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Juan Pablo Díaz

Publications and source records attributed to Juan Pablo Díaz.

6 recordsLinked to original sources
arXiv2026

A Constructive Cubical Realization of $n$-Dimensional Smooth Knots Inside the Menger $M^{n+2}_n$-continuum

We prove that every smooth $n$-dimensional knot in $\mathbb{R}^{n+2}$ can be ambiently isotoped into the Menger $n$-dimensional continuum. In contrast with classical embedding theorems for universal compacta, our construction is explicit and proceeds via cubical models, combining the cubical realization theorem of Boege--Hinojosa--Verjovsky with the affine self-similarity of the Menger continuum.

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arXiv2025

$N$-dimensional beaded necklaces and higher dimensional wild knots, invariant by a Schottky group

Starting with a smooth, non-trivial $n$-dimensional knot $K\subset\bS^{n+2}$, and a beaded $n$-dimensional necklace subordinated to $K$, we construct a wild knot with a Cantor set of wild points (\ie the knot is not locally flat in these points). The construction uses the conformal Schottky group acting on $\bS^{n+2}$, generated by inversions on the spheres which are the boundary of the ``beads''. We show that if $K$ is a fibered knot, then the wild knot is also fibered. We also study cyclic branched coverings along the wild knots. This work generalizes the result presented in [8].

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arXiv2017

Cubulated moves for 2-knots

In this paper, we prove that given two cubical links of dimension two in ${\mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respectively.

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↗