# Intrinsic linking and knotting are arbitrarily complex in directed graphs

## TL;DR

This paper demonstrates that directed graphs can have arbitrarily complex intrinsic linking and knotting, with large complete digraphs containing links of components with high linking numbers and knot complexity.

## Contribution

It extends previous results by proving that directed graphs can exhibit arbitrarily complex linking and knotting, analogous to undirected graphs, with specific bounds on linking numbers and knot invariants.

## Key findings

- Every embedding of a sufficiently large complete digraph contains links with high linking numbers.
- Such embeddings also contain knots with large second Conway polynomial coefficients.
- The results generalize known undirected graph knotting/linking complexity to 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 $Q_i$.