1. Polynomial N(L,(Subscript[R, 0]),(Subscript[R, 1]))

Clear[denomi0,polyN,equaN];denomi0[L_,R_,S_]:=512R^2S^2 Sqrt[L(R-L)(S-L)(R+S-L)](2(R-L)(S-L)-R*S);
polyN[L_,R_,S_,b02_,b12_,b13_]:=(16-2b02) R S (L^4)+(176-2b13) (R^2)(S^2)(L^2)+14(R^3)(S^3)+b02 ((R^2)+(S^2))(L^4)+(-48-b12)((R^3)+(S^3))(L^3)+b12((R^2)S+(S^2)R)(L^3)+8((R^4)+(S^4))(L^2)+b13((R^3)S+(S^3)R)(L^2)-16((R^4)S+(S^4)R)L-56((R^3)(S^2)+(S^3)(R^2))L+7((R^4)(S^2)+(S^4)(R^2));
equaN[R_,S_,b02_,b12_,b13_]:=3polyN[1,R,S,b02,b12,b13]-denomi0[1,R,S]*twist2[{1,R,0,0,S,0,0}];

Solve[equaN[1.1,1.2,b02,b12,b13]==0&&equaN[1.1,1.3,b02,b12,b13]==0&&equaN[1.2,1.3,b02,b12,b13]==0,{b02,b12,b13}]

Clear[finalN,tw2Ord0];
finalN[L_,R_,S_]:=polyN[L,R,S,8,-32,48];tw2Ord0[L_,R_,S_]:=3finalN[L,R,S]/denomi0[L,R,S];

Simplify[finalN[L,R,S]]

2. Polynomial Q(L,(Subscript[R, 0]),(Subscript[R, 1]))

Clear[denomi22,symQ,eqsymQ];
denomi22[L_,R_,S_]:=1536Sqrt[L(R-L)(S-L)(R+S-L)](R-L)^2(S-L)^2(2(R-L)(S-L)-R S)(R+S-L)^2;
symQ[L_,R_,a0_,a1_,a2_,a3_,a4_,a5_,a6_,a7_,a8_]:=a0 (L^10)+a1 R (L^9)+a2 (R^2)(L^8)+a3 (R^3)(L^7)+a4 (R^4)(L^6)+a5 (R^5)(L^5)+a6 (R^6)(L^4)+a7 (R^7)(L^3)+a8 (R^8)(L^2);
eqsymQ[R_,a0_,a1_,a2_,a3_,a4_,a5_,a6_,a7_,a8_]:=symQ[1,R,a0,a1,a2,a3,a4,a5,a6,a7,a8]-denomi22[1,R,R](twist2[{1,R,1,0,R,1,0}]+tw2Ord0[1,R,R]-twist2[{1,R,1,0,R,0,0}]-twist2[{1,R,0,0,R,1,0}]);

Solve[eqsymQ[1.1,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0&&eqsymQ[1.2,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0&&eqsymQ[1.3,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0&&eqsymQ[1.4,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0&&eqsymQ[1.6,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0&&eqsymQ[1.7,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0&&eqsymQ[1.8,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0&&eqsymQ[2.0,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0&&eqsymQ[2.1,a0,a1,a2,a3,a4,a5,a6,a7,a8]==0,{a0,a1,a2,a3,a4,a5,a6,a7,a8}]

Clear[symQfinal];
symQfinal[L_,R_]:=symQ[L,R,0,0,-64,384,-930,1160,-780,264,-34];

Simplify[symQfinal[L,R]]

Clear[polyQ,eqQ];
polyQ[L_,R_,S_,c20_,c30_,c40_,c31_,c50_,c41_,c51_,c42_,c52_,c53_]:=(L^8)(c20(R^2-2R*S+S^2)-64*R*S)+(L^7)(c30(R^3+S^3-R^2S-S^2R)+192(R^2S+S^2R))+(L^6)(c40(R^4-2R^2S^2+S^4)+c31(R^3S-2R^2S^2+S^3R)-930R^2S^2)+(L^5)(c50(R^5+S^5-R^3S^2-S^3R^2)+c41(R^4S+S^4R-R^3S^2-S^3R^2)+580(R^3S^2+S^3R^2))+(L^4)(c51(R^5S+S^5R-2R^3S^3)+c42(R^4S^2+S^4R^2-2R^3S^3)-780R^3S^3)+(L^3)(c52*(R^5S^2+S^5R^2-R^4S^3-S^4R^3)+132(R^4S^3+S^4R^3))+(L^2)(c53*(R^5S^3+S^5R^3-2R^4S^4)-34R^4S^4);
eqQ[R_,S_,c20_,c30_,c40_,c31_,c50_,c41_,c51_,c42_,c52_,c53_]:=polyQ[1,R,S,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]-denomi22[1,R,S](twist2[{1,R,1,0,S,1,0}]+tw2Ord0[1,R,S]-twist2[{1,R,1,0,S,0,0}]-twist2[{1,R,0,0,S,1,0}]);

