Any smooth knot ${\mathbb{S}}^n\hookrightarrow {\mathbb{R}}^{n+2}$ is isotopic to a cubic knot contained in the canonical scaffolding of ${\mathbb{R}}^{n+2}$

Abstract

The $n$-skeleton of the canonical cubulation $\mathcal{C}$ of ${\mathbb{R}}^{n+2}$ into unit cubes is called the {\it canonical scaffolding} $\mathcal{S}$. In this paper, we prove that any smooth, compact, closed, $n$-dimensional submanifold of ${\mathbb{R}}^{n+2}$ with trivial normal bundle can be continuously isotoped by an ambient isotopy to a cubic submanifold contained in $\mathcal{S}$. In particular, any smooth knot ${\mathbb{S}}^n\hookrightarrow {\mathbb{R}}^{n+2}$ can be continuously isotoped to a knot contained in $\mathcal{S}$.

In this revised version we include more detailed, complete and self-contained proofs of our theorems