A short exposition of S. Parsa's theorems on intrinsic linking and non-realizability

TL;DR

This paper summarizes S. Parsa's theorems on conditions for embedding graphs and complexes into Euclidean spaces, focusing on intrinsic linking and non-realizability, with implications for topological graph theory.

Contribution

It provides a concise exposition of Parsa's results on embeddings, linking, and simplicial complex size bounds in Euclidean spaces.

Findings

PL embedding of join graphs into R^4 implies embeddability into R^3 with zero linking number

Bound on the number of 2-simplices in complexes embeddable in R^4

Analogous results for intrinsic linking in complexes

Abstract

We present a short exposition of the following results by S. Parsa. Let $L$ be a graph such that the join $L*\{1,2,3\}$ (i.e. the union of three cones over $L$ along their common bases) piecewise linearly (PL) embeds into $\mathbb R^4$. Then $L$ admits a PL embedding into $\mathbb R^3$ such that any two disjoint cycles have zero linking number. There is $C$ such that every 2-dimensional simplicial complex having $n$ vertices and embeddable into $\mathbb R^4$ contains less than $Cn^{8/3}$ simplices of dimension 2. We also present the analogue of the second result for intrinsic linking.