^{-148} + q^{-150} -8 q^{-152} +10 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }

Further Quantum Invariants

Further quantum knot invariants for 10_92.

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-2 q^{-1} +4 q^{-3} -3 q^{-5} +4 q^{-7} - q^{-9} - q^{-11} +2 q^{-13} -4 q^{-15} +4 q^{-17} -3 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^6-2 q^4-q^2+9-6 q^{-2} -13 q^{-4} +24 q^{-6} + q^{-8} -34 q^{-10} +27 q^{-12} +21 q^{-14} -41 q^{-16} +11 q^{-18} +30 q^{-20} -26 q^{-22} -11 q^{-24} +22 q^{-26} +3 q^{-28} -25 q^{-30} +2 q^{-32} +33 q^{-34} -25 q^{-36} -21 q^{-38} +43 q^{-40} -12 q^{-42} -28 q^{-44} +28 q^{-46} + q^{-48} -15 q^{-50} +8 q^{-52} + q^{-54} -3 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}-2 q^{13}-q^{11}+4 q^9+6 q^7-9 q^5-19 q^3+13 q+44 q^{-1} -3 q^{-3} -77 q^{-5} -33 q^{-7} +111 q^{-9} +92 q^{-11} -114 q^{-13} -172 q^{-15} +85 q^{-17} +247 q^{-19} -14 q^{-21} -291 q^{-23} -75 q^{-25} +293 q^{-27} +168 q^{-29} -258 q^{-31} -233 q^{-33} +192 q^{-35} +269 q^{-37} -116 q^{-39} -282 q^{-41} +44 q^{-43} +259 q^{-45} +34 q^{-47} -229 q^{-49} -106 q^{-51} +176 q^{-53} +181 q^{-55} -110 q^{-57} -243 q^{-59} +20 q^{-61} +288 q^{-63} +77 q^{-65} -295 q^{-67} -168 q^{-69} +262 q^{-71} +233 q^{-73} -192 q^{-75} -251 q^{-77} +108 q^{-79} +229 q^{-81} -42 q^{-83} -174 q^{-85} - q^{-87} +109 q^{-89} +19 q^{-91} -60 q^{-93} -16 q^{-95} +30 q^{-97} +5 q^{-99} -10 q^{-101} -3 q^{-103} +5 q^{-105} + q^{-107} -3 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}-2 q^{26}-q^{24}+4 q^{22}+q^{20}+3 q^{18}-15 q^{16}-13 q^{14}+21 q^{12}+30 q^{10}+38 q^8-61 q^6-117 q^4-19 q^2+116+269 q^{-2} +28 q^{-4} -330 q^{-6} -394 q^{-8} -75 q^{-10} +664 q^{-12} +656 q^{-14} -103 q^{-16} -949 q^{-18} -1039 q^{-20} +406 q^{-22} +1467 q^{-24} +1122 q^{-26} -627 q^{-28} -2170 q^{-30} -1028 q^{-32} +1205 q^{-34} +2480 q^{-36} +964 q^{-38} -2076 q^{-40} -2550 q^{-42} -370 q^{-44} +2572 q^{-46} +2568 q^{-48} -683 q^{-50} -2837 q^{-52} -1928 q^{-54} +1465 q^{-56} +2993 q^{-58} +748 q^{-60} -2054 q^{-62} -2479 q^{-64} +263 q^{-66} +2457 q^{-68} +1487 q^{-70} -1081 q^{-72} -2308 q^{-74} -596 q^{-76} +1663 q^{-78} +1884 q^{-80} -133 q^{-82} -1952 q^{-84} -1480 q^{-86} +643 q^{-88} +2233 q^{-90} +1164 q^{-92} -1199 q^{-94} -2451 q^{-96} -951 q^{-98} +2029 q^{-100} +2574 q^{-102} +342 q^{-104} -2637 q^{-106} -2630 q^{-108} +727 q^{-110} +2948 q^{-112} +2022 q^{-114} -1453 q^{-116} -3072 q^{-118} -863 q^{-120} +1790 q^{-122} +2450 q^{-124} +104 q^{-126} -1962 