# Intrinsic linking and knotting in tournaments

## TL;DR

This paper investigates the minimum size of tournaments needed to exhibit various intrinsic linking and knotting properties, providing bounds and introducing the concept of a consistency gap between directed and undirected graphs.

## Contribution

It establishes bounds for intrinsic linking and knotting in tournaments and introduces the concept of the consistency gap, advancing understanding of topological properties in directed graphs.

## Key findings

- Intrinsic linking at 8 vertices
- Intrinsic knottedness between 9 and 12 vertices
- Bounds for m-linked tournaments

## Abstract

A directed graph $G$ is $\textit{intrinsically linked}$ if every embedding of that graph contains a non-split link $L$, where each component of $L$ is a consistently oriented cycle in $G$. A $\textit{tournament}$ is a directed graph where each pair of vertices is connected by exactly one directed edge. We consider intrinsic linking and knotting in tournaments, and study the minimum number of vertices required for a tournament to have various intrinsic linking or knotting properties. We produce the following bounds: intrinsically linked ($n=8$), intrinsically knotted ($9 \leq n \leq 12$), intrinsically 3-linked ($10 \leq n \leq 23$), intrinsically 4-linked ($12 \leq n \leq 66$), intrinsically 5-linked ($15 \leq n \leq 154$), intrinsically $m$-linked ($3m \leq n \leq 8(2m-3)^2$), intrinsically linked with knotted components ($9 \leq n \leq 107$), and the disjoint linking property ($12 \leq n \leq 14$). We also introduce the $\textit{consistency gap}$, which measures the difference in the order of a graph required for intrinsic $n$-linking in tournaments versus undirected graphs. We conjecture the consistency gap to be non-decreasing in $n$, and provide an upper bound at each $n$.