10 80

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 3 t^3-9 t^2+15 t-17+15 t^{-1} -9 t^{-2} +3 t^{-3} }
Conway polynomial	$3z^{6}+9z^{4}+6z^{2}+1$
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	{ 71, -6 }
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^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -12 q^{-8} +11 q^{-9} -10 q^{-10} +6 q^{-11} -3 q^{-12} + q^{-13} }
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^2 a^{12}+2 a^{12}-3 z^4 a^{10}-9 z^2 a^{10}-6 a^{10}+2 z^6 a^8+8 z^4 a^8+9 z^2 a^8+3 a^8+z^6 a^6+4 z^4 a^6+5 z^2 a^6+2 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 z^4 a^{16}-z^2 a^{16}+3 z^5 a^{15}-3 z^3 a^{15}+z a^{15}+5 z^6 a^{14}-5 z^4 a^{14}+2 z^2 a^{14}+6 z^7 a^{13}-8 z^5 a^{13}+6 z^3 a^{13}-2 z a^{13}+4 z^8 a^{12}-z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+2 a^{12}+z^9 a^{11}+10 z^7 a^{11}-29 z^5 a^{11}+29 z^3 a^{11}-12 z a^{11}+7 z^8 a^{10}-15 z^6 a^{10}+13 z^4 a^{10}-13 z^2 a^{10}+6 a^{10}+z^9 a^9+6 z^7 a^9-23 z^5 a^9+22 z^3 a^9-8 z a^9+3 z^8 a^8-8 z^6 a^8+8 z^4 a^8-7 z^2 a^8+3 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 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 q^{40}+q^{38}-q^{36}+q^{34}-3 q^{32}-2 q^{30}-q^{28}-3 q^{26}+3 q^{24}-q^{22}+3 q^{20}+2 q^{18}+3 q^{14}-q^{12}+q^{10}}
The G2 invariant	$q^{210}-2q^{208}+4q^{206}-6q^{204}+5q^{202}-4q^{200}-2q^{198}+11q^{196}-20q^{194}+28q^{192}-32q^{190}+25q^{188}-8q^{186}-19q^{184}+54q^{182}-78q^{180}+88q^{178}-74q^{176}+29q^{174}+32q^{172}-94q^{170}+139q^{168}-135q^{166}+89q^{164}-9q^{162}-71q^{160}+123q^{158}-119q^{156}+67q^{154}+16q^{152}-87q^{150}+111q^{148}-74q^{146}-12q^{144}+110q^{142}-174q^{140}+165q^{138}-94q^{136}-29q^{134}+143q^{132}-219q^{130}+216q^{128}-150q^{126}+31q^{124}+83q^{122}-168q^{120}+182q^{118}-135q^{116}+42q^{114}+50q^{112}-112q^{110}+114q^{108}-58q^{106}-25q^{104}+104q^{102}-135q^{100}+105q^{98}-22q^{96}-75q^{94}+152q^{92}-166q^{90}+126q^{88}-41q^{86}-48q^{84}+111q^{82}-123q^{80}+102q^{78}-49q^{76}+36q^{72}-49q^{70}+43q^{68}-25q^{66}+11q^{64}+2q^{62}-6q^{60}+7q^{58}-5q^{56}+4q^{54}-q^{52}+q^{50}$

Further Quantum Invariants

Further quantum knot invariants for 10_80.

A1 Invariants.

