10 87

From Knot Atlas
Revision as of 18:50, 23 August 2005 by ScottKnotPageRobot (talk | contribs) (Resetting knot page to basic template.)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -2 z^6-3 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 81, 0 }
Jones polynomial [math]\displaystyle{ q^6-4 q^5+7 q^4-10 q^3+13 q^2-13 q+13-10 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2+z^2 a^{-2} +z^2 a^{-4} -4 z^2+a^2+3 a^{-2} - a^{-4} -2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +10 z^8 a^{-2} +5 z^8 a^{-4} +5 z^8+6 a z^7+7 z^7 a^{-1} +5 z^7 a^{-3} +4 z^7 a^{-5} +5 a^2 z^6-21 z^6 a^{-2} -12 z^6 a^{-4} +z^6 a^{-6} -3 z^6+3 a^3 z^5-6 a z^5-21 z^5 a^{-1} -23 z^5 a^{-3} -11 z^5 a^{-5} +a^4 z^4-5 a^2 z^4+8 z^4 a^{-2} +5 z^4 a^{-4} -2 z^4 a^{-6} -5 z^4-3 a^3 z^3+2 a z^3+13 z^3 a^{-1} +15 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} +z^2 a^{-4} +z^2 a^{-6} +7 z^2+a^3 z+a z-z a^{-1} -z a^{-3} -a^2-3 a^{-2} - a^{-4} -2 }[/math]
The A2 invariant [math]\displaystyle{ q^{12}-q^{10}+q^8+q^6-3 q^4+2 q^2-2+ q^{-2} +2 q^{-4} +4 q^{-8} -2 q^{-10} -2 q^{-16} + q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-4 q^{56}-2 q^{54}+11 q^{52}-19 q^{50}+28 q^{48}-31 q^{46}+25 q^{44}-8 q^{42}-16 q^{40}+46 q^{38}-69 q^{36}+84 q^{34}-82 q^{32}+51 q^{30}+2 q^{28}-68 q^{26}+135 q^{24}-165 q^{22}+149 q^{20}-83 q^{18}-21 q^{16}+122 q^{14}-185 q^{12}+174 q^{10}-92 q^8-27 q^6+124 q^4-159 q^2+104+16 q^{-2} -140 q^{-4} +207 q^{-6} -188 q^{-8} +76 q^{-10} +86 q^{-12} -229 q^{-14} +300 q^{-16} -263 q^{-18} +141 q^{-20} +32 q^{-22} -184 q^{-24} +272 q^{-26} -261 q^{-28} +172 q^{-30} -32 q^{-32} -105 q^{-34} +188 q^{-36} -179 q^{-38} +96 q^{-40} +31 q^{-42} -140 q^{-44} +177 q^{-46} -128 q^{-48} +5 q^{-50} +128 q^{-52} -219 q^{-54} +225 q^{-56} -142 q^{-58} +4 q^{-60} +126 q^{-62} -201 q^{-64} +202 q^{-66} -135 q^{-68} +37 q^{-70} +46 q^{-72} -96 q^{-74} +99 q^{-76} -70 q^{-78} +35 q^{-80} -18 q^{-84} +20 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_87.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^9-2 q^7+3 q^5-4 q^3+3 q+3 q^{-5} -3 q^{-7} +3 q^{-9} -3 q^{-11} + q^{-13} }[/math]
2 [math]\displaystyle{ q^{26}-2 q^{24}+5 q^{20}-8 q^{18}+2 q^{16}+14 q^{14}-21 q^{12}+29 q^8-25 q^6-12 q^4+31 q^2-7-18 q^{-2} +12 q^{-4} +14 q^{-6} -14 q^{-8} -11 q^{-10} +25 q^{-12} -2 q^{-14} -28 q^{-16} +23 q^{-18} +12 q^{-20} -32 q^{-22} +9 q^{-24} +20 q^{-26} -18 q^{-28} -4 q^{-30} +12 q^{-32} -2 q^{-34} -3 q^{-36} + q^{-38} }[/math]
