10 62

10_61

10_63

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10 62 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,19,12,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8 X13,1,14,20 X19,13,20,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8
Dowker-Thistlethwaite code	4 10 14 16 2 18 20 6 8 12
Conway Notation	[4,3,21]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [0][-12] Hyperbolic Volume 10.1415 A-Polynomial See Data:10 62/A-polynomial

[edit Notes for 10 62's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math] Topological 4 genus [math]\displaystyle{ 2 }[/math] Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant 4

[edit Notes for 10 62's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-3 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+5 z^6+8 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 45, 4 }
Jones polynomial	[math]\displaystyle{ -q^9+2 q^8-4 q^7+6 q^6-7 q^5+7 q^4-6 q^3+6 q^2-3 q+2- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +18 z^4 a^{-4} -5 z^4 a^{-6} -7 z^2 a^{-2} +20 z^2 a^{-4} -8 z^2 a^{-6} -2 a^{-2} +7 a^{-4} -4 a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -z^7 a^{-3} +2 z^7 a^{-5} +4 z^7 a^{-7} -10 z^6 a^{-2} -21 z^6 a^{-4} -7 z^6 a^{-6} +4 z^6 a^{-8} -5 z^5 a^{-1} -9 z^5 a^{-3} -15 z^5 a^{-5} -8 z^5 a^{-7} +3 z^5 a^{-9} +16 z^4 a^{-2} +30 z^4 a^{-4} +6 z^4 a^{-6} -6 z^4 a^{-8} +2 z^4 a^{-10} +7 z^3 a^{-1} +15 z^3 a^{-3} +16 z^3 a^{-5} +5 z^3 a^{-7} -2 z^3 a^{-9} +z^3 a^{-11} -10 z^2 a^{-2} -23 z^2 a^{-4} -8 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} -2 z a^{-1} -5 z a^{-3} -6 z a^{-5} -z a^{-7} +z a^{-9} -z a^{-11} +2 a^{-2} +7 a^{-4} +4 a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2- q^{-2} + q^{-4} +2 q^{-6} + q^{-8} +3 q^{-10} - q^{-12} +2 q^{-14} -2 q^{-22} - q^{-26} }[/math]
The G2 invariant	[math]\displaystyle{ q^{12}-q^{10}+3 q^8-5 q^6+4 q^4-4 q^2-2+9 q^{-2} -16 q^{-4} +19 q^{-6} -18 q^{-8} +5 q^{-10} +11 q^{-12} -27 q^{-14} +30 q^{-16} -27 q^{-18} +13 q^{-20} +8 q^{-22} -23 q^{-24} +27 q^{-26} -20 q^{-28} +11 q^{-30} +12 q^{-32} -20 q^{-34} +15 q^{-36} -3 q^{-38} -3 q^{-40} +21 q^{-42} -22 q^{-44} +21 q^{-46} -5 q^{-48} -5 q^{-50} +25 q^{-52} -38 q^{-54} +33 q^{-56} -19 q^{-58} + q^{-60} +18 q^{-62} -31 q^{-64} +33 q^{-66} -22 q^{-68} +6 q^{-70} +11 q^{-72} -23 q^{-74} +17 q^{-76} -7 q^{-78} -7 q^{-80} +16 q^{-82} -13 q^{-84} +6 q^{-86} +4 q^{-88} -13 q^{-90} +16 q^{-92} -16 q^{-94} +8 q^{-96} - q^{-98} -9 q^{-100} +12 q^{-102} -13 q^{-104} +13 q^{-106} -10 q^{-108} +4 q^{-110} -9 q^{-114} +10 q^{-116} -12 q^{-118} +10 q^{-120} -5 q^{-122} +2 q^{-124} +3 q^{-126} -6 q^{-128} +6 q^{-130} -5 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_62.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^3+q- q^{-1} +3 q^{-3} + q^{-7} - q^{-11} +2 q^{-13} -2 q^{-15} + q^{-17} - q^{-19} }[/math]