Solve[eqQ[1.02,1.1,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.02,1.2,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.02,1.3,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.04,1.4,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.04,1.5,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.04,1.6,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.07,1.7,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.07,1.8,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.09,1.9,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0&&eqQ[1.09,2.0,c20,c30,c40,c31,c50,c41,c51,c42,c52,c53]==0,{c20,c30,c40,c31,c50,c41,c51,c42,c52,c53}]

Clear[finalQ,eqfinalQ];
finalQ[L_,R_,S_]:=polyQ[L,R,S,0,0,0,-192,0,64,0,-162,0,0];
eqfinalQ[R_,S_]:=finalQ[1,R,S]-denomi22[1,R,S](twist2[{1,R,1,0,S,1,0}]+tw2Ord0[1,R,S]-twist2[{1,R,1,0,S,0,0}]-twist2[{1,R,0,0,S,1,0}]);

Simplify[finalQ[L,R,S]]

3. Polynomial P(L,(Subscript[R, 0]),(Subscript[R, 1]))

3.1 Find the symmetric part symP of P

Clear[symPsU,eqsPsU]; (* use the variables U=(R_0+R_1)/2 and V=(R_0-R_1)/2 assuming U is same for each group of data*)
symPsU[U_,V_,ua6_,ua4_,ua2_,ua0_]:=(1-U)^4 (24-192 U+456 U^2-384 U^3+79 U^4)+ua6*V^2+ua4*V^4+ua2*V^6+ua0*V^8; (*symmetric part of P with same variable U*)
eqsPsU[R_,S_,ua6_,ua4_,ua2_,ua0_]:=symPsU[(R+S)/2,(R-S)/2,ua6,ua4,ua2,ua0]-((twist2[{1,R,1,0,S,1,0}]+twist2[{1,R,-1,0,S,-1,0}]-2tw2Ord0[1,R,S])denomi22[1,R,S]-2finalQ[1,R,S])/4;

Clear[dataA1,dataA2,dataA3,dataA4,dataA5,dataA6,dataA7];dataA1=Solve[eqsPsU[2.1,1.9,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.2,1.8,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.3,1.7,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.4,1.6,ua6,ua4,ua2,ua0]==0,{ua6,ua4,ua2,ua0}];
dataA2=Solve[eqsPsU[2.2,2.0,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.3,1.9,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.4,1.8,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.5,1.7,ua6,ua4,ua2,ua0]==0,{ua6,ua4,ua2,ua0}];
dataA3=Solve[eqsPsU[2.3,2.1,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.4,2.0,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.5,1.9,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.6,1.8,ua6,ua4,ua2,ua0]==0,{ua6,ua4,ua2,ua0}];
dataA4=Solve[eqsPsU[2.4,2.2,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.5,2.1,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.6,2.0,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.7,1.9,ua6,ua4,ua2,ua0]==0,{ua6,ua4,ua2,ua0}];dataA5=Solve[eqsPsU[2.5,2.3,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.6,2.2,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.7,2.1,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.8,2.0,ua6,ua4,ua2,ua0]==0,{ua6,ua4,ua2,ua0}];
dataA6=Solve[eqsPsU[2.6,2.4,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.7,2.3,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.8,2.2,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.9,2.1,ua6,ua4,ua2,ua0]==0,{ua6,ua4,ua2,ua0}];dataA7=Solve[eqsPsU[2.7,2.5,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.8,2.4,ua6,ua4,ua2,ua0]==0&&eqsPsU[2.9,2.3,ua6,ua4,ua2,ua0]==0&&eqsPsU[3.0,2.2,ua6,ua4,ua2,ua0]==0,{ua6,ua4,ua2,ua0}];

dataA1

(*(Subscript[a, 0])=-17*)

