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Intrinsic linking of a simplicial $n$-complex embedded in $\mathbb{R}^{2n}$

Ryo Nikkuni

Abstract

We demonstrate the existence of minimal simplicial $n$-complexes which inevitably contain a nonsplittable two-component link formed by an $(n-1)$-sphere and an $n$-sphere in any embedding into $\mathbb{R}^{2n}$. This provides a higher-dimensional generalization of graphs that are not non-separating planar.

Intrinsic linking of a simplicial $n$-complex embedded in $\mathbb{R}^{2n}$

Abstract

We demonstrate the existence of minimal simplicial -complexes which inevitably contain a nonsplittable two-component link formed by an -sphere and an -sphere in any embedding into . This provides a higher-dimensional generalization of graphs that are not non-separating planar.
Paper Structure (1 section, 1 theorem, 8 equations, 4 figures)

This paper contains 1 section, 1 theorem, 8 equations, 4 figures.

Table of Contents

  1. Proof of Theorem \ref{['ILR2n']}

Key Result

Theorem 1.1

(van Kampen VK33, Flores Flores32) Let $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\varphi$ of $K$ into ${\mathbb R}^{2n}$, the following holds:

Figures (4)

  • Figure 1: $K_1 \sqcup K_4$, $K_1 \sqcup K_{2,3}$ and $K_{1,1,3}$
  • Figure 1.1: $\lambda=\gamma\sqcup \gamma'\in \Lambda^{1,2}(N_{1}^{(2)})$
  • Figure 1.2: $\lambda=\gamma\sqcup \gamma'\in \Lambda^{1,2}(N_{2}^{(2)})$
  • Figure 1.3: $\mu=\gamma\sqcup \gamma'\in M^{1,2}(N_3^{(2)})$

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1.1
  • proof : Proof of Theorem \ref{['ILR2n']}
  • proof : Proof of Proposition \ref{['min']}
  • Remark 1.3