9 32

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 t^3-6 t^2+14 t-17+14 t^{-1} -6 t^{-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 z^6-z^2+1}
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 59, 2 }
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^7-3 q^6+6 q^5-9 q^4+10 q^3-10 q^2+9 q-6+4 q^{-1} - q^{-2} }
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^{-2} +3 z^4 a^{-2} -2 z^4 a^{-4} -z^4+3 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -z^2+ a^{-2} -2 a^{-4} + a^{-6} +1}
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^8 a^{-2} +2 z^8 a^{-4} +5 z^7 a^{-1} +10 z^7 a^{-3} +5 z^7 a^{-5} +6 z^6 a^{-2} +7 z^6 a^{-4} +5 z^6 a^{-6} +4 z^6+a z^5-9 z^5 a^{-1} -18 z^5 a^{-3} -5 z^5 a^{-5} +3 z^5 a^{-7} -19 z^4 a^{-2} -18 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} -8 z^4-a z^3+3 z^3 a^{-1} +9 z^3 a^{-3} +2 z^3 a^{-5} -3 z^3 a^{-7} +10 z^2 a^{-2} +12 z^2 a^{-4} +4 z^2 a^{-6} -z^2 a^{-8} +3 z^2-z a^{-1} -2 z a^{-3} +z a^{-7} - a^{-2} -2 a^{-4} - a^{-6} +1}
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 -q^6+2 q^4+1+3 q^{-2} -2 q^{-4} +2 q^{-6} -2 q^{-8} -2 q^{-14} +2 q^{-16} - q^{-18} + q^{-22} }
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^{32}-3 q^{30}+7 q^{28}-13 q^{26}+13 q^{24}-9 q^{22}-6 q^{20}+30 q^{18}-50 q^{16}+66 q^{14}-56 q^{12}+17 q^{10}+39 q^8-93 q^6+126 q^4-112 q^2+58+22 q^{-2} -92 q^{-4} +126 q^{-6} -106 q^{-8} +48 q^{-10} +29 q^{-12} -83 q^{-14} +89 q^{-16} -47 q^{-18} -23 q^{-20} +92 q^{-22} -122 q^{-24} +101 q^{-26} -35 q^{-28} -53 q^{-30} +131 q^{-32} -173 q^{-34} +158 q^{-36} -91 q^{-38} -6 q^{-40} +98 q^{-42} -157 q^{-44} +157 q^{-46} -103 q^{-48} +19 q^{-50} +58 q^{-52} -102 q^{-54} +89 q^{-56} -33 q^{-58} -39 q^{-60} +90 q^{-62} -94 q^{-64} +49 q^{-66} +22 q^{-68} -90 q^{-70} +125 q^{-72} -111 q^{-74} +63 q^{-76} + q^{-78} -59 q^{-80} +88 q^{-82} -86 q^{-84} +64 q^{-86} -26 q^{-88} -5 q^{-90} +25 q^{-92} -33 q^{-94} +30 q^{-96} -21 q^{-98} +12 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }

Further Quantum Invariants

Further quantum knot invariants for 9_32.

