An intrinsically linked simplicial $n$-complex

TL;DR

The paper investigates intrinsic linking phenomena for high-dimensional simplicial complexes embedded in ${\mathbb R}^{2n+1}$ and introduces a new explicit family $K^{(n)}$ built from Type I/II $n$-simplices, showing that every embedding yields a nonsplittable link of an $n$-tetrahedron and an $n$-octahedron with parity $1$ modulo $2$. It extends the toolkit with a higher-dimensional ${\triangle}Y(n)$-exchange and proves a transfer principle that preserves intrinsic linking under these operations, producing many new examples. Collectively, these results generalize classical 3D intrinsic linking (Conway–Gordon–Sachs, Petersen family) to the high-dimensional setting and connect to the van Kampen–Flores non-embeddability, offering a constructive framework for intrinsically linked high-dimensional simplicial complexes.

Abstract

For any positive integer $n$, Lovász-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.

Figures (7)

• Figure 1.1: Petersen family (Each arrow represents a $\triangle Y$-exchange)
• Figure 1.2: $\triangle Y$-exchange
• Figure 1.3: $|a_{0}^{0}a_{2}^{1}a_{0}^{n}|$ of Type I and $|ba_{0}^{1}a_{1}^{n}|$ of Type II ($n=2$)
• Figure 1.4: $K^{(1)}=K_{3,3,1}=P_{7}$
• Figure 1.5: $n$-tetrahedron, $n$-octahedron ($n=2$)

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 2.1
• Remark 2.2
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['newIL']}