3 [math]\displaystyle{ q^{51}-2 q^{49}+2 q^{45}+q^{43}-4 q^{41}+8 q^{37}-8 q^{35}-13 q^{33}+20 q^{31}+30 q^{29}-37 q^{27}-63 q^{25}+52 q^{23}+112 q^{21}-50 q^{19}-165 q^{17}+17 q^{15}+209 q^{13}+35 q^{11}-221 q^9-93 q^7+184 q^5+150 q^3-125 q-170 q^{-1} +48 q^{-3} +172 q^{-5} +27 q^{-7} -150 q^{-9} -90 q^{-11} +123 q^{-13} +138 q^{-15} -93 q^{-17} -173 q^{-19} +56 q^{-21} +202 q^{-23} -14 q^{-25} -215 q^{-27} -41 q^{-29} +215 q^{-31} +94 q^{-33} -181 q^{-35} -151 q^{-37} +127 q^{-39} +178 q^{-41} -57 q^{-43} -172 q^{-45} -11 q^{-47} +137 q^{-49} +57 q^{-51} -85 q^{-53} -68 q^{-55} +34 q^{-57} +55 q^{-59} -34 q^{-63} -8 q^{-65} +11 q^{-67} +7 q^{-69} -2 q^{-71} -3 q^{-73} + q^{-75} }[/math]
4 [math]\displaystyle{ q^{84}-2 q^{82}+2 q^{78}-2 q^{76}+5 q^{74}-6 q^{72}+3 q^{68}-13 q^{66}+19 q^{64}+4 q^{62}+12 q^{60}-15 q^{58}-83 q^{56}+22 q^{54}+88 q^{52}+132 q^{50}-34 q^{48}-331 q^{46}-150 q^{44}+224 q^{42}+560 q^{40}+211 q^{38}-706 q^{36}-781 q^{34}+17 q^{32}+1165 q^{30}+1047 q^{28}-620 q^{26}-1580 q^{24}-907 q^{22}+1141 q^{20}+1985 q^{18}+322 q^{16}-1569 q^{14}-1876 q^{12}+142 q^{10}+1940 q^8+1336 q^6-512 q^4-1853 q^2-942+881 q^{-2} +1504 q^{-4} +615 q^{-6} -987 q^{-8} -1323 q^{-10} -232 q^{-12} +1059 q^{-14} +1177 q^{-16} -147 q^{-18} -1272 q^{-20} -906 q^{-22} +660 q^{-24} +1430 q^{-26} +427 q^{-28} -1216 q^{-30} -1424 q^{-32} +304 q^{-34} +1636 q^{-36} +1063 q^{-38} -965 q^{-40} -1898 q^{-42} -392 q^{-44} +1450 q^{-46} +1759 q^{-48} -130 q^{-50} -1855 q^{-52} -1277 q^{-54} +510 q^{-56} +1858 q^{-58} +947 q^{-60} -910 q^{-62} -1516 q^{-64} -680 q^{-66} +983 q^{-68} +1295 q^{-70} +281 q^{-72} -763 q^{-74} -1064 q^{-76} -96 q^{-78} +660 q^{-80} +664 q^{-82} +112 q^{-84} -544 q^{-86} -408 q^{-88} -24 q^{-90} +293 q^{-92} +297 q^{-94} -35 q^{-96} -153 q^{-98} -144 q^{-100} -5 q^{-102} +102 q^{-104} +45 q^{-106} +6 q^{-108} -35 q^{-110} -24 q^{-112} +7 q^{-114} +6 q^{-116} +7 q^{-118} -2 q^{-120} -3 q^{-122} + q^{-124} }[/math]
5 [math]\displaystyle{ q^{125}-2 q^{123}+2 q^{119}-2 q^{117}+2 q^{115}+3 q^{113}-6 q^{111}-5 q^{109}+4 q^{107}+q^{105}+11 q^{103}+18 q^{101}-11 q^{99}-41 q^{97}-41 q^{95}+6 q^{93}+84 q^{91}+130 q^{89}+42 q^{87}-186 q^{85}-334 q^{83}-161 q^{81}+310 q^{79}+708 q^{77}+531 q^{75}-384 q^{73}-1380 q^{71}-1309 q^{69}+243 q^{67}+2250 q^{65}+2730 q^{63}+523 q^{61}-3118 q^{59}-4910 q^{57}-2293 q^{55}+3477 q^{53}+7555 q^{51}+5356 q^{49}-2612 q^{47}-10011 q^{45}-9571 