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^5+3 q^3-2 q+3 q^{-1} - q^{-3} + q^{-7} -3 q^{-9} +3 q^{-11} -2 q^{-13} + q^{-15} }
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^{16}-3 q^{14}-2 q^{12}+11 q^{10}-4 q^8-14 q^6+18 q^4+5 q^2-23+13 q^{-2} +13 q^{-4} -19 q^{-6} + q^{-8} +13 q^{-10} -5 q^{-12} -10 q^{-14} +6 q^{-16} +13 q^{-18} -17 q^{-20} -5 q^{-22} +25 q^{-24} -13 q^{-26} -13 q^{-28} +19 q^{-30} -3 q^{-32} -9 q^{-34} +6 q^{-36} -2 q^{-40} + q^{-42} }
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^{33}+3 q^{31}+2 q^{29}-7 q^{27}-10 q^{25}+8 q^{23}+30 q^{21}-5 q^{19}-48 q^{17}-18 q^{15}+66 q^{13}+54 q^{11}-67 q^9-93 q^7+50 q^5+124 q^3-14 q-143 q^{-1} -23 q^{-3} +139 q^{-5} +56 q^{-7} -117 q^{-9} -80 q^{-11} +93 q^{-13} +90 q^{-15} -59 q^{-17} -93 q^{-19} +26 q^{-21} +87 q^{-23} +13 q^{-25} -83 q^{-27} -52 q^{-29} +69 q^{-31} +93 q^{-33} -48 q^{-35} -125 q^{-37} +16 q^{-39} +147 q^{-41} +21 q^{-43} -146 q^{-45} -52 q^{-47} +117 q^{-49} +75 q^{-51} -83 q^{-53} -76 q^{-55} +46 q^{-57} +60 q^{-59} -16 q^{-61} -39 q^{-63} +3 q^{-65} +21 q^{-67} - q^{-69} -8 q^{-71} +3 q^{-75} -2 q^{-79} + q^{-81} }
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^{56}-3 q^{54}-2 q^{52}+7 q^{50}+6 q^{48}+6 q^{46}-24 q^{44}-31 q^{42}+11 q^{40}+45 q^{38}+83 q^{36}-29 q^{34}-139 q^{32}-103 q^{30}+35 q^{28}+282 q^{26}+170 q^{24}-155 q^{22}-379 q^{20}-280 q^{18}+334 q^{16}+561 q^{14}+220 q^{12}-448 q^{10}-794 q^8-75 q^6+684 q^4+799 q^2-33-989 q^{-2} -661 q^{-4} +321 q^{-6} +1039 q^{-8} +514 q^{-10} -697 q^{-12} -919 q^{-14} -162 q^{-16} +832 q^{-18} +751 q^{-20} -263 q^{-22} -793 q^{-24} -418 q^{-26} +472 q^{-28} +692 q^{-30} +74 q^{-32} -546 q^{-34} -518 q^{-36} +131 q^{-38} +573 q^{-40} +390 q^{-42} -277 q^{-44} -629 q^{-46} -287 q^{-48} +418 q^{-50} +776 q^{-52} +124 q^{-54} -670 q^{-56} -794 q^{-58} +49 q^{-60} +1005 q^{-62} +652 q^{-64} -376 q^{-66} -1079 q^{-68} -494 q^{-70} +763 q^{-72} +929 q^{-74} +169 q^{-76} -815 q^{-78} -774 q^{-80} +197 q^{-82} +673 q^{-84} +469 q^{-86} -269 q^{-88} -552 q^{-90} -138 q^{-92} +212 q^{-94} +330 q^{-96} +32 q^{-98} -191 q^{-100} -111 q^{-102} -5 q^{-104} +105 q^{-106} +40 q^{-108} -32 q^{-110} -20 q^{-112} -16 q^{-114} +18 q^{-116} +7 q^{-118} -6 q^{-120} + q^{-122} -3 q^{-124} +3 q^{-126} -2 q^{-130} + q^{-132} }
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^{85}+3 q^{83}+2 q^{81}-7 q^{79}-6 q^{77}-2 q^{75}+10 q^{73}+25 q^{71}+25 q^{69}-20 q^{67}-73 q^{65}-74 q^{63}-4 q^{61}+121 q^{59}+203 q^{57}+134 q^{55}-145 q^{53}-410 q^{51}-392 q^{49}-14 q^{47}+561 q^{45}+873 q^{43}+484 q^{41}-523 q^{39}-1376 q^{37}-1291 q^{35}-44 q^{33}+1653 q^{31}+2363 q^{29}+1179 q^{27}-1344 q^{25}-3271 q^{23}-2809 q^{21}+185 q^{19}+3609 q^{17}+4532 q^{15}+1734 q^{13}-3011 q^{11}-5824 q^9-4017 q^7+1422 q^5+6227 q^3+6163 q+822 q^{-1} -5611 q^{-3} -7593 q^{-5} -3196 q^{-7} +4095 q^{-9} +8075 q^{-11} +5192 q^{-13} -2147 q^{-15} -7620 q^{-17} -6426 q^{-19} +244 q^{-21} +6455 q^{-23} +6841 q^{-25} +1297 q^{-27} -5029 q^{-29} -6557 q^{-31} -2264 q^{-33} +3612 q^{-35} +5830 q^{-37} +2780 q^{-39} -2434 q^{-41} -5011 q^{-43} -2947 q^{-45} +1501 q^{-47} +4259 q^{-49} +3079 q^{-51} -734 q^{-53} -3704 q^{-55} -3331 q^{-57} -62 q^{-59} +3284 q^{-61} +3846 q^{-63} +1053 q^{-65} -2839 q^{-67} -4548 q^{-69} -2406 q^{-71} +2149 q^{-73} +5300 q^{-75} +4045 q^{-77} -1004 q^{-79} -5743 q^{-81} -5840 q^{-83} -687 q^{-85} +5612 q^{-87} +7407 q^{-89} +2741 q^{-91} -4635 q^{-93} -8311 q^{-95} -4861 q^{-97} +2869 q^{-99} +8250 q^{-101} +6537 q^{-103} -679 q^{-105} -7065 q^{-107} -7315 q^{-109} -1521 q^{-111} +5098 q^{-113} +7029 q^{-115} +3096 q^{-117} -2832 q^{-119} -5760 q^{-121} -3789 q^{-123} +819 q^{-125} +4014 q^{-127} +3582 q^{-129} +506 q^{-131} -2296 q^{-133} -2751 q^{-135} -1078 q^{-137} +971 q^{-139} +1769 q^{-141} +1069 q^{-143} -215 q^{-145} -942 q^{-147} -753 q^{-149} -98 q^{-151} +396 q^{-153} +432 q^{-155} +150 q^{-157} -139 q^{-159} -200 q^{-161} -91 q^{-163} +29 q^{-165} +73 q^{-167} +46 q^{-169} -29 q^{-173} -16 q^{-175} +5 q^{-177} +4 q^{-179} +2 q^{-181} +3 q^{-183} -2 q^{-185} -3 q^{-187} +3 q^{-189} -2 q^{-193} + q^{-195} }

