Number of connected components in the space of Pell-Abel equations admitting primitive solution of given degree

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 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.