Weight	Invariant
1	$q^{27}-2q^{25}+3q^{23}-4q^{21}+q^{19}-q^{17}-q^{15}+3q^{13}-2q^{11}+4q^{9}-q^{7}+q^{5}$
2	$q^{74}-2q^{72}+5q^{68}-9q^{66}+17q^{62}-17q^{60}-7q^{58}+28q^{56}-14q^{54}-15q^{52}+22q^{50}+q^{48}-15q^{46}+2q^{44}+13q^{42}-8q^{40}-17q^{38}+18q^{36}+5q^{34}-27q^{32}+13q^{30}+16q^{28}-22q^{26}+2q^{24}+17q^{22}-8q^{20}-3q^{18}+7q^{16}-q^{12}+q^{10}$
3	$q^{141}-2q^{139}+2q^{135}-5q^{131}+q^{129}+14q^{127}-5q^{125}-26q^{123}+6q^{121}+48q^{119}-78q^{115}-17q^{113}+109q^{111}+42q^{109}-123q^{107}-87q^{105}+125q^{103}+120q^{101}-98q^{99}-141q^{97}+53q^{95}+142q^{93}-3q^{91}-122q^{89}-45q^{87}+95q^{85}+81q^{83}-54q^{81}-112q^{79}+23q^{77}+129q^{75}+17q^{73}-142q^{71}-46q^{69}+136q^{67}+86q^{65}-126q^{63}-119q^{61}+93q^{59}+137q^{57}-53q^{55}-145q^{53}+4q^{51}+128q^{49}+33q^{47}-96q^{45}-55q^{43}+53q^{41}+61q^{39}-22q^{37}-42q^{35}+28q^{31}+8q^{29}-10q^{27}-5q^{25}+5q^{23}+3q^{21}-q^{17}+q^{15}$
4	$q^{228}-2q^{226}+2q^{222}-3q^{220}+4q^{218}-4q^{216}+4q^{214}+7q^{212}-18q^{210}+2q^{208}-5q^{206}+31q^{204}+34q^{202}-66q^{200}-54q^{198}-24q^{196}+133q^{194}+167q^{192}-119q^{190}-252q^{188}-200q^{186}+271q^{184}+538q^{182}+29q^{180}-523q^{178}-688q^{176}+154q^{174}+975q^{172}+555q^{170}-454q^{168}-1205q^{166}-400q^{164}+950q^{162}+1086q^{160}+105q^{158}-1161q^{156}-935q^{154}+337q^{152}+1054q^{150}+681q^{148}-532q^{146}-964q^{144}-341q^{142}+539q^{140}+850q^{138}+153q^{136}-622q^{134}-723q^{132}+13q^{130}+757q^{128}+604q^{126}-291q^{124}-906q^{122}-350q^{120}+636q^{118}+932q^{116}-1020q^{112}-699q^{110}+413q^{108}+1180q^{106}+424q^{104}-886q^{102}-1034q^{100}-100q^{98}+1108q^{96}+897q^{94}-336q^{92}-1032q^{90}-702q^{88}+545q^{86}+985q^{84}+342q^{82}-509q^{80}-876q^{78}-141q^{76}+527q^{74}+573q^{72}+105q^{70}-491q^{68}-377q^{66}-q^{64}+302q^{62}+286q^{60}-65q^{58}-186q^{56}-145q^{54}+24q^{52}+136q^{50}+51q^{48}-12q^{46}-60q^{44}-29q^{42}+24q^{40}+17q^{38}+13q^{36}-7q^{34}-8q^{32}+3q^{30}+q^{28}+3q^{26}-q^{22}+q^{20}$
5	$q^{335}-2q^{333}+2q^{329}-3q^{327}+q^{325}+5q^{323}-q^{321}-3q^{319}-9q^{315}-3q^{313}+20q^{311}+21q^{309}-2q^{307}-36q^{305}-56q^{303}-24q^{301}+72q^{299}+159q^{297}+88q^{295}-133q^{293}-329q^{291}-265q^{289}+140q^{287}+632q^{285}+675q^{283}-41q^{281}-1034q^{279}-1369q^{277}-390q^{275}+1401q^{273}+2433q^{271}+1337q^{269}-1506q^{267}-3722q^{265}-2907q^{263}+996q^{261}+4904q^{259}+5007q^{257}+400q^{255}-5518q^{253}-7302q^{251}-2595q^{249}+5112q^{247}+9087q^{245}+5346q^{243}-3498q^{241}-9907q^{239}-7923q^{237}+993q^{235}+9251q^{233}+9705q^{231}+1932q^{229}-7327q^{227}-10214q^{225}-4512q^{223}+4568q^{221}+9347q^{219}+6260q^{217}-1590q^{215}-7505q^{213}-6959q^{211}-970q^{209}+5191q^{207}+6729q^{205}+2872q^{203}-2962q^{201}-6023q^{199}-4022q^{197}+1153q^{195}+5199q^{193}+4725q^{191}+109q^{189}-4615q^{187}-5208q^{185}-1011q^{183}+4341q^{181}+5830q^{179}+1736q^{177}-4306q^{175}-6587q^{173}-2667q^{171}+4208q^{169}+7571q^{167}+3875q^{165}-3791q^{163}-8393q^{161}-5462q^{159}+2703q^{157}+8856q^{155}+7198q^{153}-957q^{151}-8470q^{149}-8718q^{147}-1408q^{145}+7113q^{143}+9559q^{141}+3958q^{139}-4788q^{137}-9321q^{135}-6119q^{133}+1835q^{131}+7869q^{129}+7363q^{127}+1150q^{125}-5442q^{123}-7313q^{121}-3480q^{119}+2560q^{117}+6030q^{115}+4683q^{113}+87q^{111}-3979q^{109}-4623q^{107}-1875q^{105}+1754q^{103}+3560q^{101}+2624q^{99}+9q^{97}-2136q^{95}-2386q^{93}-982q^{91}+768q^{89}+1647q^{87}+1233q^{85}+67q^{83}-831q^{81}-958q^{79}-437q^{77}+233q^{75}+564q^{73}+424q^{71}+57q^{69}-230q^{67}-277q^{65}-123q^{63}+52q^{61}+127q^{59}+101q^{57}+12q^{55}-47q^{53}-45q^{51}-13q^{49}+6q^{47}+21q^{45}+14q^{43}-3q^{41}-5q^{39}-q^{35}+q^{33}+3q^{31}-q^{27}+q^{25}$

