Intrinsic Linking of 2-complexes in $\mathbb{R}^4$

TL;DR

The paper constructs an infinite family of 2-complexes that are intrinsically linked in four-dimensional space, demonstrating that certain graph minors guarantee non-trivial linking in embeddings.

Contribution

It introduces a new class of intrinsically linked 2-complexes in four dimensions and connects graph minors to topological linking properties.

Findings

An infinite family of intrinsically linked 2-complexes is constructed.

Any embedding of the suspension of a graph with a K6 minor in R^4 contains linked cycles.

Embedding properties are linked to the presence of specific graph minors.

Abstract

We produce an infinite family of $2$-complexes that are intrinsically linked when embedded into four dimensions. In particular, we show that any embedding into $\mathbb{R}^4$ of the suspension of a graph containing $K_6$ as a minor contains a non-trivially linked 1 and 2-cycle.