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

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.

11 pages, 6 figures