10 86

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+9 t^2-19 t+25-19 t^{-1} +9 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-3 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 85, 0 }
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+6 q^4-10 q^3+13 q^2-14 q+14-11 q^{-1} +8 q^{-2} -4 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -2 z^4+a^2 z^2-4 z^2 a^{-2} +2 z^2 a^{-4} -2 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} +4 z^8 a^{-4} +6 z^8+9 a z^7+9 z^7 a^{-1} +3 z^7 a^{-3} +3 z^7 a^{-5} +8 a^2 z^6-21 z^6 a^{-2} -10 z^6 a^{-4} +z^6 a^{-6} -2 z^6+4 a^3 z^5-11 a z^5-23 z^5 a^{-1} -17 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-9 a^2 z^4+13 z^4 a^{-2} +7 z^4 a^{-4} -3 z^4 a^{-6} -7 z^4-2 a^3 z^3+4 a z^3+13 z^3 a^{-1} +15 z^3 a^{-3} +8 z^3 a^{-5} +3 a^2 z^2-6 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2 a^{-6} +z^2-a z-3 z a^{-1} -4 z a^{-3} -2 z a^{-5} +2 a^{-2} + a^{-4} +2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}+q^8+q^6-2 q^4+4 q^2-1+2 q^{-2} -2 q^{-6} +2 q^{-8} -3 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+10 q^{58}-8 q^{56}+q^{54}+17 q^{52}-36 q^{50}+57 q^{48}-65 q^{46}+49 q^{44}-16 q^{42}-40 q^{40}+104 q^{38}-150 q^{36}+163 q^{34}-127 q^{32}+39 q^{30}+74 q^{28}-178 q^{26}+236 q^{24}-219 q^{22}+126 q^{20}+8 q^{18}-137 q^{16}+206 q^{14}-179 q^{12}+77 q^{10}+65 q^8-170 q^6+187 q^4-97 q^2-59+222 q^{-2} -307 q^{-4} +278 q^{-6} -132 q^{-8} -74 q^{-10} +266 q^{-12} -372 q^{-14} +354 q^{-16} -222 q^{-18} +24 q^{-20} +161 q^{-22} -278 q^{-24} +285 q^{-26} -190 q^{-28} +35 q^{-30} +109 q^{-32} -192 q^{-34} +172 q^{-36} -65 q^{-38} -79 q^{-40} +196 q^{-42} -230 q^{-44} +158 q^{-46} -13 q^{-48} -147 q^{-50} +258 q^{-52} -268 q^{-54} +189 q^{-56} -52 q^{-58} -88 q^{-60} +178 q^{-62} -195 q^{-64} +151 q^{-66} -70 q^{-68} -8 q^{-70} +58 q^{-72} -75 q^{-74} +64 q^{-76} -38 q^{-78} +15 q^{-80} +3 q^{-82} -11 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_86.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-3 q^7+4 q^5-3 q^3+3 q- q^{-3} +3 q^{-5} -4 q^{-7} +3 q^{-9} -2 q^{-11} + q^{-13} }[/math]
2	[math]\displaystyle{ q^{26}-3 q^{24}+q^{22}+9 q^{20}-14 q^{18}-2 q^{16}+26 q^{14}-24 q^{12}-12 q^{10}+39 q^8-17 q^6-23 q^4+29 q^2+3-21 q^{-2} +2 q^{-4} +21 q^{-6} -7 q^{-8} -24 q^{-10} +26 q^{-12} +11 q^{-14} -38 q^{-16} +16 q^{-18} +24 q^{-20} -30 q^{-22} + q^{-24} +21 q^{-26} -12 q^{-28} -6 q^{-30} +8 q^{-32} - q^{-34} -2 q^{-36} + q^{-38} }[/math]
