Topological type of discriminants of some special families

We will describe the topological type of the discriminant curve of the morphism $(\mathrm{\ell },f)$, where $\mathrm{\ell }$ is a smooth curve and $f$ is an irreducible curve (branch) of multiplicity less than five or a branch that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.