2017/07/10 by Álvaro Lozano-Rojo, Lozano-Rojo, Álvaro, Rubén Vigara +1
Mathematics · #57M12 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M12
paper · pdf · doi:10.48550/arxiv.1707.02823
14 pages, 10 figures. Dedicated to Prof. Maite Lozano on her 70th anniversary
arxiv created 2017/07/10 · arxiv updated 2017/07/11
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" (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 S3 (Banchoff's sphere) diametrically splits the trefoil knot. Filling Dehn spheres, and their Johansson diagrams, are constructed for the coverings of S3 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.