q^{43}+6 q^{41}+11339 q^{39}+14087 q^{37}+4365 q^{35}-10550 q^{33}-17711 q^{31}-9846 q^{29}+7368 q^{27}+19232 q^{25}+15034 q^{23}-2228 q^{21}-17901 q^{19}-18557 q^{17}-3730 q^{15}+13964 q^{13}+19550 q^{11}+8890 q^9-8366 q^7-17767 q^5-12339 q^3+2466 q+14108 q^{-1} +13564 q^{-3} +2461 q^{-5} -9526 q^{-7} -12990 q^{-9} -5920 q^{-11} +5290 q^{-13} +11443 q^{-15} +7873 q^{-17} -2014 q^{-19} -9787 q^{-21} -8885 q^{-23} -157 q^{-25} +8681 q^{-27} +9624 q^{-29} +1544 q^{-31} -8304 q^{-33} -10621 q^{-35} -2745 q^{-37} +8381 q^{-39} +12159 q^{-41} +4325 q^{-43} -8384 q^{-45} -14072 q^{-47} -6688 q^{-49} +7672 q^{-51} +15854 q^{-53} +9828 q^{-55} -5647 q^{-57} -16837 q^{-59} -13368 q^{-61} +2183 q^{-63} +16255 q^{-65} +16452 q^{-67} +2547 q^{-69} -13634 q^{-71} -18265 q^{-73} -7599 q^{-75} +9072 q^{-77} +17831 q^{-79} +11923 q^{-81} -3195 q^{-83} -15007 q^{-85} -14338 q^{-87} -2660 q^{-89} +10124 q^{-91} +14150 q^{-93} +7201 q^{-95} -4357 q^{-97} -11487 q^{-99} -9419 q^{-101} -818 q^{-103} +7240 q^{-105} +9053 q^{-107} +4251 q^{-109} -2750 q^{-111} -6767 q^{-113} -5465 q^{-115} -676 q^{-117} +3703 q^{-119} +4749 q^{-121} +2441 q^{-123} -981 q^{-125} -3064 q^{-127} -2673 q^{-129} -629 q^{-131} +1324 q^{-133} +1921 q^{-135} +1164 q^{-137} -116 q^{-139} -1001 q^{-141} -981 q^{-143} -347 q^{-145} +291 q^{-147} +535 q^{-149} +389 q^{-151} +47 q^{-153} -216 q^{-155} -226 q^{-157} -100 q^{-159} +31 q^{-161} +86 q^{-163} +73 q^{-165} +15 q^{-167} -29 q^{-169} -25 q^{-171} -9 q^{-173} +2 q^{-175} +6 q^{-177} +7 q^{-179} -2 q^{-181} -3 q^{-183} + q^{-185} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{12}-q^{10}+q^8+q^6-3 q^4+2 q^2-2+ q^{-2} +2 q^{-4} +4 q^{-8} -2 q^{-10} -2 q^{-16} + q^{-18} }[/math]
1,1 [math]\displaystyle{ q^{36}-4 q^{34}+10 q^{32}-20 q^{30}+38 q^{28}-64 q^{26}+100 q^{24}-148 q^{22}+207 q^{20}-278 q^{18}+362 q^{16}-464 q^{14}+572 q^{12}-668 q^{10}+742 q^8-762 q^6+697 q^4-520 q^2+238+136 q^{-2} -566 q^{-4} +996 q^{-6} -1372 q^{-8} +1646 q^{-10} -1774 q^{-12} +1752 q^{-14} -1566 q^{-16} +1260 q^{-18} -851 q^{-20} +388 q^{-22} +70 q^{-24} -474 q^{-26} +771 q^{-28} -952 q^{-30} +994 q^{-32} -922 q^{-34} +773 q^{-36} -586 q^{-38} +402 q^{-40} -244 q^{-42} +133 q^{-44} -62 q^{-46} +22 q^{-48} -6 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{32}-q^{30}-q^{28}+3 q^{26}-5 q^{22}+2 q^{20}+7 q^{18}-3 q^{16}-11 q^{14}+5 q^{12}+14 q^{10}-12 q^8-9 q^6+12 q^4+7 q^2-8-4 q^{-2} +11 q^{-4} -3 