Clear[fUa2,fUa4,fUa6];
fUa2[U_,ua20_,ua21_,ua22_]:=ua20+ua21*U+ua22*U^2;
fUa4[U_,ua40_,ua41_,ua42_,ua43_,ua44_]:=ua40+ua41*U+ua42*U^2+ua43*U^3+ua44*U^4;
fUa6[U_,ua60_,ua61_,ua62_,ua63_,ua64_,ua65_,ua66_]:=ua60+ua61*U+ua62*U^2+ua63*U^3+ua64*U^4+ua65*U^5+ua66*U^6;

Clear[x1,x2,x3];
x1=ua2/.Part[Part[dataA1,1],3];
x2=ua2/.Part[Part[dataA2,1],3];
x3=ua2/.Part[Part[dataA3,1],3];

Solve[fUa2[2.0,ua20,ua21,ua22]==x1&&fUa2[2.1,ua20,ua21,ua22]==x2&&fUa2[2.2,ua20,ua21,ua22]==x3,{ua20,ua21,ua22}]

Clear[y1,y2,y3,y4,y5];
y1=ua4/.Part[Part[dataA1,1],2];
y2=ua4/.Part[Part[dataA2,1],2];
y3=ua4/.Part[Part[dataA3,1],2];
y4=ua4/.Part[Part[dataA4,1],2];
y5=ua4/.Part[Part[dataA5,1],2];

Solve[fUa4[2.0,ua40,ua41,ua42,ua43,ua44]==y1&&fUa4[2.1,ua40,ua41,ua42,ua43,ua44]==y2&&fUa4[2.2,ua40,ua41,ua42,ua43,ua44]==y3&&fUa4[2.3,ua40,ua41,ua42,ua43,ua44]==y4&&fUa4[2.4,ua40,ua41,ua42,ua43,ua44]==y5,{ua40,ua41,ua42,ua43,ua44}]

Clear[z1,z2,z3,z4,z5,z6,z7];
z1=ua6/.Part[Part[try1,1],1];
z2=ua6/.Part[Part[try2,1],1];
z3=ua6/.Part[Part[try3,1],1];
z4=ua6/.Part[Part[try4,1],1];
z5=ua6/.Part[Part[try5,1],1];
z6=ua6/.Part[Part[try6,1],1];
z7=ua6/.Part[Part[try7,1],1];

Solve[fUa6[2.0,ua60,ua61,ua62,ua63,ua64,ua65,ua66]==z1&&fUa6[2.1,ua60,ua61,ua62,ua63,ua64,ua65,ua66]==z2&&fUa6[2.2,ua60,ua61,ua62,ua63,ua64,ua65,ua66]==z3&&fUa6[2.3,ua60,ua61,ua62,ua63,ua64,ua65,ua66]==z4&&fUa6[2.4,ua60,ua61,ua62,ua63,ua64,ua65,ua66]==z5&&fUa6[2.5,ua60,ua61,ua62,ua63,ua64,ua65,ua66]==z6&&fUa6[2.6,ua60,ua61,ua62,ua63,ua64,ua65,ua66]==z7,{ua60,ua61,ua62,ua63,ua64,ua65,ua66}]

Clear[symPsUnew,eqsPsUnew]; (*doulbe chck: update our function symP*)
symPsUnew[U_,V_,a60_,a61_,a62_,a63_,a64_,a65_,a66_]:=(1-U)^4 (24-192 U+456 U^2-384 U^3+79 U^4)+(a60+a61 U +a62 (U^2)+a63 (U^3)+a64 (U^4)+a65 (U^5)+a66 (U^6))*V^2+(-41+548 U -1410 (U^2)+1196 (U^3)-310(U^4))*V^4+(114-372U +220 (U^2))*V^6-17V^8; (*symmetric part of P with same variable U*)
eqsPsUnew[R_,S_,a60_,a61_,a62_,a63_,a64_,a65_,a66_]:=symPsUnew[(R+S)/2,(R-S)/2,a60,a61,a62,a63,a64,a65,a66]-((twist2[{1,R,1,0,S,1,0}]+twist2[{1,R,-1,0,S,-1,0}]-2tw2Ord0[1,R,S])denomi22[1,R,S]-2finalQ[1,R,S])/4;

