# Intrinsically Knotted and 4-Linked Directed Graphs

## TL;DR

This paper explores intrinsic linking and knotting in directed graphs, providing constructions of such graphs, introducing new minor operations, and analyzing their properties and edge bounds.

## Contribution

It introduces new operations for directed graph minors, constructs examples of intrinsically linked and knotted directed graphs, and analyzes their embedding properties.

## Key findings

- Existence of directed graphs with knotted cycles in every embedding
- Construction of intrinsically 3-linked and 4-linked directed graphs
- Edge bounds for non-intrinsically linked directed graphs

## Abstract

We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edge contraction and H-cyclic subcontraction, as special cases of minors for digraphs, and show that the property of having a linkless embedding is closed under these operations. We analyze the relationship between the number of distinct knots and links in an undirected graph $G$ and its corresponding symmetric digraph $\overline{DG}$. Finally, we note that the maximum number of edges for a graph that is not intrinsically linked is $O(n)$ in the undirected case, but $O(n^2)$ for directed graphs.