q^{-128} -1354 q^{-130} +315 q^{-132} +1520 q^{-134} +670 q^{-136} -631 q^{-138} -778 q^{-140} -264 q^{-142} +507 q^{-144} +394 q^{-146} -67 q^{-148} -204 q^{-150} -174 q^{-152} +96 q^{-154} +98 q^{-156} - q^{-158} -15 q^{-160} -44 q^{-162} +17 q^{-164} +13 q^{-166} -6 q^{-168} +2 q^{-170} -6 q^{-172} +5 q^{-174} + q^{-176} -3 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}-2 q^{43}-q^{41}+4 q^{39}+q^{37}-2 q^{35}-3 q^{33}-9 q^{31}-5 q^{29}+24 q^{27}+36 q^{25}+8 q^{23}-40 q^{21}-97 q^{19}-89 q^{17}+41 q^{15}+232 q^{13}+284 q^{11}+74 q^9-338 q^7-670 q^5-512 q^3+244 q+1159 q^{-1} +1393 q^{-3} +420 q^{-5} -1360 q^{-7} -2638 q^{-9} -2020 q^{-11} +671 q^{-13} +3748 q^{-15} +4487 q^{-17} +1516 q^{-19} -3666 q^{-21} -7192 q^{-23} -5449 q^{-25} +1493 q^{-27} +8853 q^{-29} +10312 q^{-31} +3344 q^{-33} -7926 q^{-35} -14714 q^{-37} -10269 q^{-39} +3624 q^{-41} +16694 q^{-43} +17663 q^{-45} +3934 q^{-47} -14899 q^{-49} -23458 q^{-51} -13256 q^{-53} +9106 q^{-55} +25834 q^{-57} +22167 q^{-59} -386 q^{-61} -23990 q^{-63} -28639 q^{-65} -9350 q^{-67} +18542 q^{-69} +31428 q^{-71} +17884 q^{-73} -10903 q^{-75} -30419 q^{-77} -23837 q^{-79} +2987 q^{-81} +26597 q^{-83} +26569 q^{-85} +3629 q^{-87} -21302 q^{-89} -26442 q^{-91} -8267 q^{-93} +15967 q^{-95} +24539 q^{-97} +10786 q^{-99} -11497 q^{-101} -21849 q^{-103} -11951 q^{-105} +8041 q^{-107} +19517 q^{-109} +12632 q^{-111} -5369 q^{-113} -17777 q^{-115} -13776 q^{-117} +2565 q^{-119} +16606 q^{-121} +15936 q^{-123} +1127 q^{-125} -15198 q^{-127} -19017 q^{-129} -6419 q^{-131} +12642 q^{-133} +22214 q^{-135} +13326 q^{-137} -7954 q^{-139} -24321 q^{-141} -21034 q^{-143} +851 q^{-145} +23869 q^{-147} +28018 q^{-149} +8190 q^{-151} -20034 q^{-153} -32512 q^{-155} -17554 q^{-157} +12908 q^{-159} +32986 q^{-161} +25248 q^{-163} -3741 q^{-165} -29118 q^{-167} -29445 q^{-169} -5295 q^{-171} +21820 q^{-173} +29170 q^{-175} +12189 q^{-177} -13005 q^{-179} -25026 q^{-181} -15577 q^{-183} +4974 q^{-185} +18497 q^{-187} +15350 q^{-189} +806 q^{-191} -11613 q^{-193} -12610 q^{-195} -3736 q^{-197} +5992 q^{-199} +8795 q^{-201} +4272 q^{-203} -2239 q^{-205} -5256 q^{-207} -3493 q^{-209} +328 q^{-211} +2711 q^{-213} +2263 q^{-215} +317 q^{-217} -1146 q^{-219} -1230 q^{-221} -399 q^{-223} +416 q^{-225} +592 q^{-227} +226 q^{-229} -119 q^{-231} -217 q^{-233} -121 q^{-235} +23 q^{-237} +83 q^{-239} +48 q^{-241} -15 q^{-243} -24 q^{-245} -5 q^{-247} +2 q^{-251} +6 q^{-253} - q^{-255} -6 q^{-257} +5 q^{-259} + q^{-261} -3 q^{-263} + q^{-265} }

