///////////////////////////////////////////////////////////////////////////// // Refined Modifications for Figure Eight Graph with Vertical modification // ///////////////////////////////////////////////////////////////////////////// // The following contains an example for the modified tropical figure eight graph in R^4. The curve can be visualized by means of projections. // Since our modifications involve square roots, we take our branch points to be squares, resp. negative of squares, for computational convenience. // The parameters w2,..., w5 satisfy w2 < w3 = w4 < w5 with val(a3-a4)=val(a3) // We replace our coefficients a and b by a + t^e and b+t^e', respectively. This refinement gives a planar faithful re-embedding of the extended skeleton. LIB "all.lib"; LIB "poly.lib"; LIB "tropical.lib"; LIB "elim.lib"; ring rr = (0,t), (a2,a3,a4,a5, x,y,z),dp; poly f=y^2-x*(x-a2)*(x-a3)*(x-a4)*(x-a5); // f; // -a2*a3*a4*a5*x+a2*a3*a4*x^2+a2*a3*a5*x^2+a2*a4*a5*x^2+a3*a4*a5*x^2-a2*a3*x^3-a2*a4*x^3-a3*a4*x^3-a2*a5*x^3-a3*a5*x^3-a4*a5*x^3+a2*x^4+a3*x^4+a4*x^4+a5*x^4-x^5+y^2 setring(rr); poly b2 = (t^9+t^12); poly b3 = (t^5+t^9); poly b4 = (3*t^5+t^8); poly b5 = (1+t^5); poly A2 = b2^2; poly A3 = b3^2; poly A4 = b4^2; poly A5 = -1*b5^2; // A5*(A4+A3)*A2; // (-t^52-t^50-2*t^49-2*t^48-10*t^47-t^46-6*t^45-27*t^44-4*t^43-19*t^42-28*t^41-9*t^40-48*t^39-12*t^38-12*t^37-53*t^36-4*t^35-23*t^34-20*t^33-2*t^32-26*t^31-10*t^28) // A4+A3; // (t^18+t^16+2*t^14+6*t^13+10*t^10) map P2 = rr, A2, A3, A4, A5, x, y, z; poly ff = P2(f); ring r = (0,t),(x,y),dp; map newP2 = rr, 0,0,0,0,x,y,0; poly f2 = newP2(ff); // f2; // -x5+(t24+2t21+2t18+t16+2t14+6t13+9t10-2t5-1)*x4+(-t42-t40-2t39-2t38-8t37-t36-4t35-23t34-2t32-30t31-2t30+2t29-17t28-12t27+4t26-15t24+4t23+4t21+t20+4t19+14t18+t16+20t15+2t14+6t13+10t10)*x3+(t58+8t55+2t54+21t52+16t51+22t49+42t48-2t47+8t46+42t45-6t44-4t43-t42-10t41-11t40-50t39-12t38-24t37-66t36-8t35-42t34-44t33-26t32-34t31-11t30-36t29-31t28-12t27-t26-18t25-18t24-6t23-9t20)*x2+y2+(t68+8t65+2t64+2t63+22t62+16t61+17t60+28t59+45t58+52t57+41t56+58t55+72t54+88t53+56t52+58t51+97t50+68t49+53t48+44t47+57t46+48t45+22t44+18t43+18t42+24t41+9t38)*x drawTropicalCurve(f2,"max"); /////////////////// // XZ-projection // /////////////////// poly b2 = newP2(b2); poly b3 = newP2(b3); poly b4 = newP2(b4); poly b5 = newP2(b5); poly g2 = substitute(f2, y, y+b3*b4*b5*(1+t^3)*x-b5*(1+t^4)*x^2); // g2; // -x5+(t24+2t21+3t18+t16+4t14+8t13+10t10+4t9+t8+2t4)*x4+(-t42-t40-2t39-2t38-8t37-t36-4t35-25t34-2t32-38t31-6t30-2t29-23t28-28t27-14t26-8t25-29t24-16t23-32t22-8t21-9t20-20t19-8t18-16t17-t16+8t15-10t14-2t13+4t10)*x3+(-2t9-2t5-2t4-2)*x2y+(t58+8t55+2t54+21t52+16t51+t50+22t49+42t48+6t47+10t46+44t45+16t44+12t43+16t42+18t41+34t40+2t39+29t38+34t37+6t36+80t35+14t34+14t33+71t32+34t31+42t30+8t29+26t28+36t27+21t26+18t23)*x2+(2t25+8t22+2t21+2t20+6t19+8t18+8t17+2t16+6t15+6t14+8t13+6t10)*xy+y2+(t68+8t65+2t64+2t63+22t62+16t61+17t60+28t59+45t58+52t57+41t56+58t55+72t54+88t53+56t52+58t51+97t50+68t49+53t48+44t47+57t46+48t45+22t44+18t43+18t42+24t41+9t38)*x