On the number of links in a linearly embedded $K_{3,3,1}$

TL;DR

This paper characterizes the possible number of nontrivial 2-component links in linear embeddings of the graph $K_{3,3,1}$, establishing that the count can only be between 1 and 5 inclusive.

Contribution

It provides a complete classification of the possible numbers of nontrivial links in linear embeddings of $K_{3,3,1}$, a result not previously known.

Findings

Number of nontrivial links can only be 1, 2, 3, 4, or 5.

Existence of linear embeddings with each of these link counts.

No linear embedding can have more than 5 nontrivial links.

Abstract

We show there exists a linear embedding of $K_{3,3,1}$ with n nontrivial 2-component links if and only if n = 1, 2, 3, 4, or 5.