A2 Invariants.

Weight	Invariant
1,0	$1-q^{-2}+q^{-4}+2q^{-6}-2q^{-8}+4q^{-10}-q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}-3q^{-22}+q^{-24}+q^{-26}-2q^{-28}+q^{-30}$
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^4-4 q^2+14-36 q^{-2} +82 q^{-4} -162 q^{-6} +298 q^{-8} -484 q^{-10} +726 q^{-12} -994 q^{-14} +1244 q^{-16} -1420 q^{-18} +1479 q^{-20} -1354 q^{-22} +1034 q^{-24} -528 q^{-26} -104 q^{-28} +814 q^{-30} -1524 q^{-32} +2132 q^{-34} -2575 q^{-36} +2796 q^{-38} -2780 q^{-40} +2510 q^{-42} -2033 q^{-44} +1402 q^{-46} -700 q^{-48} +16 q^{-50} +570 q^{-52} -998 q^{-54} +1256 q^{-56} -1340 q^{-58} +1276 q^{-60} -1116 q^{-62} +916 q^{-64} -706 q^{-66} +506 q^{-68} -344 q^{-70} +226 q^{-72} -136 q^{-74} +73 q^{-76} -38 q^{-78} +18 q^{-80} -6 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-q^2-2+3 q^{-2} +5 q^{-4} -2 q^{-6} -8 q^{-8} +4 q^{-10} +14 q^{-12} -7 q^{-14} -14 q^{-16} +10 q^{-18} +15 q^{-20} -10 q^{-22} -9 q^{-24} +14 q^{-26} +7 q^{-28} -11 q^{-30} + q^{-32} +10 q^{-34} -10 q^{-36} -6 q^{-38} +10 q^{-40} -8 q^{-42} -14 q^{-44} +8 q^{-46} +14 q^{-48} -12 q^{-50} -10 q^{-52} +16 q^{-54} +8 q^{-56} -13 q^{-58} -5 q^{-60} +11 q^{-62} +2 q^{-64} -5 q^{-66} -2 q^{-68} +3 q^{-70} -2 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 1-2 q^{-2} +2 q^{-4} +4 q^{-6} -9 q^{-8} +8 q^{-10} +9 q^{-12} -21 q^{-14} +18 q^{-16} +13 q^{-18} -28 q^{-20} +20 q^{-22} +14 q^{-24} -29 q^{-26} +7 q^{-28} +11 q^{-30} -14 q^{-32} -6 q^{-34} +5 q^{-36} +11 q^{-38} -13 q^{-40} -8 q^{-42} +29 q^{-44} -16 q^{-46} -17 q^{-48} +33 q^{-50} -13 q^{-52} -17 q^{-54} +22 q^{-56} -5 q^{-58} -10 q^{-60} +9 q^{-62} -3 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} - q^{-3} +2 q^{-5} - q^{-7} +3 q^{-9} -2 q^{-11} +4 q^{-13} - q^{-15} +2 q^{-17} -3 q^{-25} +2 q^{-27} -3 q^{-29} +2 q^{-31} -2 q^{-33} +2 q^{-35} -2 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^{-4} +4 q^{-8} - q^{-10} -4 q^{-12} +6 q^{-14} +6 q^{-16} -8 q^{-18} -2 q^{-20} +18 q^{-22} +4 q^{-24} -15 q^{-26} +11 q^{-28} +26 q^{-30} -17 q^{-32} -19 q^{-34} +20 q^{-36} +3 q^{-38} -32 q^{-40} -3 q^{-42} +20 q^{-44} -14 q^{-46} -15 q^{-48} +23 q^{-50} +9 q^{-52} -24 q^{-54} +9 q^{-56} +23 q^{-58} -17 q^{-60} -15 q^{-62} +20 q^{-64} +7 q^{-66} -20 q^{-68} -2 q^{-70} +16 q^{-72} -2 q^{-74} -12 q^{-76} +4 q^{-78} +7 q^{-80} -3 q^{-82} -2 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} - q^{-4} +2 q^{-6} +3 q^{-12} -2 q^{-14} +4 q^{-16} - q^{-18} +2 q^{-20} + q^{-22} - q^{-28} -3 q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} - q^{-42} +2 q^{-44} -2 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 1-2 q^{-2} +6 q^{-4} -10 q^{-6} +17 q^{-8} -24 q^{-10} +31 q^{-12} -35 q^{-14} +38 q^{-16} -33 q^{-18} +26 q^{-20} -12 q^{-22} -4 q^{-24} +23 q^{-26} -41 q^{-28} +57 q^{-30} -68 q^{-32} +72 q^{-34} -69 q^{-36} +59 q^{-38} -45 q^{-40} +26 q^{-42} -7 q^{-44} -10 q^{-46} +23 q^{-48} -33 q^{-50} +37 q^{-52} -37 q^{-54} +32 q^{-56} -25 q^{-58} +18 q^{-60} -11 q^{-62} +6 q^{-64} -3 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^2-2 q^{-2} -2 q^{-4} +4 q^{-6} +7 q^{-8} -2 q^{-10} -13 q^{-12} -5 q^{-14} +18 q^{-16} +19 q^{-18} -13 q^{-20} -30 q^{-22} - q^{-24} +38 q^{-26} +21 q^{-28} -28 q^{-30} -34 q^{-32} +13 q^{-34} +40 q^{-36} +6 q^{-38} -34 q^{-40} -17 q^{-42} +22 q^{-44} +20 q^{-46} -16 q^{-48} -23 q^{-50} +9 q^{-52} +23 q^{-54} -6 q^{-56} -27 q^{-58} +28 q^{-62} +9 q^{-64} -28 q^{-66} -19 q^{-68} +25 q^{-70} +30 q^{-72} -15 q^{-74} -38 q^{-76} - q^{-78} +38 q^{-80} +19 q^{-82} -25 q^{-84} -30 q^{-86} +7 q^{-88} +27 q^{-90} +8 q^{-92} -15 q^{-94} -14 q^{-96} +3 q^{-98} +10 q^{-100} +3 q^{-102} -3 q^{-104} -3 q^{-106} + q^{-110} }