2	[math]\displaystyle{ q^{12}-q^{10}-2 q^8+3 q^6-6 q^2+3+5 q^{-2} -6 q^{-4} +2 q^{-6} +8 q^{-8} -3 q^{-10} -2 q^{-12} +6 q^{-14} - q^{-16} -6 q^{-18} +2 q^{-20} +4 q^{-22} -3 q^{-24} -2 q^{-26} +5 q^{-28} -7 q^{-32} +4 q^{-34} +2 q^{-36} -6 q^{-38} +3 q^{-40} + q^{-42} -3 q^{-44} +2 q^{-46} - q^{-50} + q^{-52} }[/math]
3	[math]\displaystyle{ -q^{27}+q^{25}+2 q^{23}-4 q^{19}-2 q^{17}+6 q^{15}+6 q^{13}-6 q^{11}-12 q^9+q^7+15 q^5+6 q^3-17 q-14 q^{-1} +10 q^{-3} +22 q^{-5} -2 q^{-7} -18 q^{-9} -6 q^{-11} +20 q^{-13} +15 q^{-15} -10 q^{-17} -17 q^{-19} +5 q^{-21} +18 q^{-23} -20 q^{-27} -5 q^{-29} +20 q^{-31} +7 q^{-33} -18 q^{-35} -12 q^{-37} +19 q^{-39} +14 q^{-41} -13 q^{-43} -19 q^{-45} +3 q^{-47} +17 q^{-49} +7 q^{-51} -14 q^{-53} -16 q^{-55} +7 q^{-57} +21 q^{-59} -19 q^{-63} -7 q^{-65} +16 q^{-67} +7 q^{-69} -8 q^{-71} -6 q^{-73} +2 q^{-75} +3 q^{-77} - q^{-85} + q^{-87} + q^{-89} - q^{-91} + q^{-97} - q^{-99} }[/math]
4	[math]\displaystyle{ q^{48}-q^{46}-2 q^{44}+q^{40}+6 q^{38}-6 q^{34}-6 q^{32}-5 q^{30}+15 q^{28}+14 q^{26}-15 q^{22}-30 q^{20}+3 q^{18}+26 q^{16}+33 q^{14}+10 q^{12}-47 q^{10}-42 q^8-11 q^6+45 q^4+66 q^2+1-47 q^{-2} -74 q^{-4} -16 q^{-6} +70 q^{-8} +69 q^{-10} +28 q^{-12} -67 q^{-14} -86 q^{-16} -7 q^{-18} +62 q^{-20} +98 q^{-22} +13 q^{-24} -81 q^{-26} -82 q^{-28} -13 q^{-30} +94 q^{-32} +81 q^{-34} -23 q^{-36} -93 q^{-38} -65 q^{-40} +51 q^{-42} +92 q^{-44} +20 q^{-46} -70 q^{-48} -70 q^{-50} +28 q^{-52} +83 q^{-54} +23 q^{-56} -68 q^{-58} -66 q^{-60} +20 q^{-62} +84 q^{-64} +36 q^{-66} -58 q^{-68} -78 q^{-70} -20 q^{-72} +72 q^{-74} +80 q^{-76} +4 q^{-78} -66 q^{-80} -94 q^{-82} -10 q^{-84} +88 q^{-86} +98 q^{-88} +17 q^{-90} -116 q^{-92} -108 q^{-94} +19 q^{-96} +121 q^{-98} +105 q^{-100} -55 q^{-102} -124 q^{-104} -52 q^{-106} +57 q^{-108} +106 q^{-110} +9 q^{-112} -61 q^{-114} -54 q^{-116} - q^{-118} +55 q^{-120} +19 q^{-122} -12 q^{-124} -21 q^{-126} -13 q^{-128} +16 q^{-130} +5 q^{-132} +3 q^{-134} -2 q^{-136} -8 q^{-138} +5 q^{-140} - q^{-142} + q^{-144} -3 q^{-148} +3 q^{-150} - q^{-152} - q^{-158} + q^{-160} }[/math]
5	[math]\displaystyle{ -q^{75}+q^{73}+2 q^{71}-q^{67}-3 q^{65}-4 q^{63}+8 q^{59}+8 q^{57}+2 q^{55}-6 q^{53}-16 q^{51}-16 q^{49}+2 q^{47}+25 q^{45}+31 q^{43}+16 q^{41}-16 q^{39}-50 q^{37}-50 q^{35}-9 q^{33}+50 q^{31}+82 q^{29}+63 q^{27}-10 q^{25}-95 q^{23}-123 q^{21}-64 q^{19}+53 q^{17}+151 q^{15}+154 q^{13}+43 q^{11}-115 q^9-215 q^7-171 q^5+6 q^3+193 q+260 q^{-1} +157 q^{-3} -78 q^{-5} -280 q^{-7} -292 q^{-9} -96 q^{-11} +174 q^{-13} +356 q^{-15} +295 q^{-17} +8 q^{-19} -300 q^{-21} -403 q^{-23} -228 