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

TL;DR

This paper proves that certain minimal simplicial complexes in 2n-dimensional space always contain a non-splittable link, extending classical graph embedding results to higher dimensions.

Contribution

It establishes the intrinsic linking property of minimal simplicial n-complexes in ra8^{2n}a8, generalizing known planar graph non-separability results.

Findings

Existence of unavoidable linked pairs in embedded complexes

Generalization of planar graph non-separability to higher dimensions

Intrinsic linking is inherent in minimal simplicial complexes

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.