q^{-6} -6 q^{-8} +6 q^{-10} +3 q^{-12} -8 q^{-14} +7 q^{-16} +12 q^{-18} -8 q^{-20} -7 q^{-22} +6 q^{-24} +5 q^{-26} -12 q^{-28} -8 q^{-30} +11 q^{-32} +4 q^{-34} -7 q^{-36} -2 q^{-38} +5 q^{-40} +4 q^{-42} -3 q^{-44} -2 q^{-46} + q^{-48} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{28}-2 q^{26}+5 q^{22}-7 q^{20}-3 q^{18}+15 q^{16}-9 q^{14}-10 q^{12}+25 q^{10}-8 q^8-17 q^6+23 q^4-7 q^2-16+10 q^{-2} +2 q^{-4} -4 q^{-6} -3 q^{-8} +13 q^{-10} +10 q^{-12} -17 q^{-14} +8 q^{-16} +14 q^{-18} -24 q^{-20} +2 q^{-22} +16 q^{-24} -19 q^{-26} +5 q^{-28} +10 q^{-30} -11 q^{-32} +4 q^{-34} +2 q^{-36} -3 q^{-38} + q^{-40} }[/math]
1,0,0 [math]\displaystyle{ q^{15}-q^{13}+2 q^{11}-q^9+2 q^7-3 q^5+2 q^3-3 q+ q^{-5} +3 q^{-7} + q^{-9} +4 q^{-11} -2 q^{-13} + q^{-15} -3 q^{-17} + q^{-19} -2 q^{-21} + q^{-23} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{34}-q^{32}-q^{30}+3 q^{28}-5 q^{24}-q^{22}+7 q^{20}+q^{18}-11 q^{16}+2 q^{14}+18 q^{12}-4 q^{10}-14 q^8+16 q^6+12 q^4-17 q^2-8+12 q^{-2} -7 q^{-4} -21 q^{-6} +8 q^{-8} +15 q^{-10} -14 q^{-12} +4 q^{-14} +31 q^{-16} -3 q^{-18} -16 q^{-20} +17 q^{-22} +9 q^{-24} -21 q^{-26} -10 q^{-28} +13 q^{-30} -14 q^{-34} +4 q^{-36} +10 q^{-38} -5 q^{-40} -3 q^{-42} +5 q^{-44} - q^{-46} -2 q^{-48} + q^{-50} }[/math]
1,0,0,0 [math]\displaystyle{ q^{18}-q^{16}+2 q^{14}+2 q^8-3 q^6+2 q^4-3 q^2-1- q^{-2} + q^{-6} +2 q^{-8} +4 q^{-10} + q^{-12} +4 q^{-14} -2 q^{-16} + q^{-18} -2 q^{-20} -2 q^{-22} + q^{-24} -2 q^{-26} + q^{-28} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-7 q^{22}+11 q^{20}-15 q^{18}+21 q^{16}-25 q^{14}+28 q^{12}-29 q^{10}+24 q^8-17 q^6+5 q^4+9 q^2-24+38 q^{-2} -48 q^{-4} +56 q^{-6} -57 q^{-8} +55 q^{-10} -44 q^{-12} +33 q^{-14} -16 q^{-16} +2 q^{-18} +12 q^{-20} -22 q^{-22} +28 q^{-24} -31 q^{-26} +29 q^{-28} -26 q^{-30} +19 q^{-32} -14 q^{-34} +8 q^{-36} -3 q^{-38} + q^{-40} }[/math]
1,0 [math]\displaystyle{ q^{46}-2 q^{42}-2 q^{40}+2 q^{38}+6 q^{36}+q^{34}-9 q^{32}-9 q^{30}+5 q^{28}+18 q^{26}+6 q^{24}-18 q^{22}-19 q^{20}+9 q^{18}+30 q^{16}+6 q^{14}-27 q^{12}-19 q^{10}+18 q^8+26 q^6-9 q^4-28 q^2-2+24 q^{-2} +7 q^{-4} -20 q^{-6} -12 q^{-8} +16 q^{-10} +16 q^{-12} -10 q^{-14} -15 q^{-16} +12 q^{-18} +21 q^{-20} -4 q^{-22} -25 q^{-24} -3 q^{-26} +27 q^{-28} +16 q^{-30} -24 q^{-32} -30 q^{-34} +9 q^{-36} +32 q^{-38} +6 q^{-40} -27 q^{-42} -18 q^{-44} +16 q^{-46} +21 q^{-48} -4 q^{-50} -15 q^{-52} -4 q^{-54} +9 q^{-56} +5 q^{-58} -3 q^{-60} -3 