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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))  );

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...
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...

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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);

1+y^2*w*(q+nu-1)*t+y*w*(2*w*y^3*nu^2+2*w*y^3*q^2+4*w*y^3*q*nu-4*w*y^3*q-4*w*y^3*nu+2*w*y^3-z*nu-y*x*z+nu*z*q+z*nu^2+y*x*z*q+y*x*nu^2*z)*t^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)));

series((-Q1(x)+2*z*t*nu*x*Q1(x)+z*t*nu*Q2(x))*y+O(y^2),y,2)

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

Q1sol := -z*t*nu*Q2(x)/(-1+2*x*nu*z*t)

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

Q2sol := -Q1(x)*(-1+2*x*nu*z*t)/z/t/nu

From now on,  z=w=1

>    eqQ:=subs(z=1,w=1,eqQ);

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

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

Q1sol := -t*nu*Q2(x)/(-1+2*t*nu*x)

Q2sol := -Q1(x)*(-1+2*t*nu*x)/t/nu

The kernel

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

Ker := -y^2*t*(-q+q*t*nu*x-nu+1)*Q(0,y)/(-1+t*nu*x)-y*t*x*(-1+2*nu)*Q1(x)-(nu-1)*y*t*Q2(x)+1-t/y-x*t+(nu-1)*y*t/(-1+t*nu*x)/x

The right-hand side

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

R := -(nu-1)*t*y/(1-t*nu*x)/x*Q(0,y)-t*Q1(x)+1-t/y-x*t

replace  Q2 by its expression in terms of Q1

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

Ker1 := -y^2*t*(-q+q*t*nu*x-nu+1)*Q(0,y)/(-1+t*nu*x)-y*(nu-1+t*nu*x)*Q1(x)/nu-t/y+(nu-1)*y*t/(-1+t*nu*x)/x+1-x*t

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

>    subs(t=0,numer(Ker));

-y*x

To find a second series, set  x=t*s

>    subs(x=t*s,Ker);

-y^2*t*(-q+q*t^2*s*nu-nu+1)*Q(0,y)/(-1+t^2*s*nu)-y*t^2*s*(-1+2*nu)*Q1(t*s)-(nu-1)*y*t*Q2(t*s)+1-t/y-t^2*s+(nu-1)*y/(-1+t^2*s*nu)/s

>    factor(subs(t=0,(%*y)));

-(-s+y*nu-y)*y/s

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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);

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

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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;

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;

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,%);

1-(q-4)*t*II

>    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;

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)