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{-2}-2q^{-4}+4q^{-6}-5q^{-8}+10q^{-10}-13q^{-12}+18q^{-14}-20q^{-16}+27q^{-18}-28q^{-20}+30q^{-22}-26q^{-24}+30q^{-26}-19q^{-28}+14q^{-30}-3q^{-32}-2q^{-34}+14q^{-36}-31q^{-38}+33q^{-40}-44q^{-42}+48q^{-44}-59q^{-46}+53q^{-48}-52q^{-50}+52q^{-52}-42q^{-54}+31q^{-56}-23q^{-58}+17q^{-60}-10q^{-64}+13q^{-66}-21q^{-68}+30q^{-70}-29q^{-72}+25q^{-74}-29q^{-76}+27q^{-78}-18q^{-80}+14q^{-82}-14q^{-84}+10q^{-86}-4q^{-88}+3q^{-90}-3q^{-92}+q^{-94}$

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} -2 q^{-4} +6 q^{-6} -10 q^{-8} +13 q^{-10} -12 q^{-12} +3 q^{-14} +19 q^{-16} -45 q^{-18} +75 q^{-20} -88 q^{-22} +65 q^{-24} -7 q^{-26} -85 q^{-28} +181 q^{-30} -229 q^{-32} +207 q^{-34} -97 q^{-36} -66 q^{-38} +227 q^{-40} -319 q^{-42} +300 q^{-44} -165 q^{-46} -29 q^{-48} +202 q^{-50} -276 q^{-52} +226 q^{-54} -72 q^{-56} -100 q^{-58} +223 q^{-60} -234 q^{-62} +120 q^{-64} +62 q^{-66} -250 q^{-68} +358 q^{-70} -329 q^{-72} +175 q^{-74} +56 q^{-76} -283 q^{-78} +422 q^{-80} -428 q^{-82} +287 q^{-84} -63 q^{-86} -176 q^{-88} +333 q^{-90} -352 q^{-92} +236 q^{-94} -38 q^{-96} -143 q^{-98} +230 q^{-100} -197 q^{-102} +55 q^{-104} +115 q^{-106} -235 q^{-108} +256 q^{-110} -158 q^{-112} -6 q^{-114} +173 q^{-116} -275 q^{-118} +278 q^{-120} -193 q^{-122} +61 q^{-124} +66 q^{-126} -157 q^{-128} +184 q^{-130} -156 q^{-132} +99 q^{-134} -28 q^{-136} -26 q^{-138} +54 q^{-140} -65 q^{-142} +54 q^{-144} -34 q^{-146} +16 q^{-148} + q^{-150} -8 q^{-152} +10 q^{-154} -10 q^{-156} +6 q^{-158} -3 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 92"];

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+10 t^2-20 t+25-20 t^{-1} +10 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^6-2 z^4+2 z^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]=

{ 89, 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}-4 q^9+8 q^8-12 q^7+14 q^6-15 q^5+14 q^4-10 q^3+7 q^2-3 q+1}

