Intrinsic linking and knotting are arbitrarily complex in directed graphs

Abstract

Fleming and Foisy recently proved the existence of a digraph whose every embedding contains a $4$-component link, and left open the possibility that a directed graph with an intrinsic $n$-component link might exist. We show that, indeed, this is the case. In fact, much as Flapan, Mellor, and Naimi show for graphs, knotting and linking are arbitrarily complex in directed graphs. Specifically, we prove the analog for digraphs of the main theorem of their paper: for any $n$ and $\alpha$, every embedding of a sufficiently large complete digraph in $\mathbb{R}^3$ contains an oriented link with components $Q_1, \ldots, Q_n$ such that, for every $i \neq j$, $|\mathrm{lk}(Q_i,Q_j)| \geq \alpha$ and $|a_2(Q_i)| \geq \alpha$, where $a_2(Q_i)$ denotes the second coefficient of the Conway polynomial of $Q_i$.