3	[math]\displaystyle{ q^{51}-3 q^{49}+q^{47}+6 q^{45}-2 q^{43}-13 q^{41}-q^{39}+32 q^{37}-q^{35}-54 q^{33}-3 q^{31}+87 q^{29}+23 q^{27}-135 q^{25}-55 q^{23}+172 q^{21}+109 q^{19}-190 q^{17}-170 q^{15}+171 q^{13}+225 q^{11}-122 q^9-246 q^7+48 q^5+237 q^3+31 q-195 q^{-1} -101 q^{-3} +135 q^{-5} +157 q^{-7} -71 q^{-9} -189 q^{-11} +3 q^{-13} +214 q^{-15} +58 q^{-17} -219 q^{-19} -118 q^{-21} +207 q^{-23} +172 q^{-25} -172 q^{-27} -221 q^{-29} +120 q^{-31} +243 q^{-33} -49 q^{-35} -238 q^{-37} -22 q^{-39} +205 q^{-41} +74 q^{-43} -143 q^{-45} -102 q^{-47} +78 q^{-49} +96 q^{-51} -26 q^{-53} -69 q^{-55} -4 q^{-57} +39 q^{-59} +13 q^{-61} -17 q^{-63} -9 q^{-65} +5 q^{-67} +4 q^{-69} - q^{-71} -2 q^{-73} + q^{-75} }[/math]
4	[math]\displaystyle{ q^{84}-3 q^{82}+q^{80}+6 q^{78}-5 q^{76}-q^{74}-12 q^{72}+11 q^{70}+30 q^{68}-23 q^{66}-16 q^{64}-40 q^{62}+50 q^{60}+109 q^{58}-74 q^{56}-119 q^{54}-128 q^{52}+196 q^{50}+387 q^{48}-90 q^{46}-451 q^{44}-530 q^{42}+328 q^{40}+1057 q^{38}+332 q^{36}-802 q^{34}-1451 q^{32}-100 q^{30}+1682 q^{28}+1379 q^{26}-426 q^{24}-2225 q^{22}-1219 q^{20}+1349 q^{18}+2197 q^{16}+731 q^{14}-1863 q^{12}-2048 q^{10}+93 q^8+1859 q^6+1649 q^4-567 q^2-1820-1039 q^{-2} +727 q^{-4} +1697 q^{-6} +650 q^{-8} -974 q^{-10} -1531 q^{-12} -298 q^{-14} +1299 q^{-16} +1376 q^{-18} -222 q^{-20} -1686 q^{-22} -987 q^{-24} +895 q^{-26} +1871 q^{-28} +425 q^{-30} -1709 q^{-32} -1618 q^{-34} +319 q^{-36} +2168 q^{-38} +1243 q^{-40} -1262 q^{-42} -2098 q^{-44} -697 q^{-46} +1807 q^{-48} +1980 q^{-50} -132 q^{-52} -1836 q^{-54} -1671 q^{-56} +615 q^{-58} +1866 q^{-60} +1011 q^{-62} -689 q^{-64} -1701 q^{-66} -570 q^{-68} +822 q^{-70} +1180 q^{-72} +392 q^{-74} -805 q^{-76} -790 q^{-78} -128 q^{-80} +522 q^{-82} +569 q^{-84} -38 q^{-86} -324 q^{-88} -294 q^{-90} +10 q^{-92} +230 q^{-94} +108 q^{-96} -9 q^{-98} -104 q^{-100} -59 q^{-102} +31 q^{-104} +28 q^{-106} +22 q^{-108} -11 q^{-110} -15 q^{-112} +2 q^{-114} + q^{-116} +4 q^{-118} - q^{-120} -2 q^{-122} + q^{-124} }[/math]
5	[math]\displaystyle{ q^{125}-3 q^{123}+q^{121}+6 q^{119}-5 q^{117}-4 q^{115}+9 q^{109}+14 q^{107}-10 q^{105}-33 q^{103}-6 q^{101}+34 q^{99}+47 q^{97}-6 q^{95}-81 q^{93}-111 q^{91}+15 q^{89}+268 q^{87}+288 q^{85}-72 q^{83}-580 q^{81}-712 q^{79}-52 q^{77}+1112 q^{75}+1650 q^{73}+526 q^{71}-1751 q^{69}-3171 q^{67}-1795 q^{65}+2062 q^{63}+5301 q^{61}+4267 q^{59}-1540 q^{57}-7597 q^{55}-7883 q^{53}-568 q^{51}+9155 q^{49}+12358 q^{47}+4502 q^{45}-9089 q^{43}-16513 q^{41}-9915 q^{39}+6591 