> restart:

Quasi-triangulations, weighted by their Potts polynomial, divided by q.
Catalytic variables catalytiques : x for internal digons,  y for the degree of the root face
Edges t, non-root vertices w, non-root- faces z.

nu for monochromatic edges, q for the number of colours

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Denote Q1(x)=coeff(Q(x,y),y,1), Q2(x)=coeff(Q(x,y),y,2)

 > eqQ:=-Q(x,y)+1+z*t/y*(Q(x,y)-1-y*Q1(x))+q*y^2*w*t*Q(x,y)*Q(0,y) +x*y*z*t*Q1(x)*Q(x,y)+x*z*t*(Q(x,y)-1) +(nu-1)*(y*z*t*Q(x,y)*(2*x*Q1(x)+Q2(x)) +y^2*w*t/(1-x*nu*z*t)*Q(0,y)*Q(x,y) +w*t*y/(1-x*nu*z*t)/x*(Q(x,y)-Q(0,y))  );

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Series expansion of Q

 > Qser:=proc(n) option remember: if n=0 then 1 else convert(factor(series(subs(Q(x,y)=Qser(n-1), Q(0,y)=subs(x=0,Qser(n-1)), Q1(x)=coeff(Qser(n-1),y,1),Q2(x)=coeff(Qser(n-1),y,2), eqQ+Q(x,y)),t,n+1)),polynom): fi: end:

 > Qser(2);

The coefficient of t is  (q+nu-1)y ^2 w t for the isthmus

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If we weight instead by the Tutte polynomial. The case of spanning trees

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Connection between  Q1 and Q2 : perform an expansion in y

 > factor(subs(Q(x,0)=1,D[2](Q)(x,0)=Q1(x),Q(0,0)=1,D[2,2](Q)(x,0)=2*Q2(x),series(eqQ,y,3)));

 > Q1sol:=solve(coeff(%,y,1),Q1(x));

 > Q2sol:=solve(coeff(%%,y,1),Q2(x));

From now on,  z=w=1

 > eqQ:=subs(z=1,w=1,eqQ);

 > Q1sol:=subs(z=1,w=1,Q1sol); Q2sol:=subs(z=1,w=1,Q2sol);

The kernel

 > Ker:=map(factor,collect(-coeff(eqQ,Q(x,y),1),[Q,Q1,Q2]));

The right-hand side

 > R:=collect(coeff(eqQ,Q(x,y),0),[Q,Q1]);

replace  Q2 by its expression in terms of Q1

 > Ker1:=map(factor,collect(subs(Q2(x)=Q2sol,Ker),[Q,Q1]));

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Let us look for series  Y(t) that cancel the kernel

 > subs(t=0,numer(Ker));

To find a second series, set  x=t*s

 > subs(x=t*s,Ker);

 > factor(subs(t=0,(%*y)));

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Expansion of the séeries Y_i

 > n:=6:res:=subs(x=t*s,subs(Q(0,y)=subs(x=0,Qser(n)), Q1(x)=coeff(Qser(n),y,1),Q2(x)=coeff(Qser(n),y,2),w=1,z=1,numer(Ker))):

 > with(gfun):

 > algeqtoseries(res,t,y,2,true);

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Invariants (Section 7)

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First  invariant

 > I1:=t*y*Q(0,y)*q-1/y+t/y^2;

The second one is  C^(m/2) T_m(xsubs) with

 > CC:=t^2*q*(1+beta)^2*II^2+beta*(4*beta+q)*t*II+q*beta*(1+beta)*(q-4)*t^3+beta^2;

 > subs(beta=-1,%);

 > xsubs:=1/2*(-q+q*(1+beta)+t*q*(1+beta)*II-2*beta)/C^(1/2)-1/2*t*(q-4)*beta/C^(1/2)/y;

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The case q=1 (m=3,k=1) (Section 11.1)

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Some minor calculations in Section 11 (Lemmas 19 and 20)

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The case q=2 (m=4,k=1) (Section 12.2)

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The case q=3 (m=6,k=1) (Section 13.2)

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Back to Tutte's original equation (Section 14.2)