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Any smooth $n$-dimensional knot $\mathbb{S}^n\hookrightarrow\mathbb{R}^{n+2}$ is isotopic to an $n$-knot contained in the Menger $M^{n+2}_n$-continuum

Juan Pablo Díaz, Gabriela Hinojosa, Alberto Verjovsky

TL;DR

The paper addresses whether smooth $n$-knots in $\mathbb{R}^{n+2}$ can be realized within the Menger $M^{n+2}_n$-continuum. It combines the cubical-isotopy result of Boege–Hinojosa–Verjovski, which places a smooth $N^n$ into the canonical scaffolding as a cubical submanifold, with an iterative subdivision construction that produces an inverse-limit embedding of $N^n$ into $M^{n+2}_n$ while preserving isotopy. The main theorem proves that any cubical $N^n$ in the $n$-skeleton admits an isotopic copy inside $M^{n+2}_n$, which implies that every smooth knot $\mathbb{S}^n \hookrightarrow \mathbb{R}^{n+2}$ is isotopic to an $n$-knot contained in the Menger continuum. This result extends the known universality of Menger spaces from topological spaces to the smooth category up to isotopy, revealing a rich family of smooth $n$-knots within fractal continua.

Abstract

In this paper, we say that an $n$-dimensional closed submanifold $N$ embedded in $\mathbb{R}^{n+2}$ is cubical if it is contained in the $n$-skeleton of the canonical cubulation of $\mathbb{R}^{n+2}$. It has been shown by the last two authors and M. Boege that any smooth $n$-dimensional closed submanifold of $\mathbb{R}^{n+2}$ can be deformed to a cubical $ n$-manifold by a global continuous isotopy of $\mathbb{R}^{n+2}$. Here, we prove that there is an isotopic copy of $N^n$ contained in the Menger $M^{n+2}_n$-continuum. In particular, any smooth knot $\mathbb{S}^n\hookrightarrow\mathbb{R}^{n+2}$ is isotopic to a knot contained in the Menger $M^{n+2}_n$-continuum.

Any smooth $n$-dimensional knot $\mathbb{S}^n\hookrightarrow\mathbb{R}^{n+2}$ is isotopic to an $n$-knot contained in the Menger $M^{n+2}_n$-continuum

TL;DR

The paper addresses whether smooth -knots in can be realized within the Menger -continuum. It combines the cubical-isotopy result of Boege–Hinojosa–Verjovski, which places a smooth into the canonical scaffolding as a cubical submanifold, with an iterative subdivision construction that produces an inverse-limit embedding of into while preserving isotopy. The main theorem proves that any cubical in the -skeleton admits an isotopic copy inside , which implies that every smooth knot is isotopic to an -knot contained in the Menger continuum. This result extends the known universality of Menger spaces from topological spaces to the smooth category up to isotopy, revealing a rich family of smooth -knots within fractal continua.

Abstract

In this paper, we say that an -dimensional closed submanifold embedded in is cubical if it is contained in the -skeleton of the canonical cubulation of . It has been shown by the last two authors and M. Boege that any smooth -dimensional closed submanifold of can be deformed to a cubical -manifold by a global continuous isotopy of . Here, we prove that there is an isotopic copy of contained in the Menger -continuum. In particular, any smooth knot is isotopic to a knot contained in the Menger -continuum.
Paper Structure (7 sections, 1 theorem, 10 figures)

This paper contains 7 sections, 1 theorem, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regular cubic honeycombs
  4. Cubical submanifolds
  5. The Menger $M^m_n$-continuum
  6. Main construction
  7. Final remarks

Key Result

Theorem 1

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

Figures (10)

  • Figure 1: The 3-dimensional cubical kaleidoscopic honeycomb $\{4,3,4\}$. This figure is courtesy of Roice Nelson RN.
  • Figure 2: Cubical figure eight knot.
  • Figure 3: First steps in the construction of Menger's sponge.
  • Figure 4: Schematic figure of the first steps in the construction of the $m$-dimensional Menger $M^m_n$-continuum.
  • Figure 5: $N^n$ embedded in the $n$-skeleton of the hypercube $Q^{n+2}(2^{m}-1)$.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Example 1
  • Theorem 1: Theorem A, Boege–Hinojosa–Verjovsky BHV
  • Remark 1
  • proof : Proof of Theorem \ref{['main1']}
  • proof : Proof of Corollary \ref{['main2']}