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

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 2 z^9 a^{-5} +2 z^9 a^{-7} +4 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8 a^{-8} +3 z^7 a^{-3} +5 z^7 a^{-5} +12 z^7 a^{-7} +10 z^7 a^{-9} +z^6 a^{-2} -8 z^6 a^{-4} -22 z^6 a^{-6} -5 z^6 a^{-8} +8 z^6 a^{-10} -8 z^5 a^{-3} -22 z^5 a^{-5} -32 z^5 a^{-7} -14 z^5 a^{-9} +4 z^5 a^{-11} -3 z^4 a^{-2} +2 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} +6 z^3 a^{-3} +18 z^3 a^{-5} +21 z^3 a^{-7} +7 z^3 a^{-9} -2 z^3 a^{-11} +3 z^2 a^{-2} +z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-3} -5 z a^{-5} -5 z a^{-7} -z a^{-9} - a^{-2} + a^{-4} + a^{-6} }

Vassiliev invariants

V2 and V3:

(2, 3)

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 24} 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{316}{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{92}{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 192} 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 464} $128$ 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 88} 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 288} 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{2528}{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{736}{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{29311}{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{1276}{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{44644}{45}} ${\frac {161}{9}}$ 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 \frac{2191}{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=} 4 is the signature of 10 92. 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										3		-3
17									5	1		4
15								7	3			-4
13							7	5				2
11						8	7					-1
9					6	7						-1
7				4	8							4
5			3	6								-3
3		1	5									4
1		2										-2
-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, 92]]
Out[2]=	10
In[3]:=	PD[Knot[10, 92]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], X[16, 12, 17, 11], X[18, 7, 19, 8], X[12, 18, 13, 17], X[6, 19, 7, 20], X[8, 14, 9, 13], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 92]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -8, 6, -9, 10, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, -4]
In[5]:=	BR[Knot[10, 92]]
Out[5]=	BR[4, {1, 1, 1, 2, 2, -3, 2, -1, 2, -3, 2}]
In[6]:=	alex = Alexander[Knot[10, 92]][t]
Out[6]=	2 10 20 2 3 25 - -- + -- - -- - 20 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 92]][z]
Out[7]=	2 4 6 1 + 2 z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 92], Knot[11, Alternating, 153], Knot[11, Alternating, 224], Knot[11, NonAlternating, 35], Knot[11, NonAlternating, 43]}
In[9]:=	{KnotDet[Knot[10, 92]], KnotSignature[Knot[10, 92]]}
Out[9]=	{89, 4}
In[10]:=	J=Jones[Knot[10, 92]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 3 q + 7 q - 10 q + 14 q - 15 q + 14 q - 12 q + 8 q - 4 q + 10 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 92]}
In[12]:=	A2Invariant[Knot[10, 92]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 1 - q + q + 2 q - 2 q + 4 q - q + q + q - 3 q + 2 q - 22 24 26 28 30 3 q + q + q - 2 q + q
In[13]:=	Kauffman[Knot[10, 92]][a, z]
Out[13]=	2 2 2 2 -6 -4 -2 z 5 z 5 z z 2 z 2 z 2 z z a + a - a - -- - --- - --- - -- + ---- + ---- - ---- + -- + 9 7 5 3 10 8 6 4 a a a a a a a a 2 3 3 3 3 3 4 4 4 3 z 2 z 7 z 21 z 18 z 6 z z 8 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 10 z 2 z 3 z 4 z 14 z 32 z 22 z 8 z 8 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 5 z 22 z 8 z z 10 z 12 z 5 z 3 z 7 z ---- - ----- - ---- + -- + ----- + ----- + ---- + ---- + ---- + 8 6 4 2 9 7 5 3 8 a a a a a a a a a 8 8 9 9 11 z 4 z 2 z 2 z ----- + ---- + ---- + ---- 6 4 7 5 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 92]], Vassiliev[3][Knot[10, 92]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[10, 92]][q, t]
Out[15]=	3 3 5 1 2 q q 5 7 7 2 9 2 5 q + 3 q + ---- + --- + -- + 6 q t + 4 q t + 8 q t + 6 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 7 q t + 8 q t + 7 q t + 7 q t + 5 q t + 7 q t + 15 6 17 6 17 7 19 7 21 8 3 q t + 5 q t + q t + 3 q t + q t