q^{-25} +135 q^{-27} +432 q^{-29} +417 q^{-31} +81 q^{-33} -331 q^{-35} -517 q^{-37} -306 q^{-39} +161 q^{-41} +516 q^{-43} +467 q^{-45} +38 q^{-47} -430 q^{-49} -554 q^{-51} -219 q^{-53} +294 q^{-55} +557 q^{-57} +344 q^{-59} -150 q^{-61} -498 q^{-63} -397 q^{-65} +36 q^{-67} +410 q^{-69} +389 q^{-71} +27 q^{-73} -324 q^{-75} -346 q^{-77} -39 q^{-79} +267 q^{-81} +279 q^{-83} +17 q^{-85} -254 q^{-87} -251 q^{-89} +24 q^{-91} +277 q^{-93} +250 q^{-95} -27 q^{-97} -297 q^{-99} -307 q^{-101} -26 q^{-103} +292 q^{-105} +372 q^{-107} +147 q^{-109} -205 q^{-111} -414 q^{-113} -320 q^{-115} +31 q^{-117} +390 q^{-119} +486 q^{-121} +209 q^{-123} -257 q^{-125} -574 q^{-127} -472 q^{-129} +37 q^{-131} +566 q^{-133} +657 q^{-135} +215 q^{-137} -425 q^{-139} -736 q^{-141} -437 q^{-143} +230 q^{-145} +686 q^{-147} +550 q^{-149} -25 q^{-151} -538 q^{-153} -566 q^{-155} -121 q^{-157} +366 q^{-159} +482 q^{-161} +188 q^{-163} -209 q^{-165} -358 q^{-167} -189 q^{-169} +98 q^{-171} +239 q^{-173} +151 q^{-175} -37 q^{-177} -141 q^{-179} -102 q^{-181} +3 q^{-183} +75 q^{-185} +67 q^{-187} +9 q^{-189} -37 q^{-191} -38 q^{-193} -9 q^{-195} +12 q^{-197} +19 q^{-199} +10 q^{-201} -4 q^{-203} -11 q^{-205} -5 q^{-207} +3 q^{-209} + q^{-211} +2 q^{-213} +2 q^{-215} -3 q^{-217} +2 q^{-221} - q^{-223} - q^{-225} + q^{-227} + q^{-233} - q^{-235} }[/math]
6	[math]\displaystyle{ q^{108}-q^{106}-2 q^{104}+q^{100}+3 q^{98}+q^{96}+4 q^{94}-2 q^{92}-10 q^{90}-7 q^{88}-2 q^{86}+8 q^{84}+9 q^{82}+21 q^{80}+10 q^{78}-14 q^{76}-29 q^{74}-34 q^{72}-15 q^{70}+q^{68}+55 q^{66}+70 q^{64}+44 q^{62}-8 q^{60}-70 q^{58}-102 q^{56}-116 q^{54}-17 q^{52}+89 q^{50}+171 q^{48}+173 q^{46}+91 q^{44}-55 q^{42}-247 q^{40}-282 q^{38}-203 q^{36}+16 q^{34}+247 q^{32}+411 q^{30}+385 q^{28}+96 q^{26}-229 q^{24}-519 q^{22}-551 q^{20}-331 q^{18}+135 q^{16}+586 q^{14}+740 q^{12}+578 q^{10}+57 q^8-520 q^6-942 q^4-861 q^2-335+378 q^{-2} +1012 q^{-4} +1153 q^{-6} +736 q^{-8} -168 q^{-10} -1003 q^{-12} -1400 q^{-14} -1133 q^{-16} -205 q^{-18} +911 q^{-20} +1665 q^{-22} +1519 q^{-24} +606 q^{-26} -723 q^{-28} -1808 q^{-30} -1964 q^{-32} -1021 q^{-34} +590 q^{-36} +1918 q^{-38} +2312 q^{-40} +1416 q^{-42} -395 q^{-44} -2090 q^{-46} -2632 q^{-48} -1644 q^{-50} +292 q^{-52} +2186 q^{-54} +2872 q^{-56} +1831 q^{-58} -365 q^{-60} -2361 q^{-62} -2948 q^{-64} -1810 q^{-66} +481 q^{-68} +2522 q^{-70} +2988 q^{-72} +1571 q^{-74} -774 q^{-76} -2595 q^{-78} -2809 q^{-80} -1231 q^{-82} +1113 q^{-84} +2660 q^{-86} +2420 q^{-88} +688 q^{-90} -1400 q^{-92} -2508 q^{-94} -1920 q^{-96} -101 q^{-98} +1635 q^{-100} +2122 q^{-102} +1210 q^{-104} -423 q^{-106} -1612 q^{-108} -1575 q^{-110} -440 q^{-112} +845 q^{-114} +1317 q^{-116} +802 q^{-118} -256 q^{-120} -975 