Solve[eqsPsUnew[1.03,1.1,a60,a61,a62,a63,a64,a65,a66]==0&&eqsPsUnew[1.03,1.2,a60,a61,a62,a63,a64,a65,a66]==0&&eqsPsUnew[1.05,1.3,a60,a61,a62,a63,a64,a65,a66]==0&&eqsPsUnew[1.05,1.4,a60,a61,a62,a63,a64,a65,a66]==0&&eqsPsUnew[1.07,1.5,a60,a61,a62,a63,a64,a65,a66]==0&&eqsPsUnew[1.07,1.6,a60,a61,a62,a63,a64,a65,a66]==0&&eqsPsUnew[1.09,1.7,a60,a61,a62,a63,a64,a65,a66]==0,{a60,a61,a62,a63,a64,a65,a66}]

Clear[symPU,symP];
symPU[U_,V_]:=(1-U)^4 (24-192 U+456 U^2-384 U^3+79 U^4)+(-216+1152 U -2222 (U^2)+1784 (U^3)-402(U^4)-124 (U^5)+28 (U^6))*V^2+(-41+548 U -1410 (U^2)+1196 (U^3)-310(U^4))*V^4+(114-372U +220 (U^2))*V^6-17V^8; 
symP[R_,S_]:=symPU[(R+S)/2,(R-S)/2];

3.2 Find the anti-symmetric part of P

Clear[antiPsU,eqaPsU];
antiPsU[V_,ub7_,ub5_,ub3_,ub1_]:=ub7*V+ub5*V^3+ub3*V^5+ub1*V^7;
eqaPsU[R_,S_,ub7_,ub5_,ub3_,ub1_]:=antiPsU[(R-S)/2,ub7,ub5,ub3,ub1]-((twist2[{1,R,1,0,S,0,0}]+twist2[{1,R,-1,0,S,0,0}]-2tw2Ord0[1,R,S])denomi22[1,R,S])/2-symP[1,R,S];

Clear[dataB1,dataB2,dataB3,dataB4,dataB5,dataB6,dataB7,dataB8];
dataB1=Solve[eqaPsU[2.1,1.9,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.2,1.8,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.3,1.7,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.4,1.6,ub7,ub5,ub3,ub1]==0,{ub7,ub5,ub3,ub1}];
dataB2=Solve[eqaPsU[2.2,2.0,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.3,1.9,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.4,1.8,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.5,1.7,ub7,ub5,ub3,ub1]==0,{ub7,ub5,ub3,ub1}];
dataB3=Solve[eqaPsU[2.3,2.1,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.4,2.0,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.5,1.9,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.6,1.8,ub7,ub5,ub3,ub1]==0,{ub7,ub5,ub3,ub1}];
dataB4=Solve[eqaPsU[2.4,2.2,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.5,2.1,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.6,2.0,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.7,1.9,ub7,ub5,ub3,ub1]==0,{ub7,ub5,ub3,ub1}];dataB5=Solve[eqaPsU[2.5,2.3,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.6,2.2,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.7,2.1,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.8,2.0,ub7,ub5,ub3,ub1]==0,{ub7,ub5,ub3,ub1}];
dataB6=Solve[eqaPsU[2.6,2.4,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.7,2.3,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.8,2.2,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.9,2.1,ub7,ub5,ub3,ub1]==0,{ub7,ub5,ub3,ub1}];dataB7=Solve[eqaPsU[2.7,2.5,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.8,2.4,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.9,2.3,ub7,ub5,ub3,ub1]==0&&eqaPsU[3.0,2.2,ub7,ub5,ub3,ub1]==0,{ub7,ub5,ub3,ub1}];
dataB8=Solve[eqaPsU[2.8,2.6,ub7,ub5,ub3,ub1]==0&&eqaPsU[2.9,2.5,ub7,ub5,ub3,ub1]==0&&eqaPsU[3.0,2.4,ub7,ub5,ub3,ub1]==0&&eqaPsU[3.1,2.3,ub7,ub5,ub3,ub1]==0,{ub7,ub5,ub3,ub1}];

dataB1

Clear[fUb1,fUb3,fUb5,fUb7];
fUb1[U_,ub10_,ub11_]:=ub10+ub11*U;
fUb3[U_,ub30_,ub31_,ub32_,ub33_]:=ub30+ub31*U+ub32*U^2+ub33*U^3;
fUb5[U_,ub50_,ub51_,ub52_,ub53_,ub54_,ub55_]:=ub50+ub51*U+ub52*U^2+ub53*U^3+ub54*U^4+ub55*U^5;
fUb7[U_,ub70_,ub71_,ub72_,ub73_,ub74_,ub75_,ub76_,ub77_]:=ub70+ub71*U+ub72*U^2+ub73*U^3+ub74*U^4+ub75*U^5+ub76*U^6+ub77*U^7;

