Graphs of 20 edges are 2–apex, hence unknotted

Mattman, Thomas W

doi:10.2140/agt.2011.11.691

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Graphs of $20$ edges are $2$–apex, hence unknotted

Thomas W Mattman

Algebraic & Geometric Topology 11 (2011) 691–718

DOI: 10.2140/agt.2011.11.691

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

Mathematical Subject Classification 2000

Primary: 05C10

Secondary: 57M15

References

Publication

Received: 29 October 2009

Accepted: 15 October 2010

Published: 11 March 2011

Authors

Thomas W Mattman
Department of Mathematics and Statistics California State University at Chico Chico CA 95929-0525 USA

Volume 11, issue 2 (2011)