Banchoff's sphere and branched covers over the trefoil

TL;DR

This paper explores the use of filling Dehn spheres, particularly Banchoff's sphere, to analyze branched coverings over the trefoil knot, providing explicit constructions and new proofs of known results.

Contribution

It introduces a method using filling Dehn spheres to study branched coverings over knots, with explicit diagrams and detailed constructions for the trefoil case.

Findings

Banchoff's sphere diametrically splits the trefoil knot.

Explicit Johansson diagrams for cyclic and irregular coverings are constructed.

New proofs of known results about branched coverings over the trefoil are provided.

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.