Volume 11, issue 2 (2011)

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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
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