Graphs of 20 edges are 2-apex, hence unknotted

TL;DR

This paper proves that all graphs with 20 or fewer edges are 2-apex and not intrinsically knotted, and it classifies certain minimal IK graphs, establishing new bounds and classifications in graph theory.

Contribution

It establishes that graphs with at most 20 edges are 2-apex, providing a new proof that IK graphs require at least 21 edges, and classifies IK graphs on nine vertices and 21 edges.

Findings

Graphs with 20 or fewer edges are 2-apex.

An example of a non-IK, non-2-apex graph on nine vertices and 21 edges is identified.

No new minor minimal IK graphs are found among nine-vertex, 21-edge graphs.

Abstract

A graph is 2-apex if it is planar after the deletion of at most two vertices. Such graphs are not intrinsically knotted, IK. We investigate the converse, does not IK imply 2-apex? We determine the simplest possible counterexample, a graph on nine vertices and 21 edges that is neither IK nor 2-apex. In the process, we show that every graph of 20 or fewer edges is 2-apex. This provides a new proof that an IK graph must have at least 21 edges. We also classify IK graphs on nine vertices and 21 edges and find no new examples of minor minimal IK graphs in this set.