drawTropicalCurve(g2,"max"); /////////////////// // UY-projection // /////////////////// poly guy = substitute(f2, x, x + b4^2); // guy; // -x5+(t24+2t21+2t18-4t16+2t14-24t13-36t10-2t5-1)*x4+(-t42+3t40-2t39-2t38+24t37-t36-4t35+69t34-8t32+90t31+6t30-70t29+55t28+36t27-320t26+57t24-644t23-4t21-485t20+4t19-34t18-3t16-52t15+2t14-18t13-26t10)*x3+(-2t58+3t56-16t55-4t54+42t53-45t52-32t51+240t50-50t49-94t48+718t47-13t46-168t45+1212t44+68t43-547t42+1142t41+322t40-2072t39+501t38+678t37-4743t36+4t35+495t34-5750t33+43t32-262t31-2894t30+36t29-589t28+24t27-160t26-450t25+36t24-312t23-225t20)*x2+y2+(-t74+t72-14t71-2t70+20t69-80t68-28t67+173t66-236t65-161t64+844t63-371t62-494t61+2549t60-262t59-971t58+4928t57+151t56-1950t55+6057t54+1028t53-5407t52+4428t51+2485t50-12854t49+1533t48+3060t47-19657t46+150t45+947t44-17058t43+280t42-1968t41-6372t40-204t39-2547t38+216t37-1161t36-1296t35+162t34-1350t33-648t30)*x drawTropicalCurve(guy,"max"); /////////////////// // UZ-projection // /////////////////// poly guz = substitute(substitute(f2, y, y+b3*b4*b5*(1+t^3)*x-b5*(1+t^4)*x^2),x, x + b4^2); // guz; // -x5+(t24+2t21+3t18-4t16+4t14-22t13-35t10+4t9+t8+2t4)*x4+(-t42+3t40-2t39-2t38+24t37-t36-4t35+71t34-8t32+106t31+10t30-66t29+85t28+68t27-286t26+8t25+119t24-568t23+64t22+16t21-451t20+124t19+28t18+32t17-t16+8t15+62t14-2t13+4t10)*x3+(-2t9-2t5-2t4-2)*x2y+(-2t58+3t56-16t55-4t54+42t53-45t52-32t51+241t50-50t49-94t48+738t47-11t46-166t45+1342t44+108t43-506t42+1494t41+583t40-1792t39+914t38+1396t37-3915t36+524t35+1331t34-4696t33+1436t32+346t31-2301t30+1388t29+440t28+720t27-30t26+216t25+702t24+36t23+108t20)*x2+(-2t25-16t22-2t21-2t20-30t19-16t18-16t17-2t16-30t15-30t14-16t13-30t10)*xy+y2+(-t74+t72-14t71-2t70+20t69-80t68-28t67+173t66-236t65-161t64+848t63-371t62-494t61+2617t60-254t59-963t58+5384t57+287t56-1810t55+7585t54+1944t53-4427t52+7184t51+5585t50-9366t49+4977t48+8620t47-13041t46+6202t45+6611t44-10422t43+10284t42+3540t41-1728t40+6732t39+4041t38+5184t37+351t36+1944t35+3402t34+1134t33+972t30)*x+(-4t38-36t35-4t34-4t33-108t32-36t31-36t30-112t29-108t28-108t27-36t26-108t25-108t24-108t23-108t20)*y+(4t76+72t73+8t72+8t71+540t70+144t69+148t68+2176t67+1084t66+1152t65+5148t64+4400t63+4868t62+7992t61+10404t60+12024t59+11560t58+14904t57+17604t56+19512t55+15012t54+15984t53+23868t52+15552t51+12636t50+13824t49+14580t48+11664t47+4860t46+5832t45+5832t44+5832t43+2916t40) drawTropicalCurve(guz,"max"); /////////////////// // ZY-projection // /////////////////// ring s = (0,t),(x,y,z),dp; map P = r, x,y; poly ff= P(f2); poly b3 = P(b3); poly b4 = P(b4); poly b5 = P(b5); ideal I = (ff,z-y+b3*b4*b5*(1+t^3)*x-b5*(1+t^4)*x^2); poly fnox= eliminate(I,x)[1]; // Replace z by y and y by x so that the pictures are not flipped. ring r2 = (0,t),(x,y),dp; setring(r2); map PP = s,0,x,y; poly newfnox = PP(fnox); // newfnox; // 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drawTropicalCurve(newfnox,"max");