q^{-122} -842 q^{-124} +14 q^{-126} +829 q^{-128} +885 q^{-130} +198 q^{-132} -740 q^{-134} -1144 q^{-136} -699 q^{-138} +375 q^{-140} +1303 q^{-142} +1363 q^{-144} +508 q^{-146} -775 q^{-148} -1646 q^{-150} -1587 q^{-152} -522 q^{-154} +961 q^{-156} +1974 q^{-158} +1906 q^{-160} +711 q^{-162} -988 q^{-164} -2316 q^{-166} -2405 q^{-168} -1044 q^{-170} +1076 q^{-172} +2735 q^{-174} +2889 q^{-176} +1336 q^{-178} -1200 q^{-180} -3200 q^{-182} -3303 q^{-184} -1366 q^{-186} +1472 q^{-188} +3479 q^{-190} +3417 q^{-192} +1200 q^{-194} -1800 q^{-196} -3590 q^{-198} -3103 q^{-200} -736 q^{-202} +1968 q^{-204} +3351 q^{-206} +2525 q^{-208} +154 q^{-210} -2040 q^{-212} -2737 q^{-214} -1677 q^{-216} +270 q^{-218} +1841 q^{-220} +2002 q^{-222} +853 q^{-224} -580 q^{-226} -1393 q^{-228} -1189 q^{-230} -307 q^{-232} +632 q^{-234} +935 q^{-236} +546 q^{-238} -52 q^{-240} -471 q^{-242} -487 q^{-244} -209 q^{-246} +173 q^{-248} +314 q^{-250} +190 q^{-252} +7 q^{-254} -133 q^{-256} -149 q^{-258} -82 q^{-260} +56 q^{-262} +99 q^{-264} +51 q^{-266} +8 q^{-268} -35 q^{-270} -45 q^{-272} -38 q^{-274} +16 q^{-276} +32 q^{-278} +13 q^{-280} +9 q^{-282} -6 q^{-284} -12 q^{-286} -17 q^{-288} +5 q^{-290} +9 q^{-292} + q^{-294} +5 q^{-296} - q^{-298} - q^{-300} -6 q^{-302} +2 q^{-304} +2 q^{-306} -2 q^{-308} +2 q^{-310} - q^{-312} + q^{-314} - q^{-316} - q^{-322} + q^{-324} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^2- q^{-2} + q^{-4} +2 q^{-6} + q^{-8} +3 q^{-10} - q^{-12} +2 q^{-14} -2 q^{-22} - q^{-26} }[/math]
1,1	[math]\displaystyle{ q^{12}-2 q^{10}+6 q^8-14 q^6+25 q^4-40 q^2+58-80 q^{-2} +85 q^{-4} -92 q^{-6} +82 q^{-8} -62 q^{-10} +33 q^{-12} +14 q^{-14} -38 q^{-16} +90 q^{-18} -104 q^{-20} +138 q^{-22} -144 q^{-24} +140 q^{-26} -135 q^{-28} +98 q^{-30} -80 q^{-32} +42 q^{-34} -10 q^{-36} -14 q^{-38} +32 q^{-40} -38 q^{-42} +39 q^{-44} -42 q^{-46} +36 q^{-48} -34 q^{-50} +34 q^{-52} -32 q^{-54} +30 q^{-56} -28 q^{-58} +24 q^{-60} -20 q^{-62} +16 q^{-64} -12 q^{-66} +9 q^{-68} -6 q^{-70} +4 q^{-72} -2 q^{-74} + q^{-76} }[/math]
2,0	[math]\displaystyle{ q^{10}-q^6+q^2-2-4 q^{-2} - q^{-4} + q^{-6} - q^{-8} - q^{-10} +6 q^{-12} +5 q^{-14} +3 q^{-16} +4 q^{-18} +4 q^{-20} -2 q^{-22} -2 q^{-24} - q^{-30} +3 q^{-32} +5 q^{-34} -2 q^{-36} -2 q^{-38} + q^{-40} -2 q^{-42} -5 q^{-44} -3 q^{-46} - q^{-48} - q^{-50} - q^{-52} + q^{-56} + q^{-60} + q^{-62} + q^{-66} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^6-q^4+q^2-4 q^{-2} + q^{-4} -2 q^{-6} -4 q^{-8} +3 q^{-10} + q^{-12} - q^{-14} +9 q^{-16} +4 q^{-18} +2 q^{-20} +9 q^{-22} +4 q^{-24} -2 q^{-26} - q^{-28} -2 q^{-30} -4 q^{-32} -7 q^{-34} - q^{-36} +2 q^{-38} -5 q^{-40} + q^{-42} +5 q^{-44} -5 q^{-46} - q^{-48} +5 q^{-50} -3 q^{-52} -2 q^{-54} +3 q^{-56} - q^{-60} + q^{-62} }[/math]
