An intrinsically linked simplicial $n$-complex

Abstract

For any positive integer $n$, Lov\'{a}sz-Schrijver, Taniyama and Skopenkov provided examples of simplicial $n$-complexes that inevitably contain a nonsplittable two-component link of $n$-spheres, no matter how they are embedded into the Euclidean $(2n+1)$-space. In this paper, we introduce a new example of such a simplicial $n$-complex through a simple argument in piecewise linear topology and an application of the van Kampen--Flores theorem. Furthermore, we demonstrate the existence of additional such complexes through higher dimensional generalizations of the $\triangle Y$-exchange on graphs.