A2 Invariants.

Weight	Invariant
1,0	$q^{40}+q^{38}-q^{36}+q^{34}-3q^{32}-2q^{30}-q^{28}-3q^{26}+3q^{24}-q^{22}+3q^{20}+2q^{18}+3q^{14}-q^{12}+q^{10}$
1,1	$q^{108}-4q^{106}+10q^{104}-20q^{102}+38q^{100}-66q^{98}+102q^{96}-154q^{94}+223q^{92}-302q^{90}+386q^{88}-474q^{86}+548q^{84}-578q^{82}+552q^{80}-460q^{78}+289q^{76}-44q^{74}-250q^{72}+562q^{70}-849q^{68}+1084q^{66}-1220q^{64}+1260q^{62}-1181q^{60}+1008q^{58}-748q^{56}+436q^{54}-129q^{52}-184q^{50}+414q^{48}-596q^{46}+658q^{44}-662q^{42}+594q^{40}-474q^{38}+360q^{36}-234q^{34}+154q^{32}-76q^{30}+43q^{28}-16q^{26}+8q^{24}-2q^{22}+q^{20}$
2,0	$q^{100}+q^{98}-q^{94}-2q^{92}-2q^{90}-5q^{88}+5q^{84}-q^{82}-q^{80}+6q^{78}+9q^{76}-3q^{74}-5q^{72}+12q^{70}+6q^{68}-8q^{66}-2q^{64}+6q^{62}-8q^{60}-12q^{58}-q^{56}-3q^{54}-11q^{52}-2q^{50}+8q^{48}-7q^{46}-5q^{44}+12q^{42}+6q^{40}-6q^{38}+3q^{36}+11q^{34}+3q^{32}-5q^{30}+3q^{28}+5q^{26}-q^{24}-q^{22}+q^{20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{88}-2q^{86}+5q^{82}-7q^{80}-3q^{78}+14q^{76}-11q^{74}-9q^{72}+21q^{70}-11q^{68}-9q^{66}+22q^{64}-4q^{60}+10q^{58}+2q^{56}-8q^{54}-16q^{52}-2q^{48}-23q^{46}+7q^{44}+13q^{42}-16q^{40}+10q^{38}+15q^{36}-11q^{34}+9q^{32}+7q^{30}-4q^{28}+4q^{26}+2q^{24}-q^{22}+q^{20}$
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^{53}+q^{51}+2 q^{49}-q^{47}+q^{45}-5 q^{43}-q^{41}-5 q^{39}-q^{37}-2 q^{35}+q^{33}+2 q^{31}+q^{29}+4 q^{27}+4 q^{23}-q^{21}+3 q^{19}-q^{17}+q^{15}}

