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

The $n$-skeleton of the canonical cubulation $\cal C$ of $\mathbb{ R}^{ n+2}$ into unit cubes is called the { \it canonical scaffolding} ${ \cal{ S}}$. 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 ${ \cal{ S}}$. In particular, any smooth knot $\mathbb{ S}^{ n}\hookrightarrow\mathbb{ R}^{ n+2}$ can be continuously isotoped to a knot contained in ${ \cal{ S}}$.

Categories: math.GT math.MG