More intrinsically knotted graphs with 22 edges and the restoring method

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 $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ 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 triangle-free intrinsically knotted graphs with 22 edges having exactly one degree 5 vertex. These are Cousin 29 of the $K_{3,3,1,1}$ family, Cousins 97 and 99 of the $E_9+e$ family and two others that were previously unknown.