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Banchoff’s sphere and branched covers over the trefoil

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

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” (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.

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Notes

  1. Of course, these and other similar questions became automatically solved after Perelman’s proof of the Poincaré Conjecture [21].

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Acknowledgements

Partially supported by the European Social Fund and Diputación General de Aragón (Grant E15 Geometría) and by MINECO grants MTM2013-46337-C2, MTM2016-77642-C2, MTM2013-45710-C2 and MTM2016-76868-C2-2-P.

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Authors and Affiliations

  1. Centro Universitario de la Defensa Zaragoza, Academia General Militar, Carretera de Huesca s/n., 50090, Zaragoza, Spain

    Álvaro Lozano Rojo & Rubén Vigara

  2. IUMA, Universidad de Zaragoza, Zaragoza, Spain

    Álvaro Lozano Rojo & Rubén Vigara

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  1. Álvaro Lozano Rojo
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  2. Rubén Vigara
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Correspondence to Rubén Vigara.

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Lozano Rojo, Á., Vigara, R. Banchoff’s sphere and branched covers over the trefoil. RACSAM 112, 751–765 (2018). https://doi.org/10.1007/s13398-017-0477-5

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