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+2q^{4}+1+3q^{-2}-2q^{-4}+2q^{-6}-2q^{-8}-2q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
1,1	$q^{20}-6q^{18}+20q^{16}-50q^{14}+97q^{12}-170q^{10}+268q^{8}-368q^{6}+464q^{4}-522q^{2}+534-462q^{-2}+323q^{-4}-120q^{-6}-132q^{-8}+394q^{-10}-641q^{-12}+832q^{-14}-960q^{-16}+998q^{-18}-942q^{-20}+802q^{-22}-590q^{-24}+342q^{-26}-79q^{-28}-154q^{-30}+334q^{-32}-448q^{-34}+491q^{-36}-470q^{-38}+408q^{-40}-330q^{-42}+247q^{-44}-170q^{-46}+110q^{-48}-70q^{-50}+40q^{-52}-20q^{-54}+10q^{-56}-4q^{-58}+q^{-60}$
2,0	$q^{18}-2q^{16}-3q^{14}+4q^{12}+4q^{10}-3q^{8}-5q^{6}+7q^{4}+10q^{2}-7-5q^{-2}+10q^{-4}+q^{-6}-8q^{-8}-2q^{-10}+6q^{-12}-4q^{-14}-4q^{-16}+5q^{-18}-6q^{-22}+5q^{-24}+8q^{-26}-10q^{-28}-2q^{-30}+9q^{-32}+3q^{-34}-8q^{-36}-2q^{-38}+9q^{-40}-6q^{-44}-q^{-46}+2q^{-48}-q^{-52}+q^{-56}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-3q^{12}+q^{10}+5q^{8}-10q^{6}+7q^{4}+10q^{2}-14+11q^{-2}+11q^{-4}-17q^{-6}+4q^{-8}+8q^{-10}-10q^{-12}-3q^{-14}+4q^{-16}+3q^{-18}-5q^{-20}-5q^{-22}+14q^{-24}-6q^{-26}-12q^{-28}+19q^{-30}-5q^{-32}-11q^{-34}+14q^{-36}-2q^{-38}-7q^{-40}+5q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$-q^{7}+2q^{5}-q^{3}+3q+3q^{-3}-q^{-5}+q^{-7}-q^{-11}-2q^{-15}+q^{-17}-3q^{-19}+2q^{-21}-q^{-23}+q^{-25}+q^{-29}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}-2q^{14}-q^{12}+4q^{10}-3q^{8}-4q^{6}+9q^{4}+6q^{2}-7+4q^{-2}+16q^{-4}-13q^{-8}+7q^{-10}+10q^{-12}-17q^{-14}-11q^{-16}+13q^{-18}-6q^{-20}-13q^{-22}+11q^{-24}+8q^{-26}-10q^{-28}+q^{-30}+14q^{-32}-4q^{-34}-12q^{-36}+8q^{-38}+9q^{-40}-11q^{-42}-4q^{-44}+10q^{-46}+q^{-48}-6q^{-50}+3q^{-54}-2q^{-58}+q^{-62}$
1,0,0,0	$-q^{8}+2q^{6}-q^{4}+2q^{2}+2+3q^{-4}-q^{-6}+2q^{-8}-q^{-10}+q^{-12}-q^{-14}-2q^{-18}-q^{-20}-3q^{-24}+2q^{-26}-q^{-28}+q^{-30}+q^{-32}+q^{-36}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+3q^{12}-7q^{10}+11q^{8}-14q^{6}+19q^{4}-18q^{2}+18-13q^{-2}+9q^{-4}+q^{-6}-10q^{-8}+20q^{-10}-28q^{-12}+33q^{-14}-36q^{-16}+33q^{-18}-29q^{-20}+21q^{-22}-12q^{-24}+2q^{-26}+6q^{-28}-13q^{-30}+17q^{-32}-19q^{-34}+18q^{-36}-14q^{-38}+11q^{-40}-7q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
1,0	$q^{24}-3q^{20}-3q^{18}+4q^{16}+8q^{14}-2q^{12}-12q^{10}-5q^{8}+16q^{6}+14q^{4}-9q^{2}-18+2q^{-2}+21q^{-4}+10q^{-6}-16q^{-8}-14q^{-10}+8q^{-12}+14q^{-14}-4q^{-16}-14q^{-18}-q^{-20}+12q^{-22}+3q^{-24}-13q^{-26}-6q^{-28}+11q^{-30}+10q^{-32}-9q^{-34}-13q^{-36}+7q^{-38}+17q^{-40}-2q^{-42}-19q^{-44}-5q^{-46}+18q^{-48}+13q^{-50}-11q^{-52}-18q^{-54}+2q^{-56}+16q^{-58}+6q^{-60}-8q^{-62}-9q^{-64}+q^{-66}+6q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}$