1,0,0	[math]\displaystyle{ -q-2 q^{-3} + q^{-5} - q^{-7} +3 q^{-9} + q^{-11} +3 q^{-13} +2 q^{-15} +2 q^{-17} +2 q^{-19} + q^{-23} -3 q^{-25} -3 q^{-29} - q^{-33} }[/math]
1,0,1	[math]\displaystyle{ q^{12}-2 q^{10}+5 q^8-7 q^6+6 q^4-9+23 q^{-2} -32 q^{-4} +27 q^{-6} -25 q^{-8} -5 q^{-10} +15 q^{-12} -47 q^{-14} +48 q^{-16} -49 q^{-18} +42 q^{-20} -7 q^{-22} +22 q^{-24} +31 q^{-26} +2 q^{-28} +55 q^{-30} -31 q^{-32} +50 q^{-34} -51 q^{-36} +17 q^{-38} -43 q^{-40} -17 q^{-42} +6 q^{-44} -43 q^{-46} +46 q^{-48} -38 q^{-50} +30 q^{-52} -6 q^{-54} -10 q^{-56} +16 q^{-58} -18 q^{-60} +8 q^{-62} +4 q^{-64} -5 q^{-66} +7 q^{-68} +3 q^{-70} -10 q^{-72} +10 q^{-74} -6 q^{-76} -3 q^{-78} +10 q^{-80} -10 q^{-82} +4 q^{-84} +3 q^{-86} -7 q^{-88} +7 q^{-90} -3 q^{-92} - q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^4+1+ q^{-2} - q^{-4} -2 q^{-6} - q^{-8} -5 q^{-10} -5 q^{-12} -3 q^{-14} -5 q^{-16} -2 q^{-18} +3 q^{-20} +7 q^{-22} +6 q^{-24} +14 q^{-26} +18 q^{-28} +14 q^{-30} +5 q^{-32} +10 q^{-34} +2 q^{-36} -12 q^{-38} -9 q^{-40} -6 q^{-42} -12 q^{-44} -9 q^{-46} + q^{-48} -2 q^{-50} -4 q^{-52} + q^{-54} +3 q^{-56} -4 q^{-58} -3 q^{-60} +4 q^{-62} + q^{-64} -3 q^{-66} +2 q^{-68} +3 q^{-70} + q^{-76} }[/math]
1,0,0,0	[math]\displaystyle{ -1-2 q^{-4} - q^{-8} +2 q^{-12} + q^{-14} +4 q^{-16} +2 q^{-18} +5 q^{-20} +2 q^{-22} +3 q^{-24} -2 q^{-30} -3 q^{-32} - q^{-34} -3 q^{-36} - q^{-40} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^6+q^4-3 q^2+4-6 q^{-2} +7 q^{-4} -8 q^{-6} +8 q^{-8} -7 q^{-10} +7 q^{-12} - q^{-14} + q^{-16} +6 q^{-18} -6 q^{-20} +13 q^{-22} -14 q^{-24} +16 q^{-26} -15 q^{-28} +12 q^{-30} -12 q^{-32} +7 q^{-34} -5 q^{-36} +3 q^{-40} -5 q^{-42} +7 q^{-44} -7 q^{-46} +7 q^{-48} -7 q^{-50} +5 q^{-52} -4 q^{-54} +3 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }[/math]
1,0	[math]\displaystyle{ q^{12}-q^8-q^6+2 q^4+2 q^2-3-5 q^{-2} +5 q^{-6} + q^{-8} -7 q^{-10} -5 q^{-12} +5 q^{-14} +8 q^{-16} + q^{-18} -8 q^{-20} - q^{-22} +8 q^{-24} +9 q^{-26} -2 q^{-28} -5 q^{-30} + q^{-32} +8 q^{-34} +3 q^{-36} -3 q^{-38} - q^{-40} +5 q^{-42} +2 q^{-44} -5 q^{-46} -4 q^{-48} +2 q^{-50} +4 q^{-52} -4 q^{-54} -8 q^{-56} -2 q^{-58} +6 q^{-60} +2 q^{-62} -6 q^{-64} -5 q^{-66} +3 q^{-68} +7 q^{-70} -6 q^{-74} -4 q^{-76} +3 q^{-78} +6 q^{-80} -4 q^{-84} -3 q^{-86} + q^{-88} +3 q^{-90} + q^{-92} - q^{-94} - q^{-96} + q^{-100} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^6-q^4+2 q^2-3+4 q^{-2} -6 q^{-4} +4 q^{-6} -8 q^{-8} +5 q^{-10} -9 q^{-12} +3 q^{-14} -6 q^{-16} +5 q^{-18} +3 q^{-22} +8 q^{-24} +4 q^{-26} +15 q^{-28} - q^{-30} +16 q^{-32} -8 q^{-34} +13 q^{-36} -14 q^{-38} +7 q^{-40} -15 q^{-42} +3 q^{-44} -12 q^{-46} +2 q^{-48} -5 