Intrinsically triple-linked graphs in RP^3

TL;DR

This paper explores the intrinsic triple-linking properties of graphs in real projective 3-space, showing that while some graphs can be embedded without triple links, $K_{10}$ is inherently triple-linked in $ pt$, extending known results.

Contribution

It demonstrates that $K_{10}$ is intrinsically triple-linked in $ pt$, contrasting with its embedding properties in $ r^3$, and extends understanding of linkings in different 3-manifolds.

Findings

$K_{10}$ is intrinsically triple-linked in $ pt$

Some graphs can be embedded 3-linklessly in $ pt$

Extension of intrinsic linking results to projective space

Abstract

Flapan--Naimi--Pommersheim showed that every spatial embedding of $K_{10}$, the complete graph on ten vertices, contains a non-split three-component link; that is, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}^3$. The work of Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known intrinsically triple-linked graphs in $\mathbb{R}^3$ to include several other families of graphs. In this paper, we will show that while some of these graphs can be embedded 3-linklessly in $\mathbb{R}P^3$, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}P^3$.