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^{18}-3 q^{16}+4 q^{14}-6 q^{12}+9 q^{10}-12 q^8+14 q^6-13 q^4+18 q^2-12+13 q^{-2} -7 q^{-4} +9 q^{-6} - q^{-8} -7 q^{-10} +10 q^{-12} -14 q^{-14} +19 q^{-16} -26 q^{-18} +24 q^{-20} -26 q^{-22} +28 q^{-24} -26 q^{-26} +19 q^{-28} -19 q^{-30} +15 q^{-32} -7 q^{-34} + q^{-38} -6 q^{-40} +14 q^{-42} -12 q^{-44} +13 q^{-46} -15 q^{-48} +16 q^{-50} -10 q^{-52} +8 q^{-54} -9 q^{-56} +6 q^{-58} -3 q^{-60} +2 q^{-62} -2 q^{-64} + q^{-66} }

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^{32}-3 q^{30}+7 q^{28}-13 q^{26}+13 q^{24}-9 q^{22}-6 q^{20}+30 q^{18}-50 q^{16}+66 q^{14}-56 q^{12}+17 q^{10}+39 q^8-93 q^6+126 q^4-112 q^2+58+22 q^{-2} -92 q^{-4} +126 q^{-6} -106 q^{-8} +48 q^{-10} +29 q^{-12} -83 q^{-14} +89 q^{-16} -47 q^{-18} -23 q^{-20} +92 q^{-22} -122 q^{-24} +101 q^{-26} -35 q^{-28} -53 q^{-30} +131 q^{-32} -173 q^{-34} +158 q^{-36} -91 q^{-38} -6 q^{-40} +98 q^{-42} -157 q^{-44} +157 q^{-46} -103 q^{-48} +19 q^{-50} +58 q^{-52} -102 q^{-54} +89 q^{-56} -33 q^{-58} -39 q^{-60} +90 q^{-62} -94 q^{-64} +49 q^{-66} +22 q^{-68} -90 q^{-70} +125 q^{-72} -111 q^{-74} +63 q^{-76} + q^{-78} -59 q^{-80} +88 q^{-82} -86 q^{-84} +64 q^{-86} -26 q^{-88} -5 q^{-90} +25 q^{-92} -33 q^{-94} +30 q^{-96} -21 q^{-98} +12 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }

.

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["9 32"];

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 t^3-6 t^2+14 t-17+14 t^{-1} -6 t^{-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 z^6-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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 59, 2 }

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^7-3 q^6+6 q^5-9 q^4+10 q^3-10 q^2+9 q-6+4 q^{-1} - q^{-2} }

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^{-2} +3 z^4 a^{-2} -2 z^4 a^{-4} -z^4+3 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -z^2+ a^{-2} -2 a^{-4} + a^{-6} +1}

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^8 a^{-2} +2 z^8 a^{-4} +5 z^7 a^{-1} +10 z^7 a^{-3} +5 z^7 a^{-5} +6 z^6 a^{-2} +7 z^6 a^{-4} +5 z^6 a^{-6} +4 z^6+a z^5-9 z^5 a^{-1} -18 z^5 a^{-3} -5 z^5 a^{-5} +3 z^5 a^{-7} -19 z^4 a^{-2} -18 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} -8 z^4-a z^3+3 z^3 a^{-1} +9 z^3 a^{-3} +2 z^3 a^{-5} -3 z^3 a^{-7} +10 z^2 a^{-2} +12 z^2 a^{-4} +4 z^2 a^{-6} -z^2 a^{-8} +3 z^2-z a^{-1} -2 z a^{-3} +z a^{-7} - a^{-2} -2 a^{-4} - a^{-6} +1}