#### Volume 11, issue 2 (2011)

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 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Graphs of $20$ edges are $2$–apex, hence unknotted

### Thomas W Mattman

Algebraic & Geometric Topology 11 (2011) 691–718
##### 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.

##### Keywords
spatial graph, intrinsic knotting, apex graph
Primary: 05C10
Secondary: 57M15