Clear[wb1,wb2];
wb1=ub1/.Part[Part[dataB1,1],4];
wb2=ub1/.Part[Part[dataB2,1],4];

Solve[fUb1[2.0,ub10,ub11]==wb1&&fUb1[2.1,ub10,ub11]==wb2,{ub10,ub11}]

Clear[xb1,xb2,xb3,xb4];
xb1=ub3/.Part[Part[dataB1,1],3];
xb2=ub3/.Part[Part[dataB2,1],3];
xb3=ub3/.Part[Part[dataB3,1],3];
xb4=ub3/.Part[Part[dataB4,1],3];

Solve[fUb3[2.0,ub30,ub31,ub32,ub33]==xb1&&fUb3[2.1,ub30,ub31,ub32,ub33]==xb2&&fUb3[2.2,ub30,ub31,ub32,ub33]==xb3&&fUb3[2.3,ub30,ub31,ub32,ub33]==xb4,{ub30,ub31,ub32,ub33}]

Clear[yb1,yb2,yb3,yb4,yb5,yb6];
yb1=ub5/.Part[Part[dataB1,1],2];
yb2=ub5/.Part[Part[dataB2,1],2];
yb3=ub5/.Part[Part[dataB3,1],2];
yb4=ub5/.Part[Part[dataB4,1],2];
yb5=ub5/.Part[Part[dataB5,1],2];
yb6=ub5/.Part[Part[dataB6,1],2];

Solve[fUb5[2.0,ub50,ub51,ub52,ub53,ub54,ub55]==yb1&&fUb5[2.1,ub50,ub51,ub52,ub53,ub54,ub55]==yb2&&fUb5[2.2,ub50,ub51,ub52,ub53,ub54,ub55]==yb3&&fUb5[2.3,ub50,ub51,ub52,ub53,ub54,ub55]==yb4&&fUb5[2.4,ub50,ub51,ub52,ub53,ub54,ub55]==yb5&&fUb5[2.5,ub50,ub51,ub52,ub53,ub54,ub55]==yb6,{ub50,ub51,ub52,ub53,ub54,ub55}]

Clear[zb1,zb2,zb3,zb4,zb5,zb6,zb7,zb8];
zb1=ub7/.Part[Part[dataB1,1],1];
zb2=ub7/.Part[Part[dataB2,1],1];
zb3=ub7/.Part[Part[dataB3,1],1];
zb4=ub7/.Part[Part[dataB4,1],1];
zb5=ub7/.Part[Part[dataB5,1],1];
zb6=ub7/.Part[Part[dataB6,1],1];
zb7=ub7/.Part[Part[dataB7,1],1];
zb8=ub7/.Part[Part[dataB8,1],1];

Solve[fUb7[2.0,ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77]==zb1&&fUb7[2.1,ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77]==zb2&&fUb7[2.2,ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77]==zb3&&fUb7[2.3,ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77]==zb4&&fUb7[2.4,ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77]==zb5&&fUb7[2.5,ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77]==zb6&&fUb7[2.6,ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77]==zb7&&fUb7[2.7,ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77]==zb8,{ub70,ub71,ub72,ub73,ub74,ub75,ub76,ub77}]

Clear[antiPU,antiP,finalP];
antiPU[U_,V_]:=(-288U+1872U^2-4896U^3+6556U^4-4692U^5+1668U^6-220U^7)V+(-336+2016U-4472U^2+4584U^3-2204U^4+412U^5)V^3+(220-660U+556U^2-164U^3)V^5+(-20-28U)V^7;
antiP[R_,S_]:=antiPU[(R+S)/2,(R-S)/2];
finalP[R_,S_]:=symP[R,S]+antiP[R,S];

Simplify[finalP[R,S]]

4. The polynomial S(L,(Subscript[R, 0]),(Subscript[R, 1]))

4.1. The symmetric part of S(L,R,S)

Clear[denoS,symSsU,eqsSsU]; 
denoS[R_,S_]:=-768Sqrt[(R-1)(S-1)(R+S-1)](R-1)(S-1)(R+S-1)(2(R-1)(S-1)-R*S);
symSsU[U_,V_,sa5_,sa3_,sa1_]:=2(27U^2-40 U+10)(U-1)^2(2(U-1)^2-U^2)+sa5*V^2+sa3*V^4+sa1*V^6;
eqsSsU[R_,S_,sa5_,sa3_,sa1_]:=symSsU[(R+S)/2,(R-S)/2,sa5,sa3,sa1]-((twist2[{1,R,1,0,S,0,0}]-tw2Ord0[1,R,S]-finalP[R,S]/denomi22[1,R,S])R+(twist2[{1,R,0,0,S,1,0}]-tw2Ord0[1,R,S]-finalP[S,R]/denomi22[1,R,S])S)denoS[R,S]/2;