q^{-50} + q^{-54} -3 q^{-56} +4 q^{-58} -4 q^{-60} +6 q^{-62} -6 q^{-64} +4 q^{-66} -5 q^{-68} +6 q^{-70} -4 q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{12}-q^{10}+3 q^8-5 q^6+4 q^4-4 q^2-2+9 q^{-2} -16 q^{-4} +19 q^{-6} -18 q^{-8} +5 q^{-10} +11 q^{-12} -27 q^{-14} +30 q^{-16} -27 q^{-18} +13 q^{-20} +8 q^{-22} -23 q^{-24} +27 q^{-26} -20 q^{-28} +11 q^{-30} +12 q^{-32} -20 q^{-34} +15 q^{-36} -3 q^{-38} -3 q^{-40} +21 q^{-42} -22 q^{-44} +21 q^{-46} -5 q^{-48} -5 q^{-50} +25 q^{-52} -38 q^{-54} +33 q^{-56} -19 q^{-58} + q^{-60} +18 q^{-62} -31 q^{-64} +33 q^{-66} -22 q^{-68} +6 q^{-70} +11 q^{-72} -23 q^{-74} +17 q^{-76} -7 q^{-78} -7 q^{-80} +16 q^{-82} -13 q^{-84} +6 q^{-86} +4 q^{-88} -13 q^{-90} +16 q^{-92} -16 q^{-94} +8 q^{-96} - q^{-98} -9 q^{-100} +12 q^{-102} -13 q^{-104} +13 q^{-106} -10 q^{-108} +4 q^{-110} -9 q^{-114} +10 q^{-116} -12 q^{-118} +10 q^{-120} -5 q^{-122} +2 q^{-124} +3 q^{-126} -6 q^{-128} +6 q^{-130} -5 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/math]

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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 62"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^4-3 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+5 z^6+8 z^4+5 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^9+2 q^8-4 q^7+6 q^6-7 q^5+7 q^4-6 q^3+6 q^2-3 q+2- q^{-1} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +18 z^4 a^{-4} -5 z^4 a^{-6} -7 z^2 a^{-2} +20 z^2 a^{-4} -8 z^2 a^{-6} -2 a^{-2} +7 a^{-4} -4 a^{-6} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -z^7 a^{-3} +2 z^7 a^{-5} +4 z^7 a^{-7} -10 z^6 a^{-2} -21 z^6 a^{-4} -7 z^6 a^{-6} +4 z^6 a^{-8} -5 z^5 a^{-1} -9 z^5 a^{-3} -15 z^5 a^{-5} -8 z^5 a^{-7} +3 z^5 a^{-9} +16 z^4 a^{-2} +30 z^4 a^{-4} +6 z^4 a^{-6} -6 z^4 a^{-8} +2 z^4 a^{-10} +7 z^3 a^{-1} +15 z^3 a^{-3} +16 z^3 a^{-5} +5 z^3 a^{-7} -2 z^3 a^{-9} +z^3 a^{-11} -10 z^2 a^{-2} -23 z^2 a^{-4} -8 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} -2 z a^{-1} -5 z a^{-3} -6 z a^{-5} -z a^{-7} +z a^{-9} -z a^{-11} +2 a^{-2} +7 a^{-4} +4 a^{-6} }[/math]

Vassiliev invariants

V2 and V3:

(5, 9)

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 [math]\displaystyle{ 20 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1126}{3} }[/math] [math]\displaystyle{ \frac{98}{3} }[/math] [math]\displaystyle{ 1440 }[/math] [math]\displaystyle{ 2192 }[/math] [math]\displaystyle{ 352 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 2592 }[/math] [math]\displaystyle{ \frac{22520}{3} }[/math] [math]\displaystyle{ \frac{1960}{3} }[/math] [math]\displaystyle{ \frac{79231}{6} }[/math] [math]\displaystyle{ \frac{3514}{3} }[/math] [math]\displaystyle{ \frac{32246}{9} }[/math] [math]\displaystyle{ \frac{2149}{18} }[/math] [math]\displaystyle{ \frac{1855}{6} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]4 is the signature of 10 62. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                     1 -1 17                   1   1 15                 3 1   -2 13               3 1     2 11             4 3       -1 9           3 3         0 7         3 4           1 5       3 3             0 3     1 4               3 1   1 2                 -1 -1   1                   1 -3 1                     -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 62]]
Out[2]=	10
In[3]:=	PD[Knot[10, 62]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 19, 12, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8], X[13, 1, 14, 20], X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[10, 62]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8]
In[5]:=	BR[Knot[10, 62]]
Out[5]=	BR[3, {1, 1, 1, 1, -2, 1, 1, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 62]][t]
Out[6]=	-4 3 6 8 2 3 4 9 + t - -- + -- - - - 8 t + 6 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 62]][z]
Out[7]=	2 4 6 8 1 + 5 z + 8 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 62], Knot[11, NonAlternating, 76], Knot[11, NonAlternating, 78]}
In[9]:=	{KnotDet[Knot[10, 62]], KnotSignature[Knot[10, 62]]}
Out[9]=	{45, 4}
In[10]:=	J=Jones[Knot[10, 62]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 2 - - - 3 q + 6 q - 6 q + 7 q - 7 q + 6 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 62]}
In[12]:=	A2Invariant[Knot[10, 62]][q]
Out[12]=	-2 2 4 6 8 10 12 14 22 26 -q - q + q + 2 q + q + 3 q - q + 2 q - 2 q - q
In[13]:=	Kauffman[Knot[10, 62]][a, z]
Out[13]=	2 2 2 4 7 2 z z z 6 z 5 z 2 z z 4 z 8 z -- + -- + -- - --- + -- - -- - --- - --- - --- - --- + ---- - ---- - 6 4 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 23 z 10 z z 2 z 5 z 16 z 15 z 7 z 2 z ----- - ----- + --- - ---- + ---- + ----- + ----- + ---- + ---- - 4 2 11 9 7 5 3 a 10 a a a a a a a a 4 4 4 4 5 5 5 5 5 6 z 6 z 30 z 16 z 3 z 8 z 15 z 9 z 5 z ---- + ---- + ----- + ----- + ---- - ---- - ----- - ---- - ---- + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 4 z 7 z 21 z 10 z 4 z 2 z z z 3 z 5 z ---- - ---- - ----- - ----- + ---- + ---- - -- + -- + ---- + ---- + 8 6 4 2 7 5 3 a 6 4 a a a a a a a a a 8 9 9 2 z z z ---- + -- + -- 2 5 3 a a a
In[14]:=	{Vassiliev[2][Knot[10, 62]], Vassiliev[3][Knot[10, 62]]}
Out[14]=	{0, 9}
In[15]:=	Kh[Knot[10, 62]][q, t]
Out[15]=	3 3 5 1 1 q 2 q q 5 7 4 q + 3 q + ----- + ---- + -- + --- + -- + 3 q t + 3 q t + 3 3 2 2 t t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 4 q t + 3 q t + 3 q t + 4 q t + 3 q t + 3 q t + 13 5 15 5 15 6 17 6 19 7 q t + 3 q t + q t + q t + q t