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^{114}+q^{112}-q^{110}-2 q^{108}+2 q^{106}+3 q^{104}-7 q^{102}-8 q^{100}+7 q^{98}-16 q^{94}+q^{92}+15 q^{90}-q^{88}+25 q^{84}+21 q^{82}-q^{80}+8 q^{78}+15 q^{76}-20 q^{74}-29 q^{72}-4 q^{70}-20 q^{68}-38 q^{66}-5 q^{64}+9 q^{62}-9 q^{60}-4 q^{58}+19 q^{56}+12 q^{54}-3 q^{52}+7 q^{50}+14 q^{48}+q^{46}+10 q^{42}+2 q^{40}+3 q^{36}+2 q^{34}-q^{32}+q^{30}}
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^{66}+q^{64}+2 q^{62}+2 q^{60}-q^{58}+q^{56}-5 q^{54}-3 q^{52}-4 q^{50}-5 q^{48}-q^{46}-2 q^{44}+2 q^{42}+4 q^{38}+q^{36}+4 q^{34}+q^{32}+2 q^{30}+3 q^{28}-q^{26}+3 q^{24}-q^{22}+q^{20}}

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 q^{88}-2 q^{86}+4 q^{84}-7 q^{82}+11 q^{80}-15 q^{78}+20 q^{76}-23 q^{74}+23 q^{72}-21 q^{70}+15 q^{68}-7 q^{66}-4 q^{64}+16 q^{62}-28 q^{60}+36 q^{58}-44 q^{56}+44 q^{54}-44 q^{52}+36 q^{50}-28 q^{48}+15 q^{46}-3 q^{44}-7 q^{42}+16 q^{40}-20 q^{38}+25 q^{36}-21 q^{34}+21 q^{32}-15 q^{30}+12 q^{28}-6 q^{26}+4 q^{24}-q^{22}+q^{20}}
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^{142}-2 q^{138}-2 q^{136}+2 q^{134}+6 q^{132}+q^{130}-9 q^{128}-9 q^{126}+5 q^{124}+17 q^{122}+4 q^{120}-19 q^{118}-16 q^{116}+11 q^{114}+24 q^{112}-23 q^{108}-9 q^{106}+20 q^{104}+18 q^{102}-9 q^{100}-16 q^{98}+7 q^{96}+19 q^{94}+q^{92}-18 q^{90}-5 q^{88}+12 q^{86}+4 q^{84}-17 q^{82}-13 q^{80}+9 q^{78}+11 q^{76}-13 q^{74}-23 q^{72}+2 q^{70}+24 q^{68}+9 q^{66}-21 q^{64}-18 q^{62}+14 q^{60}+26 q^{58}+q^{56}-17 q^{54}-8 q^{52}+14 q^{50}+13 q^{48}-2 q^{46}-8 q^{44}+5 q^{40}+3 q^{38}-q^{36}-q^{34}+q^{30}}

