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2011 Graphs of $20$ edges are $2$–apex, hence unknotted
Thomas W Mattman
Algebr. Geom. Topol. 11(2): 691-718 (2011). 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.

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Thomas W Mattman. "Graphs of $20$ edges are $2$–apex, hence unknotted." Algebr. Geom. Topol. 11 (2) 691 - 718, 2011. https://doi.org/10.2140/agt.2011.11.691

Information

Received: 29 October 2009; Accepted: 15 October 2010; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1216.05017
MathSciNet: MR2782541
Digital Object Identifier: 10.2140/agt.2011.11.691

Subjects:
Primary: 05C10
Secondary: 57M15

Rights: Copyright © 2011 Mathematical Sciences Publishers

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Vol.11 • No. 2 • 2011
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