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A graph is called intrinsically knotted if every embedding of the graphcontains 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, showedthat $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves are theonly intrinsically knotted graphs with 21 edges. Also Kim, Lee, Lee, Mattmanand Oh showed that there are exactly three triangle-free intrinsically knottedgraphs 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 avertex with degree larger than 5. In this paper we show that there are exactlyfive triangle-free intrinsically knotted graphs with 22 edges having exactlyone degree 5 vertex. These are Cousin 29 of the $K_{3,3,1,1}$ family, Cousins97 and 99 of the $E_9+e$ family and two others that were previously unknown.

More intrinsically knotted graphs with 22 edges and the restoring method