Clear[dataC1,dataC2,dataC3,dataC4,dataC5,dataC6];dataC1=Solve[eqsSsU[2.1,1.9,sa5,sa3,sa1]==0&&eqsSsU[2.2,1.8,sa5,sa3,sa1]==0&&eqsSsU[2.3,1.7,sa5,sa3,sa1]==0,{sa5,sa3,sa1}];
dataC2=Solve[eqsSsU[2.2,2.0,sa5,sa3,sa1]==0&&eqsSsU[2.3,1.9,sa5,sa3,sa1]==0&&eqsSsU[2.4,1.8,sa5,sa3,sa1]==0,{sa5,sa3,sa1}];
dataC3=Solve[eqsSsU[2.3,2.1,sa5,sa3,sa1]==0&&eqsSsU[2.4,2.0,sa5,sa3,sa1]==0&&eqsSsU[2.5,1.9,sa5,sa3,sa1]==0,{sa5,sa3,sa1}];
dataC4=Solve[eqsSsU[2.4,2.2,sa5,sa3,sa1]==0&&eqsSsU[2.5,2.1,sa5,sa3,sa1]==0&&eqsSsU[2.6,2.0,sa5,sa3,sa1]==0,{sa5,sa3,sa1}];
dataC5=Solve[eqsSsU[2.5,2.3,sa5,sa3,sa1]==0&&eqsSsU[2.6,2.2,sa5,sa3,sa1]==0&&eqsSsU[2.7,2.1,sa5,sa3,sa1]==0,{sa5,sa3,sa1}];
dataC6=Solve[eqsSsU[2.6,2.4,sa5,sa3,sa1]==0&&eqsSsU[2.7,2.3,sa5,sa3,sa1]==0&&eqsSsU[2.8,2.2,sa5,sa3,sa1]==0,{sa5,sa3,sa1}];

Clear[fSa1,fSa3,fSa5];
fSa1[U_,sa10_,sa11_]:=sa10+sa11*U;
fSa3[U_,sa30_,sa31_,sa32_,sa33_]:=sa30+sa31*U+sa32*U^2+sa33*U^3;
fSa5[U_,sa50_,sa51_,sa52_,sa53_,sa54_,sa55_]:=sa50+sa51*U+sa52*U^2+sa53*U^3+sa54*U^4+sa55*U^5;

Clear[xs1,xs2];
xs1=sa1/.Part[Part[dataC1,1],3];
xs2=sa1/.Part[Part[dataC2,1],3];

Solve[fSa1[2.0,sa10,sa11]==xs1&&fSa1[2.1,sa10,sa11]==xs2,{sa10,sa11}]

Clear[ys1,ys2,ys3,ys4];
ys1=sa3/.Part[Part[dataC1,1],2];
ys2=sa3/.Part[Part[dataC2,1],2];
ys3=sa3/.Part[Part[dataC3,1],2];
ys4=sa3/.Part[Part[dataC4,1],2];

Solve[fSa3[2.0,sa30,sa31,sa32,sa33]==ys1&&fSa3[2.1,sa30,sa31,sa32,sa33]==ys2&&fSa3[2.2,sa30,sa31,sa32,sa33]==ys3&&fSa3[2.3,sa30,sa31,sa32,sa33]==ys4,{sa30,sa31,sa32,sa33}]

Clear[zs1,zs2,zs3,zs4,zs5,zs6];
zs1=sa5/.Part[Part[dataC1,1],1];
zs2=sa5/.Part[Part[dataC2,1],1];
zs3=sa5/.Part[Part[dataC3,1],1];
zs4=sa5/.Part[Part[dataC4,1],1];
zs5=sa5/.Part[Part[dataC5,1],1];
zs6=sa5/.Part[Part[dataC6,1],1];

