Title: Banchoff's sphere and branched covers over the trefoil,
Abstract: A filling Dehn surface in a $3$-manifold $M$ is a generically immersed
surface 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 Dehn
surfaces in the coverings of $M$ branched over $L$. It is shown that one of the
simplest filling Dehn spheres of $S^3$ (Banchoff's sphere) diametrically splits
the trefoil knot. Filling Dehn spheres, and their Johansson diagrams, are
constructed for the coverings of $S^3$ branched over the trefoil. The
construction is explained in detail. Johansson diagrams for generic cyclic
coverings and for the simplest locally cyclic and irregular ones are
constructed explicitly, providing new proofs of known results about cyclic
coverings and the $3$-fold irregular covering over the trefoil.