q^{-62} + q^{-66} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{38}-2 q^{36}+2 q^{34}-3 q^{32}+6 q^{30}-9 q^{28}+8 q^{26}-11 q^{24}+18 q^{22}-18 q^{20}+18 q^{18}-20 q^{16}+27 q^{14}-19 q^{12}+16 q^{10}-15 q^8+8 q^6+q^4-11 q^2+13-29 q^{-2} +33 q^{-4} -37 q^{-6} +42 q^{-8} -44 q^{-10} +48 q^{-12} -36 q^{-14} +41 q^{-16} -29 q^{-18} +24 q^{-20} -13 q^{-22} +6 q^{-24} - q^{-26} -12 q^{-28} +16 q^{-30} -21 q^{-32} +21 q^{-34} -25 q^{-36} +26 q^{-38} -22 q^{-40} +19 q^{-42} -16 q^{-44} +12 q^{-46} -8 q^{-48} +5 q^{-50} -3 q^{-52} + q^{-54} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-4 q^{56}-2 q^{54}+11 q^{52}-19 q^{50}+28 q^{48}-31 q^{46}+25 q^{44}-8 q^{42}-16 q^{40}+46 q^{38}-69 q^{36}+84 q^{34}-82 q^{32}+51 q^{30}+2 q^{28}-68 q^{26}+135 q^{24}-165 q^{22}+149 q^{20}-83 q^{18}-21 q^{16}+122 q^{14}-185 q^{12}+174 q^{10}-92 q^8-27 q^6+124 q^4-159 q^2+104+16 q^{-2} -140 q^{-4} +207 q^{-6} -188 q^{-8} +76 q^{-10} +86 q^{-12} -229 q^{-14} +300 q^{-16} -263 q^{-18} +141 q^{-20} +32 q^{-22} -184 q^{-24} +272 q^{-26} -261 q^{-28} +172 q^{-30} -32 q^{-32} -105 q^{-34} +188 q^{-36} -179 q^{-38} +96 q^{-40} +31 q^{-42} -140 q^{-44} +177 q^{-46} -128 q^{-48} +5 q^{-50} +128 q^{-52} -219 q^{-54} +225 q^{-56} -142 q^{-58} +4 q^{-60} +126 q^{-62} -201 q^{-64} +202 q^{-66} -135 q^{-68} +37 q^{-70} +46 q^{-72} -96 q^{-74} +99 q^{-76} -70 q^{-78} +35 q^{-80} -18 q^{-84} +20 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94} }[/math]

.

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 87"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -2 z^6-3 z^4+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]= { 81, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^6-4 q^5+7 q^4-10 q^3+13 q^2-13 q+13-10 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2+z^2 a^{-2} +z^2 a^{-4} -4 z^2+a^2+3 a^{-2} - a^{-4} -2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +10 z^8 a^{-2} +5 z^8 a^{-4} +5 z^8+6 a z^7+7 z^7 a^{-1} +5 z^7 a^{-3} +4 z^7 a^{-5} +5 a^2 z^6-21 z^6 a^{-2} -12 z^6 a^{-4} +z^6 a^{-6} -3 z^6+3 a^3 z^5-6 a z^5-21 z^5 a^{-1} -23 z^5 a^{-3} -11 z^5 a^{-5} +a^4 z^4-5 a^2 z^4+8 z^4 a^{-2} +5 z^4 a^{-4} -2 z^4 a^{-6} -5 z^4-3 a^3 z^3+2 a z^3+13 z^3 a^{-1} +15 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+3 a^2 z^2+3 z^2 a^{-2} +z^2 a^{-4} +z^2 a^{-6} +7 z^2+a^3 z+a z-z a^{-1} -z a^{-3} -a^2-3 a^{-2} - a^{-4} -2 }[/math]
Retrieved from "https://katlas.org/index.php?title=10_87&oldid=4439"