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A filling Dehn surface in a $3$-manifold $M$ is a generically immersedsurface in $M$ that induces a cellular decomposition of $M$. Given a tame link$L$ in $M$ there is a filling Dehn sphere of $M$ that "trivializes"(\emph{diametrically splits}) it. This allows to construct filling Dehnsurfaces in the coverings of $M$ branched over $L$. It is shown that one of thesimplest filling Dehn spheres of $S^3$ (Banchoff's sphere) diametrically splitsthe trefoil knot. Filling Dehn spheres, and their Johansson diagrams, areconstructed for the coverings of $S^3$ branched over the trefoil. Theconstruction is explained in detail. Johansson diagrams for generic cycliccoverings and for the simplest locally cyclic and irregular ones areconstructed explicitly, providing new proofs of known results about cycliccoverings and the $3$-fold irregular covering over the trefoil.

Banchoff's sphere and branched covers over the trefoil