Graphs of 20 edges are 2-apex, hence unknotted
THOMAS W MATTMANA 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.
05C10; 57M15
IntroductionWe say that a graph is intrinsically knotted or IK if every tame embedding of the graph in R 3 contains a nontrivially knotted cycle. Blain et al [1] and Ozawa and Tsutsumi [9] independently discovered an important criterion for intrinsic knotting. Let H K 2 denote the join of the graph H and the complete graph on two vertices, K 2 . Proposition 1.1 [1; 9] A graph of the form H K 2 is IK if and only if H is nonplanar.A graph is called l -apex if it becomes planar after the deletion of at most l vertices (and their edges). The proposition shows that 2-apex graphs are not IK.It's known that many non-IK graphs are 2-apex. As part of their proof that intrinsic knotting requires 21 edges, Johnson, Kidwell and Michael [5] showed that every triangle-free graph on 20 or fewer edges is 2-apex and, therefore, not knotted. In the current paper, we show: This suggests the following: Question 1.5 Is every non-IK graph 2-apex?We answer the question in the negative by giving an example of a graph, E 9 , having nine vertices and 21 edges that is neither IK nor 2-apex. (We thank Ramin Naimi [8] for providing an unknotted embedding of E 9 , which appears as Figure 8 in Section 3. This graph is called N 9 in [4].) Further, we show that no graph on fewer than 21 edges, no graph on fewer than nine vertices, and no other graph on 21 edges and nine vertices has this property. In this sense, E 9 is the simplest possible counterexample to our Question.The notation E 9 is meant to suggest that this graph is a "cousin" to the set of 14 graphs derived from K 7 by triangle-Y moves (see Kohara and Suzuki [6]). Indeed, E 9 arises from a Y-triangle move on the graph F 10 in the K 7 family. Although intrinsic knotting is preserved under triangle-Y moves by Motwani, Raghunathan and Saran [7], it is not, in general, preserved under Y-triangle moves. For example, although F 10 is derived from K 7 by triangle-Y moves and, therefore, intrinsically knotted, the graph E 9 , obtained by a Y-triangle move on F 10 , has an unknotted embedding.Our analysis includes a classification of IK and 2-apex graphs on nine vertices and at most 21 edges. Such a graph is 2-apex unless it is E 9 , or, up to addition of degree zero vertices, one of four graphs derived from K 7 by triangle-Y moves [6]. (Here jGj denotes the number of vertices in the graph G and kGk is the number of edges.) Proposition 1.6 Let G be a graph with ...
