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

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

2009-05-25

arxiv.org/abs/0905.4053v2

Mathematics

Geometric Topology

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

Scientific paper

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 ${\cal{S}}$.

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