More intrinsically knotted graphs with 22 edges and the restoring method

Abstract

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that K7 and the 13 graphs obtained from K7 by ∇Y moves are the only intrinsically knotted graphs with 21 edges. Also Kim, Lee, Lee, Mattman and Oh showed that there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5. In this paper we show that there are exactly five trianglefree intrinsically knotted graphs with 22 edges having exactly one degree 5 vertex. These are Cousin 29 of the K3,3,1,1 family, Cousins 97 and 99 of the E9 + e family and two others that were previously unknown.

Bipartite intrinsically knotted graphs with 23 edges

“…There are 92 known examples of size 22: 58 in the K 3,3,1,1 family, 33 in the E 9 +e family (Figure 2) and a 4-regular example due to Schwartz (see [3]). We are in the process of determining whether or not this is a complete list [7,9,10].…”