Solve[fSa5[2.0,sa50,sa51,sa52,sa53,sa54,sa55]==zs1&&fSa5[2.1,sa50,sa51,sa52,sa53,sa54,sa55]==zs2&&fSa5[2.2,sa50,sa51,sa52,sa53,sa54,sa55]==zs3&&fSa5[2.3,sa50,sa51,sa52,sa53,sa54,sa55]==zs4&&fSa5[2.4,sa50,sa51,sa52,sa53,sa54,sa55]==zs5&&fSa5[2.5,sa50,sa51,sa52,sa53,sa54,sa55]==zs6,{sa50,sa51,sa52,sa53,sa54,sa55}]

Clear[symSU];
symSU[U_,V_]:=2(27U^2-40 U+10)(U-1)^2(2(U-1)^2-U^2)+(-152+600U-822U^2+488U^3-108U^4)V^2+(48-84U+54U^2)V^4;

4.2. The antisymmetric part of S

Clear[antiSsU,eqaSsU]; 
antiSsU[V_,sb6_,sb4_,sb2_,sb0_]:=sb6*V+sb4*V^3+sb2*V^5+sb0*V^7;
eqaSsU[R_,S_,sb6_,sb4_,sb2_,sb0_]:=antiSsU[(R-S)/2,sb6,sb4,sb2,sb0]-((twist2[{1,R,1,0,S,0,0}]-tw2Ord0[1,R,S]-finalP[R,S]/denomi22[1,R,S])R-(twist2[{1,R,0,0,S,1,0}]-tw2Ord0[1,R,S]-finalP[S,R]/denomi22[1,R,S])S)denoS[R,S]/2;

Clear[dataD1,dataD2,dataD3,dataD4,dataD5,dataD6,dataD7,dataD8];
dataD1=Solve[diffCstUb[2.1,1.9,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.2,1.8,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.3,1.7,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.4,1.6,sb6,sb4,sb2,sb0]==0,{sb6,sb4,sb2,sb0}];
dataD2=Solve[eqaSsU[2.2,2.0,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.3,1.9,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.4,1.8,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.5,1.7,sb6,sb4,sb2,sb0]==0,{sb6,sb4,sb2,sb0}];
dataD3=Solve[eqaSsU[2.3,2.1,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.4,2.0,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.5,1.9,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.6,1.8,sb6,sb4,sb2,sb0]==0,{sb6,sb4,sb2,sb0}];
dataD4=Solve[eqaSsU[2.4,2.2,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.5,2.1,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.6,2.0,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.7,1.9,sb6,sb4,sb2,sb0]==0,{sb6,sb4,sb2,sb0}];
dataD5=Solve[eqaSsU[2.5,2.3,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.6,2.2,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.7,2.1,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.8,2.0,sb6,sb4,sb2,sb0]==0,{sb6,sb4,sb2,sb0}];
dataD6=Solve[eqaSsU[2.6,2.4,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.7,2.3,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.8,2.2,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.9,2.1,sb6,sb4,sb2,sb0]==0,{sb6,sb4,sb2,sb0}];
dataD7=Solve[eqaSsU[2.7,2.5,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.8,2.4,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.9,2.3,sb6,sb4,sb2,sb0]==0&&eqaSsU[3.0,2.2,sb6,sb4,sb2,sb0]==0,{sb6,sb4,sb2,sb0}];
dataD8=Solve[eqaSsU[2.8,2.6,sb6,sb4,sb2,sb0]==0&&eqaSsU[2.9,2.5,sb6,sb4,sb2,sb0]==0&&eqaSsU[3.0,2.4,sb6,sb4,sb2,sb0]==0&&eqaSsU[3.1,2.3,sb6,sb4,sb2,sb0]==0,{sb6,sb4,sb2,sb0}];

Clear[wt1];
wt1=sb0/.Part[Part[dataD2,1],4]

Clear[fSb2,fSb4,fSb6];
fSb2[U_,sb20_,sb21_,sb22_]:=sb20+sb21*U+sb22*U^2;
fSb4[U_,sb40_,sb41_,sb42_,sb43_,sb44_]:=sb40+sb41*U+sb42*U^2+sb43*U^3+sb44*U^4;
fSb6[U_,sb60_,sb61_,sb62_,sb63_,sb64_,sb65_,sb66_]:=sb60+sb61*U+sb62*U^2+sb63*U^3+sb64*U^4+sb65*U^5+sb66*U^6;

Clear[xt1,xt2,xt3];
xt1=sb2/.Part[Part[dataD4,1],3];
xt2=sb2/.Part[Part[dataD2,1],3];
xt3=sb2/.Part[Part[dataD3,1],3];

