Geodesic
Graphs of 20 edges are 2–apex, hence unknotted
Mattman, Thomas W1
1Department of Mathematics and Statistics, California State University at Chico, Chico CA 95929-0525, USA
Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 691-718
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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.

DOI : 10.2140/agt.2011.11.691
Keywords: spatial graph, intrinsic knotting, apex graph

Mattman, Thomas W  1

1 Department of Mathematics and Statistics, California State University at Chico, Chico CA 95929-0525, USA 
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Mattman, Thomas W. Graphs of 20 edges are 2–apex, hence unknotted. Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 691-718. doi: 10.2140/agt.2011.11.691

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