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