q^{37}+19104 q^{35}+15772 q^{33}-1810 q^{31}-19011 q^{29}-20551 q^{27}-4382 q^{25}+15920 q^{23}+22884 q^{21}+10544 q^{19}-10418 q^{17}-22187 q^{15}-15204 q^{13}+3871 q^{11}+18666 q^9+17460 q^7+2381 q^5-13423 q^3-17290 q-7168 q^{-1} +7751 q^{-3} +15294 q^{-5} +10139 q^{-7} -2690 q^{-9} -12540 q^{-11} -11599 q^{-13} -1091 q^{-15} +9934 q^{-17} +12136 q^{-19} +3701 q^{-21} -8012 q^{-23} -12584 q^{-25} -5544 q^{-27} +6935 q^{-29} +13321 q^{-31} +7258 q^{-33} -6222 q^{-35} -14583 q^{-37} -9386 q^{-39} +5327 q^{-41} +16022 q^{-43} +12178 q^{-45} -3551 q^{-47} -17039 q^{-49} -15465 q^{-51} +462 q^{-53} +16824 q^{-55} +18717 q^{-57} +3913 q^{-59} -14777 q^{-61} -20928 q^{-63} -9066 q^{-65} +10574 q^{-67} +21240 q^{-69} +13995 q^{-71} -4718 q^{-73} -19008 q^{-75} -17365 q^{-77} -1874 q^{-79} +14294 q^{-81} +18273 q^{-83} +7721 q^{-85} -8016 q^{-87} -16293 q^{-89} -11451 q^{-91} +1500 q^{-93} +11978 q^{-95} +12437 q^{-97} +3674 q^{-99} -6636 q^{-101} -10727 q^{-103} -6539 q^{-105} +1662 q^{-107} +7387 q^{-109} +6961 q^{-111} +1732 q^{-113} -3723 q^{-115} -5546 q^{-117} -3197 q^{-119} +819 q^{-121} +3395 q^{-123} +3099 q^{-125} +784 q^{-127} -1467 q^{-129} -2150 q^{-131} -1229 q^{-133} +225 q^{-135} +1113 q^{-137} +1012 q^{-139} +274 q^{-141} -394 q^{-143} -576 q^{-145} -322 q^{-147} +35 q^{-149} +243 q^{-151} +216 q^{-153} +51 q^{-155} -74 q^{-157} -91 q^{-159} -44 q^{-161} +4 q^{-163} +35 q^{-165} +25 q^{-167} -3 q^{-169} -9 q^{-171} -4 q^{-173} -2 q^{-175} + q^{-177} +4 q^{-179} - q^{-181} -2 q^{-183} + q^{-185} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-2 q^{10}+q^8+q^6-2 q^4+4 q^2-1+2 q^{-2} -2 q^{-6} +2 q^{-8} -3 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }[/math]
1,1	[math]\displaystyle{ q^{36}-6 q^{34}+18 q^{32}-38 q^{30}+75 q^{28}-142 q^{26}+236 q^{24}-352 q^{22}+504 q^{20}-680 q^{18}+848 q^{16}-992 q^{14}+1085 q^{12}-1088 q^{10}+968 q^8-708 q^6+319 q^4+188 q^2-754+1320 q^{-2} -1804 q^{-4} +2160 q^{-6} -2332 q^{-8} +2294 q^{-10} -2065 q^{-12} +1666 q^{-14} -1152 q^{-16} +570 q^{-18} +3 q^{-20} -498 q^{-22} +892 q^{-24} -1134 q^{-26} +1220 q^{-28} -1168 q^{-30} +1020 q^{-32} -818 q^{-34} +595 q^{-36} -400 q^{-38} +246 q^{-40} -136 q^{-42} +68 q^{-44} -30 q^{-46} +12 q^{-48} -4 q^{-50} + q^{-52} }[/math]
