Planar maps weighted by their Potts polynomial, divided by q.
Catalytic variables: x (degree of the root vertex) and y (degree of the root face).
Other variables: t (edges), w (non-root vertices) and z (non-root faces).

Finally, nu counts monochromatic edges, and q is the number of colors.

>    restart:

The equation of Proposition 1.

>    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)+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)+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)

Expansion of the series M

>    Mser:=proc(n)
option remember:
if n=0 then 1 else
factor(series(subs(M(x,y)=Mser(n-1), M(x,1)=subs(y=1,Mser(n-1)), M(1,y)=subs(x=1,Mser(n-1)) ,eqM+M(x,y)),t,n+1));
fi:end:

>    Mser(1);

series(1+x*y*(z*nu*x+y*w*nu-y*w+y*w*q)*t,t)

For one-edge maps, one should find:  
   (q-1 + nu)  w x y^2  for the isthmus (2 vertices,  degree 1 for the root-vertex, degree 2 for the root face)
    nu x^2 y  z  for the loop (2 faces, degree 2 for the root-vertex, degree 1 for the root face).
Let us check.

>    normal((q-1+nu)*w*x*y^2+nu*x^2*y*z-coeff(Mser(1),t));

0

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From now on, we set z, the face variable, to 1.

>    eqM:=subs(z=1,eqM);

eqM := 1+x*y*w*t*((nu-1)*(y-1)+q*y)*M(x,y)*M(1,y)+x*y*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*t*(y*M(x,y)-M(x,1))/(y-1)-M(x,y)
eqM := 1+x*y*w*t*((nu-1)*(y-1)+q*y)*M(x,y)*M(1,y)+x*y*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*t*(y*M(x,y)-M(x,1))/(y-1)-M(x,y)

Expansion of the series M

>    Mser:=proc(n)
option remember:
if n=0 then 1 else
factor(series(subs(M(x,y)=Mser(n-1), M(x,1)=subs(y=1,Mser(n-1)), M(1,y)=subs(x=1,Mser(n-1)) ,eqM+M(x,y)),t,n+1));
fi:end:

>    Mser(1);

series(1+x*y*(x*nu+y*w*nu-y*w+y*w*q)*t,t)

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The kernel

>    Ker:=-coeff(eqM,M(x,y));

Ker := -x*y*w*t*((nu-1)*(y-1)+q*y)*M(1,y)-x*y*t*(x*nu-1)*M(x,1)-x^2*y*w*t*(nu-1)/(x-1)-x*y^2*t/(y-1)+1

The right-hand side

>    R:=coeff(eqM,M(x,y),0);

R := 1-x*y*w*t*(nu-1)*M(1,y)/(x-1)-x*y*t*M(x,1)/(y-1)

How many roots has Ker (when solved for y)? There exists only one power series in t, denoted Y_1, that cancels Ker.

>    factor(subs(t=0,numer(Ker)));

(x-1)*(y-1)

To obtain another root, we have to set x=1+t*s

>    factor(subs(t=0,numer(subs(x=1+t*s,Ker))));

-(y-1)*(-y*w+y*w*nu-s)

>    factor([solve(%,y)]);

[1, s/w/(nu-1)]

Expansion of the series  Y_i (Lemma 5). We replace M by its expansion (first terms) and then solve for y, using the package GFUN (available in the Algolib library).

>    n:=6:
res:=normal(subs(M(1,y)=convert(subs(x=1,Mser(n)),polynom), M(x,1)=convert(subs(y=1,Mser(n)),polynom),x=1+t*s,numer(Ker))):

>   

>    with(gfun):

>    algeqtoseries(res,t,y,2,true);

[series(1+(-s/(-w+w*nu-s))*t+O(t^2),t,2), series(s/w/(nu-1)+(-(-nu^2*s+nu*s+nu*q*w*s-s*q*w+s^2-s^2*nu-s^2*q+w*nu^3-3*w*nu^2+3*w*nu-3*w^2*nu^2+3*w^2*nu-w^2-w+w^2*nu^3)*s^2/w^2/(-w+w*nu-s)/(nu-1)^3)*t+O(...
[series(1+(-s/(-w+w*nu-s))*t+O(t^2),t,2), series(s/w/(nu-1)+(-(-nu^2*s+nu*s+nu*q*w*s-s*q*w+s^2-s^2*nu-s^2*q+w*nu^3-3*w*nu^2+3*w*nu-3*w^2*nu^2+3*w^2*nu-w^2-w+w^2*nu^3)*s^2/w^2/(-w+w*nu-s)/(nu-1)^3)*t+O(...

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The case q=2 et nu=0 (Section 2)

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Construction of the two invariants (Section 6)

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The two  invariants. The first one is I1:

>    I1:=y*t*w*q*M(1,y)-(1-y)/y+t*y/(y-1);

I1 := y*t*w*q*M(1,y)-(1-y)/y+t*y/(y-1)

The second is J=CC^(m/2) T_m(xsubs) with

>    xsubs:=-1/2*(nu-1)*(q-4)/C^(1/2)*yinv+1/2*(-2+2*nu+q)/C^(1/2)*II+1/2*(-2*q-4*nu+4+q*nu)/C^(1/2);

xsubs := -1/2*(nu-1)*(q-4)/C^(1/2)*yinv+1/2*(-2+2*nu+q)/C^(1/2)*II+1/2*(-2*q-4*nu+4+q*nu)/C^(1/2)

>    CC:=(-2*nu+q*nu+1+nu^2)*II^2-q*(nu+1)*II+(nu-1)*(q-4)*(w*q+nu-1)*t+q;

CC := (-2*nu+q*nu+1+nu^2)*II^2-q*(nu+1)*II+(nu-1)*(q-4)*(w*q+nu-1)*t+q

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The case q=1 (m=3, k=1)  (Section 10.1)

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A derivative we have to evaluate in Lemma 17

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The case q=2 (m=4,k=1) (Section 12.1)

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The case q=3 (m=6,k=1) : an equation with a single catalytic variable (Section 13.1)

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Interlude : planar maps by guessing and checking (Section 13.1, continued)

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The case q=3 and nu=0 : guessing the three unknown series M(1), M'(1) and M"(1)   (Section 13.1, continued)

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The case q=3 and nu=0 : conclusion (Section 13.1, the end)

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The case q=3 and nu=0 : asymptotics (Theorem 25)

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From separable maps to non-separable ones (Section 14.1)

>