The so-called "pinched disk" model of the Mandelbrot set is due to A.~Douady,
J.~H.~Hubbard and W.~P.~Thurston. It can be described in the language of
geodesic laminations. The combinatorial model is the quotient space of the unit
disk under an equivalence relation that, loosely speaking, "pinches" the disk
in the plane (whence the name of the model). The significance of the model lies
in particular in the fact that this quotient is planar and therefore can be
easily visualized. The conjecture that the Mandelbrot set is actually
homeomorphic to this model is equivalent to the celebrated MLC conjecture
stating that the Mandelbrot set is locally connected.
For parameter spaces of higher degree polynomials no combinatorial model is
known. One possible reason may be that the higher degree analog of the MLC
conjecture is known to be false. We investigate to which extent a geodesic
lamination is determined by the location of its critical sets and when
different choices of critical sets lead to essentially the same lamination.
This yields models of various parameter spaces of laminations similar to the
"pinched disk" model of the Mandelbrot set.