Number of connected components in the space of Pell–Abel equations admitting fixed degree primitive solution

Pell–Abel equation is the functional reincarnation of the known diophantine equation
$$ P^2-DQ^2=1, $$
where $P,Q$ and $D$ are complex polynomials. Monic $D$ is known and has no multiple roots; $P$ and $Q$ have to be found. Given $D$, the set of nontrivial solutions $(P,Q)\neq (1,0)$ is generated by the so called primitive solution with minimal $\operatorname{deg}P>0$. We use pictorial calculus of weighted planar graphs to calculate the number of connected components in the space of equations with fixed degrees of $D$ and the primitive solution.