A Short Exposition of Salman Parsa’s Theorems on Intrinsic Linking and Non-realizability

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.