2,0	[math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+6 q^{26}-q^{24}-8 q^{22}+q^{20}+12 q^{18}-2 q^{16}-17 q^{14}+6 q^{12}+18 q^{10}-9 q^8-13 q^6+13 q^4+7 q^2-8-3 q^{-2} +9 q^{-4} -4 q^{-6} -7 q^{-8} +11 q^{-10} -13 q^{-14} +6 q^{-16} +14 q^{-18} -10 q^{-20} -10 q^{-22} +10 q^{-24} +11 q^{-26} -9 q^{-28} -10 q^{-30} +9 q^{-32} +5 q^{-34} -6 q^{-36} -4 q^{-38} +2 q^{-40} +3 q^{-42} - q^{-44} - q^{-46} + q^{-48} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}-3 q^{26}+9 q^{22}-10 q^{20}-4 q^{18}+21 q^{16}-17 q^{14}-11 q^{12}+29 q^{10}-16 q^8-14 q^6+28 q^4-5 q^2-9+11 q^{-2} +6 q^{-4} -5 q^{-6} -14 q^{-8} +11 q^{-10} +6 q^{-12} -27 q^{-14} +13 q^{-16} +17 q^{-18} -26 q^{-20} +12 q^{-22} +15 q^{-24} -19 q^{-26} +7 q^{-28} +6 q^{-30} -9 q^{-32} +3 q^{-34} + q^{-36} -2 q^{-38} + q^{-40} }[/math]
1,0,0	[math]\displaystyle{ q^{15}-2 q^{13}+2 q^{11}-2 q^9+2 q^7-2 q^5+4 q^3+2 q^{-1} + q^{-3} - q^{-5} -3 q^{-9} +2 q^{-11} -3 q^{-13} +2 q^{-15} - q^{-17} +2 q^{-19} - q^{-21} + q^{-23} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{34}-2 q^{32}-2 q^{30}+6 q^{28}-8 q^{24}+3 q^{22}+12 q^{20}-5 q^{18}-16 q^{16}+8 q^{14}+15 q^{12}-16 q^{10}-13 q^8+25 q^6+7 q^4-17 q^2+11+21 q^{-2} -10 q^{-4} -11 q^{-6} +15 q^{-8} -2 q^{-10} -28 q^{-12} +3 q^{-14} +18 q^{-16} -20 q^{-18} -12 q^{-20} +26 q^{-22} +8 q^{-24} -17 q^{-26} +3 q^{-28} +17 q^{-30} -3 q^{-32} -12 q^{-34} +5 q^{-36} +6 q^{-38} -7 q^{-40} -2 q^{-42} +4 q^{-44} - q^{-46} - q^{-48} + q^{-50} }[/math]
1,0,0,0	[math]\displaystyle{ q^{18}-2 q^{16}+2 q^{14}-q^{12}-q^{10}+2 q^8-2 q^6+4 q^4+3+ q^{-2} + q^{-4} - q^{-6} - q^{-8} - q^{-10} -3 q^{-12} +2 q^{-14} -3 q^{-16} +2 q^{-18} +2 q^{-24} - q^{-26} + q^{-28} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{28}-3 q^{26}+6 q^{24}-11 q^{22}+18 q^{20}-24 q^{18}+29 q^{16}-33 q^{14}+33 q^{12}-29 q^{10}+20 q^8-6 q^6-8 q^4+27 q^2-41+55 q^{-2} -62 q^{-4} +65 q^{-6} -60 q^{-8} +49 q^{-10} -36 q^{-12} +19 q^{-14} -3 q^{-16} -13 q^{-18} +24 q^{-20} -30 q^{-22} +33 q^{-24} -31 q^{-26} +27 q^{-28} -22 q^{-30} +15 q^{-32} -9 q^{-34} +5 q^{-36} -2 q^{-38} + q^{-40} }[/math]
1,0	[math]\displaystyle{ q^{46}-3 q^{42}-3 q^{40}+3 q^{38}+10 q^{36}+3 q^{34}-14 q^{32}-14 q^{30}+9 q^{28}+25 q^{26}+5 q^{24}-28 q^{22}-22 q^{20}+18 q^{18}+33 q^{16}-2 q^{14}-34 q^{12}-12 q^{10}+27 q^8+23 q^6-17 q^4-23 q^2+10+26 q^{-2} - q^{-4} -23 q^{-6} -4 q^{-8} +21 q^{-10} +7 q^{-12} -21 q^{-14} -13 q^{-16} +19 q^{-18} +19 q^{-20} -17 q^{-22} -30 q^{-24} +6 q^{-26} +36 q^{-28} +10 q^{-30} -31 q^{-32} -25 q^{-34} +19 q^{-36} +33 q^{-38} - q^{-40} -27 q^{-42} -12 q^{-44} +16 q^{-46} +16 q^{-48} -5 q^{-50} -12 q^{-52} -2 q^{-54} +6 q^{-56} +3 q^{-58} -2 q^{-60} -2 