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

TL;DR

This paper presents an explicit cubical construction demonstrating that every smooth n-dimensional knot in Euclidean space can be embedded into the Menger n-dimensional continuum, utilizing cubical models and self-similarity.

Contribution

It introduces a constructive cubical realization method for embedding smooth knots into the Menger continuum, extending classical embedding theorems with an explicit approach.

Findings

Every smooth n-dimensional knot can be ambiently isotoped into the Menger n-dimensional continuum.

The construction combines cubical realization theorems with the affine self-similarity of the Menger continuum.

The method provides an explicit, constructive embedding process.

Abstract

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.