arXiv subjects

Rubén Vigara

Publications and source records attributed to Rubén Vigara.

4 recordsLinked to original sources

arXiv2026

On the Egyptian knotted rope

The Egyptian knotted rope has been historically considered a measuring/geometric tool. A naive change in the way the rope is held to tighten it turns the rope into a primitive calculating tool --- a sort of primitive abacus. The arithmetic skills of the knotted rope have a double interest. From an historical point of view, they fit perfectly with some of the basic ancient Egyptian computing techniques. From an educational point of view, they enhance the potential interest of the knotted rope as a useful manipulative in mathematical education.

math.HO↗

arXiv2017

Banchoff's sphere and branched covers over the trefoil

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.

math.GT↗

arXiv2014

On the subadditivity of Montesinos complexity of closed orientable 3-manifolds

A filling Dehn sphere $Σ$ in a closed 3-manifold $M$ is a sphere transversely immersed in $M$ that defines a cell decomposition of $M$. Every closed 3-manifold has a filling Dehn sphere. The Montesinos complexity of a $3$-manifold $M$ is defined as the minimal number of triple points among all the filling Dehn spheres of $M$. A sharp upper bound for the Montesinos complexity of the connected sum of two 3-manifolds is given.

math.GT↗

arXiv2004

A set of moves for Johansson representation of 3-manifolds. An outline

A Dehn sphere in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The Dehn sphere S fills M if it defines a cell-decomposition of M. The inverse image in S^{2} of the double curves of S is the Johansson diagram of S and if S fills M it is possible to reconstruct M from the diagram. A Johansson representation of M is the Johansson diagram of a filling Dehn sphere of M. In a recent paper of J. M. Montesinos it is proved that every closed 3-manifold has a Johansson representation coming from a nulhomotopic filling Dehn sphere. In this paper a set of moves for Johansson representations of 3-manifolds is given. In a forthcoming paper it is proved that this set of moves suffices for relating different Johansson representations of the same 3-manifold coming from nulhomotopic filling Dehn spheres. The proof of this result is outlined here.

math.GT↗