Any smooth knot \(\mathbb S^{n} \hookrightarrow \mathbb R^{n+2}\) is isotopic to a cubic knot contained in the canonical scaffolding of \(\mathbb R^{n+2}\)
type: Item
instance of: scholarly article
title: Any smooth knot \(\mathbb S^{n} \hookrightarrow \mathbb R^{n+2}\) is isotopic to a cubic knot contained in the canonical scaffolding of \(\mathbb R^{n+2}\)
zbMATH Open document ID: 1241.57030
author: Margareta Boege
author: Gabriela Hinojosa
author: Alberto Verjovsky
published in: Revista Matemática Complutense
publication date: 2011-04-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.4053
review text: The canonical cubulation \(\mathcal{C}\) of \(\mathbb{R}^{n+2}\) is the decomposition into hypercubes which are the images of the unit cube \(I^{n+2}=\{(x_1, \ldots, x_{n+2}) \mid 0\leq x_i \leq1\}\) by translations by vectors with integer coefficients. The canonical scaffolding of \(\mathbb{R}^{n+2}\), denoted by \(\mathcal{S}\), is the \(n\)-skeleton of \(\mathcal{C}\) i.e. the union of all cubes of dimension \(n\) contained in the faces of the \((n+2)\)-cubes in \(\mathcal{C}\). The main results are as follows. Any smooth, compact, closed, \(n\)-dimensional submanifold of \(\mathbb{R}^{n+2}\) with trivial normal bundle can be continuously isotoped by an ambient isotopy of \(\mathbb{R}^{n+2}\) onto a cubic submanifold contained in \(\mathcal{S}\) (Theorem 2.9). In particular, any smooth knot \(\mathbb{S}^n \hookrightarrow \mathbb{R}^{n+2}\) can be isotoped onto a cubic knot contained in \(\mathcal{S}\) (Theorem 3.1). Note that Theorem 3.1 implies that smooth knots can be embedded as cubic sub-complexes of \(\mathbb{R}^{n+2}\), which in turn implies the fact that smooth knots can be triangulated by a PL triangulation; see \textit{S. Cairns} [Bull. Am. Math. Soc. 67, 389--390 (1961; Zbl 0192.29901)].
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57R52
zbMATH DE Number: 5875639
zbMATH Keywords: high dimensional knots
zbMATH Keywords: cubic complexes
reviewed by: Inasa Nakamura
MaRDI profile type: Publication
OpenAlex ID: W2090334051
arXiv ID: 0905.4053
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DOI: 10.1007/S13163-010-0037-4
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description: scientific article