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^{122}-2 q^{120}+2 q^{118}-3 q^{116}+6 q^{114}-9 q^{112}+8 q^{110}-11 q^{108}+17 q^{106}-18 q^{104}+15 q^{102}-18 q^{100}+19 q^{98}-14 q^{96}+8 q^{94}-6 q^{92}+3 q^{90}+14 q^{88}-9 q^{86}+24 q^{84}-22 q^{82}+35 q^{80}-33 q^{78}+29 q^{76}-44 q^{74}+22 q^{72}-37 q^{70}+14 q^{68}-26 q^{66}+6 q^{64}-3 q^{62}+10 q^{58}-9 q^{56}+22 q^{54}-14 q^{52}+21 q^{50}-15 q^{48}+20 q^{46}-11 q^{44}+14 q^{42}-7 q^{40}+8 q^{38}-2 q^{36}+3 q^{34}-q^{32}+q^{30}}

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^{210}-2 q^{208}+4 q^{206}-6 q^{204}+5 q^{202}-4 q^{200}-2 q^{198}+11 q^{196}-20 q^{194}+28 q^{192}-32 q^{190}+25 q^{188}-8 q^{186}-19 q^{184}+54 q^{182}-78 q^{180}+88 q^{178}-74 q^{176}+29 q^{174}+32 q^{172}-94 q^{170}+139 q^{168}-135 q^{166}+89 q^{164}-9 q^{162}-71 q^{160}+123 q^{158}-119 q^{156}+67 q^{154}+16 q^{152}-87 q^{150}+111 q^{148}-74 q^{146}-12 q^{144}+110 q^{142}-174 q^{140}+165 q^{138}-94 q^{136}-29 q^{134}+143 q^{132}-219 q^{130}+216 q^{128}-150 q^{126}+31 q^{124}+83 q^{122}-168 q^{120}+182 q^{118}-135 q^{116}+42 q^{114}+50 q^{112}-112 q^{110}+114 q^{108}-58 q^{106}-25 q^{104}+104 q^{102}-135 q^{100}+105 q^{98}-22 q^{96}-75 q^{94}+152 q^{92}-166 q^{90}+126 q^{88}-41 q^{86}-48 q^{84}+111 q^{82}-123 q^{80}+102 q^{78}-49 q^{76}+36 q^{72}-49 q^{70}+43 q^{68}-25 q^{66}+11 q^{64}+2 q^{62}-6 q^{60}+7 q^{58}-5 q^{56}+4 q^{54}-q^{52}+q^{50}}

.

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 80"];

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 3 t^3-9 t^2+15 t-17+15 t^{-1} -9 t^{-2} +3 t^{-3} }

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+9z^{4}+6z^{2}+1$

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]=

{ 71, -6 }

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^{-3} -2 q^{-4} +6 q^{-5} -8 q^{-6} +11 q^{-7} -12 q^{-8} +11 q^{-9} -10 q^{-10} +6 q^{-11} -3 q^{-12} + q^{-13} }

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^2 a^{12}+2 a^{12}-3 z^4 a^{10}-9 z^2 a^{10}-6 a^{10}+2 z^6 a^8+8 z^4 a^8+9 z^2 a^8+3 a^8+z^6 a^6+4 z^4 a^6+5 z^2 a^6+2 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 z^4 a^{16}-z^2 a^{16}+3 z^5 a^{15}-3 z^3 a^{15}+z a^{15}+5 z^6 a^{14}-5 z^4 a^{14}+2 z^2 a^{14}+6 z^7 a^{13}-8 z^5 a^{13}+6 z^3 a^{13}-2 z a^{13}+4 z^8 a^{12}-z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+2 a^{12}+z^9 a^{11}+10 z^7 a^{11}-29 z^5 a^{11}+29 z^3 a^{11}-12 z a^{11}+7 z^8 a^{10}-15 z^6 a^{10}+13 z^4 a^{10}-13 z^2 a^{10}+6 a^{10}+z^9 a^9+6 z^7 a^9-23 z^5 a^9+22 z^3 a^9-8 z a^9+3 z^8 a^8-8 z^6 a^8+8 z^4 a^8-7 z^2 a^8+3 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 a^6}