q^{-62} + q^{-66} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{38}-3 q^{36}+3 q^{34}-4 q^{32}+10 q^{30}-14 q^{28}+14 q^{26}-17 q^{24}+25 q^{22}-26 q^{20}+22 q^{18}-26 q^{16}+26 q^{14}-18 q^{12}+10 q^{10}-7 q^8+19 q^4-20 q^2+32-37 q^{-2} +49 q^{-4} -46 q^{-6} +47 q^{-8} -53 q^{-10} +42 q^{-12} -39 q^{-14} +28 q^{-16} -27 q^{-18} +11 q^{-20} - q^{-22} -4 q^{-24} +13 q^{-26} -18 q^{-28} +27 q^{-30} -23 q^{-32} +26 q^{-34} -25 q^{-36} +23 q^{-38} -19 q^{-40} +15 q^{-42} -13 q^{-44} +8 q^{-46} -5 q^{-48} +3 q^{-50} -2 q^{-52} + q^{-54} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+10 q^{58}-8 q^{56}+q^{54}+17 q^{52}-36 q^{50}+57 q^{48}-65 q^{46}+49 q^{44}-16 q^{42}-40 q^{40}+104 q^{38}-150 q^{36}+163 q^{34}-127 q^{32}+39 q^{30}+74 q^{28}-178 q^{26}+236 q^{24}-219 q^{22}+126 q^{20}+8 q^{18}-137 q^{16}+206 q^{14}-179 q^{12}+77 q^{10}+65 q^8-170 q^6+187 q^4-97 q^2-59+222 q^{-2} -307 q^{-4} +278 q^{-6} -132 q^{-8} -74 q^{-10} +266 q^{-12} -372 q^{-14} +354 q^{-16} -222 q^{-18} +24 q^{-20} +161 q^{-22} -278 q^{-24} +285 q^{-26} -190 q^{-28} +35 q^{-30} +109 q^{-32} -192 q^{-34} +172 q^{-36} -65 q^{-38} -79 q^{-40} +196 q^{-42} -230 q^{-44} +158 q^{-46} -13 q^{-48} -147 q^{-50} +258 q^{-52} -268 q^{-54} +189 q^{-56} -52 q^{-58} -88 q^{-60} +178 q^{-62} -195 q^{-64} +151 q^{-66} -70 q^{-68} -8 q^{-70} +58 q^{-72} -75 q^{-74} +64 q^{-76} -38 q^{-78} +15 q^{-80} +3 q^{-82} -11 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }[/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 86"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^3+9 t^2-19 t+25-19 t^{-1} +9 t^{-2} -2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^6-3 z^4-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]=

{ 85, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-3 q^5+6 q^4-10 q^3+13 q^2-14 q+14-11 q^{-1} +8 q^{-2} -4 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^4-3 z^4 a^{-2} +z^4 a^{-4} -2 z^4+a^2 z^2-4 z^2 a^{-2} +2 z^2 a^{-4} -2 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} +4 z^8 a^{-4} +6 z^8+9 a z^7+9 z^7 a^{-1} +3 z^7 a^{-3} +3 z^7 a^{-5} +8 a^2 z^6-21 z^6 a^{-2} -10 z^6 a^{-4} +z^6 a^{-6} -2 z^6+4 a^3 z^5-11 a z^5-23 z^5 a^{-1} -17 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-9 a^2 z^4+13 z^4 a^{-2} +7 z^4 a^{-4} -3 z^4 a^{-6} -7 z^4-2 a^3 z^3+4 a z^3+13 z^3 a^{-1} +15 z^3 a^{-3} +8 z^3 a^{-5} +3 a^2 z^2-6 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2 a^{-6} +z^2-a z-3 z a^{-1} -4 z a^{-3} -2 z a^{-5} +2 a^{-2} + a^{-4} +2 }[/math]