Solve[fSb2[2.3,sb20,sb21,sb22]==xt1&&fSb2[2.1,sb20,sb21,sb22]==xt2&&fSb2[2.2,sb20,sb21,sb22]==xt3,{sb20,sb21,sb22}]

Clear[yt1,yt2,yt3,yt4,yt5];
yt1=sb4/.Part[Part[dataD6,1],2];
yt2=sb4/.Part[Part[dataD2,1],2];
yt3=sb4/.Part[Part[dataD3,1],2];
yt4=sb4/.Part[Part[dataD4,1],2];
yt5=sb4/.Part[Part[dataD5,1],2];

Solve[fSb4[2.5,sb40,sb41,sb42,sb43,sb44]==yt1&&fSb4[2.1,sb40,sb41,sb42,sb43,sb44]==yt2&&fSb4[2.2,sb40,sb41,sb42,sb43,sb44]==yt3&&fSb4[2.3,sb40,sb41,sb42,sb43,sb44]==yt4&&fSb4[2.4,sb40,sb41,sb42,sb43,sb44]==yt5,{sb40,sb41,sb42,sb43,sb44}]

Clear[zt1,zt2,zt3,zt4,zt5,zt6,zt7];
zt1=sb6/.Part[Part[dataD8,1],1];
zt2=sb6/.Part[Part[dataD2,1],1];
zt3=sb6/.Part[Part[dataD3,1],1];
zt4=sb6/.Part[Part[dataD4,1],1];
zt5=sb6/.Part[Part[dataD5,1],1];
zt6=sb6/.Part[Part[dataD6,1],1];
zt7=sb6/.Part[Part[dataD7,1],1];

Solve[fSb6[2.7,sb60,sb61,sb62,sb63,sb64,sb65,sb66]==zt1&&fSb6[2.1,sb60,sb61,sb62,sb63,sb64,sb65,sb66]==zt2&&fSb6[2.2,sb60,sb61,sb62,sb63,sb64,sb65,sb66]==zt3&&fSb6[2.3,sb60,sb61,sb62,sb63,sb64,sb65,sb66]==zt4&&fSb6[2.4,sb60,sb61,sb62,sb63,sb64,sb65,sb66]==zt5&&fSb6[2.5,sb60,sb61,sb62,sb63,sb64,sb65,sb66]==zt6&&fSb6[2.6,sb60,sb61,sb62,sb63,sb64,sb65,sb66]==zt7,{sb60,sb61,sb62,sb63,sb64,sb65,sb66}]

Clear[antiSU];
antiSU[U_,V_]:=(-144U+552U^2-714U^3+360U^4-54U^5)V+(-104+394U-400U^2+108U^3)V^3+(40-54U)V^5;
finalS[R_,S_]:=symSU[(R+S)/2,(R-S)/2]+antiSU[(R+S)/2,(R-S)/2];

Simplify[finalS[R,S]]

5. The polynomial T(L,(Subscript[R, 0]),(Subscript[R, 1]))

Clear[denoR4,polyT,eqpolyT];
denoR4[L_,R_,S_]:=-192Sqrt[L(R-L)(S-L)(R+S-L)](R-L)(R+S-L);
polyT[L_,R_,S_,a0_,a1_,a2_,a3_,a4_]:=S^2L^2R+S(a0*L^4+a1*L^3R+a2*L^2R^2+a3*R^3*L+a4*R^4)+R((1-a0)*L^4+(-2-a1)*L^3R-a2*L^2R^2-a3*R^3*L-a4*R^4);
eqpolyT[R_,S_,a0_,a1_,a2_,a3_,a4_]:=polyT[1,R,S,a0,a1,a2,a3,a4]-denoR4[1,R,S](twist2[{1,R,0,1,S,0,0}]-tw2Ord0[1,R,S]);

Solve[eqpolyT[2.0,1.5,a0,a1,a2,a3,a4]==0&&eqpolyT[2.5,1.5,a0,a1,a2,a3,a4]==0&&eqpolyT[3.0,1.5,a0,a1,a2,a3,a4]==0&&eqpolyT[3.5,1.5,a0,a1,a2,a3,a4]==0&&eqpolyT[4.0,1.5,a0,a1,a2,a3,a4]==0,{a0,a1,a2,a3,a4}]

Clear[finalT];
finalT[L_,R_,S_]:=polyT[L,R,S,0,-2,0,0,0];

Simplify[finalT[L,R,S]]