arXiv subjects

Rogelio Valdez

Publications and source records attributed to Rogelio Valdez.

3 recordsLinked to original sources

arXiv2026

Wild knots embedded in the Menger Sponge

In this paper, we provide explicit recursive constructions of infinitely many non-equivalent wild knots contained in the Menger sponge, in such a way that we can control their set of wild points that lies in a usual Cantor set contained in the Menger sponge. Furthermore, we show that wild knots of dynamically defined type arising from Kleinian group actions can be isotoped into the sponge. We want to emphasize that our approach is constructive and geometric.

math.GT↗

arXiv2025

Minimal area of the spun trefoil knot on the canonical cubulation of $\mathbb{R}^4$

We say that a \emph{cubical 2-knot} $K^{2}$ is an embedding of the 2-sphere in the 2-skeleton of the canonical cubulation of $\mathbb{R}^4$; in particular, $K^{2}$ is the union of $m(K^{2})$ unit squares, hence $m(K^{2})$ is its area. We define the minimal area of $K^{2}$ as the minimum over all the areas of cubical 2-knots isotopic to the given knot type. The minimal area of a cubical 2-knot is an invariant, and the following natural question arose: Given a knot type, what area is needed for a cubical 2-knot in the canonical cubulation of $\mathbb{R}^4$ to realise that type with minimal area? In this paper, we answer this question for the spun trefoil knot in the weakly minimal case.

math.GT↗

arXiv2017

Smoothing closed gridded surfaces embedded in ${\mathbb R}^4$

We say that a topological $n$-manifold $N$ is a cubical $n$-manifold if it is contained in the $n$-skeleton of the canonical cubulation $\mathcal{C}$ of ${\mathbb{R}}^{n+k}$ ($k\geq1$). In this paper, we prove that any closed, oriented cubical $2$-manifold has a transverse field of 2-planes in the sense of Whitehead and therefore it is smoothable by a small ambient isotopy.

math.GT↗