From separable maps to non-separable ones (Section 14.1)
Planar maps weighted by their Potts polynomial (divided by q)
> | eqM:=1+x*y*w*t*((nu-1)*(y-1)+q*y)*M(x,y)*M(1,y) + x*y*z*t*(x*nu-1)*M(x,y)*M(x,1) +x*y*w*t*(nu-1)*(x*M(x,y)-M(1,y))/(x-1)+ x*y*z*t*(y*M(x,y)-M(x,1))/(y-1)-M(x,y); |
Connection with non-separable maps (series N, Prop. 28)
> | eqNM:=1+M(x,y)*M(1,1)/M(x,1)/M(1,y)*subs(t=t*M(1,1)^2, x=x*M(x,1)/M(1,1), y=y*M(1,y)/M(1,1), N(t,x,y)) -M(x,y); |
From M to N
> | subs(M(x,1)=1+N(u,1), M(1,y)=1+N(1,v),M(1,1)=1+N(1,1), M(x,y)= 1/(1-(1+N(1,1))*N(u,v)/(1+N(u,1))/(1+N(1,v)) ), t=s/M(1,1)^2, x=u*M(1,1)/M(x,1), y=v*M(1,1)/M(1,y), M(x,1)=1+N(u,1), M(1,y)=1+N(1,v),M(1,1)=1+N(1,1), eqM); |
> | eqN0:=numer(factor(%));nops(%); |
> | op(1,eqN0);op(2,eqN0); |
> | eqN1:=collect(op(3,eqN0),N,factor); |
> | factor(subs(s=0,%)); |
Version of the paper
> | eqN2:=(q+nu-1)*s*w*u*v^2+nu*s*z*u^2*v + u*v*z*s*(N(u,v)-v*N(u,1))/(v-1-N(1,v)+v*N(1,1)) +(nu-1)* u*v*w*s* (N(u,v)-u*N(1,v))/(u-1-N(u,1)+u*N(1,1)) -N(u,v); |
> | factor(eqN1/eqN2); |
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