A new intrinsically knotted graph with 22 edges

TL;DR

This paper identifies a new intrinsically knotted graph with 22 edges, expanding the classification of such graphs and revealing structural constraints related to vertex degrees.

Contribution

It introduces a previously unknown triangle-free intrinsically knotted graph with 22 edges and degree constraints, extending the understanding of intrinsically knotted graphs.

Findings

Exactly three triangle-free intrinsically knotted graphs with 22 edges are identified.

Two of these graphs are related to the $E_9+e$ family, and one is a new graph $M_{11}.

No such graphs exist with vertices of degree larger than 5.

Abstract

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves are the only intrinsically knotted graphs with 21 edges. In this paper we present the following results: there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Two are the cousins 94 and 110 of the $E_9+e$ family and the third is a previously unknown graph named $M_{11}$. These graphs are shown in Figure 3 and 4. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5.