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);

eqM := 1+x*y*w*t*((nu-1)*(y-1)+y*q)*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)
eqM := 1+x*y*w*t*((nu-1)*(y-1)+y*q)*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);

eqNM := 1+M(x,y)*M(1,1)/M(x,1)/M(1,y)*N(t*M(1,1)^2,x*M(x,1)/M(1,1),y*M(1,y)/M(1,1))-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);

1+u/(1+N(u,1))*v*w*s*((nu-1)*(v*(1+N(1,1))/(1+N(1,v))-1)+v*(1+N(1,1))/(1+N(1,v))*q)/(1-(1+N(1,1))*N(u,v)/(1+N(u,1))/(1+N(1,v)))+u*v/(1+N(1,v))*z*s*(u*(1+N(1,1))/(1+N(u,1))*nu-1)/(1-(1+N(1,1))*N(u,v)/(1...
1+u/(1+N(u,1))*v*w*s*((nu-1)*(v*(1+N(1,1))/(1+N(1,v))-1)+v*(1+N(1,1))/(1+N(1,v))*q)/(1-(1+N(1,1))*N(u,v)/(1+N(u,1))/(1+N(1,v)))+u*v/(1+N(1,v))*z*s*(u*(1+N(1,1))/(1+N(u,1))*nu-1)/(1-(1+N(1,1))*N(u,v)/(1...
1+u/(1+N(u,1))*v*w*s*((nu-1)*(v*(1+N(1,1))/(1+N(1,v))-1)+v*(1+N(1,1))/(1+N(1,v))*q)/(1-(1+N(1,1))*N(u,v)/(1+N(u,1))/(1+N(1,v)))+u*v/(1+N(1,v))*z*s*(u*(1+N(1,1))/(1+N(u,1))*nu-1)/(1-(1+N(1,1))*N(u,v)/(1...

>    eqN0:=numer(factor(%));nops(%);

eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...
eqN0 := -(1+N(1,1))*(-N(u,v)-u*v^2*w*s+u^2*v^2*w*s+u*v^3*w*s-u^2*v^3*w*s+u*v^2*w*s*q+u*v^2*w*s*nu-u^2*v^2*w*s*q-u^2*v^2*w*s*nu+u^2*v^3*w*s*q+u^2*v^3*w*s*nu-u*v^3*w*s*q-u*v^3*w*s*nu+u^2*v*z*s*nu+u^3*v^2...

3

>    op(1,eqN0);op(2,eqN0);

-1

1+N(1,1)

>    eqN1:=collect(op(3,eqN0),N,factor);

eqN1 := (-N(u,v)*u*v+u^2*v^2*s*(z*nu*u+v*w*q+v*w*nu-w*v))*N(1,1)^2+((N(u,v)*u-u^2*v*s*(z*nu*u+v*w*q+2*v*w*nu-2*w*v))*N(1,v)+(N(u,v)*v-u*v^2*s*(z*u+z*nu*u-w*v+v*w*nu+v*w*q))*N(u,1)+(u^2*v*z*s+u+v-u*v^2*...
eqN1 := (-N(u,v)*u*v+u^2*v^2*s*(z*nu*u+v*w*q+v*w*nu-w*v))*N(1,1)^2+((N(u,v)*u-u^2*v*s*(z*nu*u+v*w*q+2*v*w*nu-2*w*v))*N(1,v)+(N(u,v)*v-u*v^2*s*(z*u+z*nu*u-w*v+v*w*nu+v*w*q))*N(u,1)+(u^2*v*z*s+u+v-u*v^2*...
eqN1 := (-N(u,v)*u*v+u^2*v^2*s*(z*nu*u+v*w*q+v*w*nu-w*v))*N(1,1)^2+((N(u,v)*u-u^2*v*s*(z*nu*u+v*w*q+2*v*w*nu-2*w*v))*N(1,v)+(N(u,v)*v-u*v^2*s*(z*u+z*nu*u-w*v+v*w*nu+v*w*q))*N(u,1)+(u^2*v*z*s+u+v-u*v^2*...
eqN1 := (-N(u,v)*u*v+u^2*v^2*s*(z*nu*u+v*w*q+v*w*nu-w*v))*N(1,1)^2+((N(u,v)*u-u^2*v*s*(z*nu*u+v*w*q+2*v*w*nu-2*w*v))*N(1,v)+(N(u,v)*v-u*v^2*s*(z*u+z*nu*u-w*v+v*w*nu+v*w*q))*N(u,1)+(u^2*v*z*s+u+v-u*v^2*...
eqN1 := (-N(u,v)*u*v+u^2*v^2*s*(z*nu*u+v*w*q+v*w*nu-w*v))*N(1,1)^2+((N(u,v)*u-u^2*v*s*(z*nu*u+v*w*q+2*v*w*nu-2*w*v))*N(1,v)+(N(u,v)*v-u*v^2*s*(z*u+z*nu*u-w*v+v*w*nu+v*w*q))*N(u,1)+(u^2*v*z*s+u+v-u*v^2*...
eqN1 := (-N(u,v)*u*v+u^2*v^2*s*(z*nu*u+v*w*q+v*w*nu-w*v))*N(1,1)^2+((N(u,v)*u-u^2*v*s*(z*nu*u+v*w*q+2*v*w*nu-2*w*v))*N(1,v)+(N(u,v)*v-u*v^2*s*(z*u+z*nu*u-w*v+v*w*nu+v*w*q))*N(u,1)+(u^2*v*z*s+u+v-u*v^2*...

>    factor(subs(s=0,%));

N(u,v)*(v+v*N(1,1)-1-N(1,v))*(-u-u*N(1,1)+1+N(u,1))

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);

eqN2 := (q+nu-1)*s*w*u*v^2+u^2*v*z*s*nu+u*v*z*s*(N(u,v)-v*N(u,1))/(v+v*N(1,1)-1-N(1,v))+(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);

-(-u-u*N(1,1)+1+N(u,1))*(v+v*N(1,1)-1-N(1,v))

>   

>   

>   

>