Any smooth knot \mathbbSⁿ\hookrightarrowℝⁿ⁺² is isotopic to a cubic knot contained in the canonical scaffolding of ℝⁿ⁺²

2009/05/25 by Margareta Boege, Boege, Margareta, Gabriela Hinojosa +3
Computer Science · Mathematics · #52B7 #57Q45 #57R40 #57R52. #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #math.GT #math.MG #msc:52B7 #msc:57Q45 #msc:57R40 #msc:57R52.

paper · pdf · doi:10.48550/arxiv.0905.4053

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

openalex publication_date 2009/05/25 · arxiv created 2009/11/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The n-skeleton of the canonical cubulation \cal C of ℝⁿ⁺² into unit cubes is called the \it canonical scaffolding \calS. In this paper, we prove that any smooth, compact, closed, n-dimensional submanifold of ℝⁿ⁺² with trivial normal bundle can be continuously isotoped by an ambient isotopy to a cubic submanifold contained in \calS. In particular, any smooth knot \mathbbSⁿ\hookrightarrowℝⁿ⁺² can be continuously isotoped to a knot contained in \calS.

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