Exactly fourteen intrinsically knotted graphs have $21$ edges

Abstract

Johnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least $21$ edges. Also it is known that $K_7$ and the thirteen graphs obtained from $K_7$ by $\nabla Y$ moves are intrinsically knotted graphs with $21$ edges. We prove that these 14 graphs are the only intrinsically knotted graphs with $21$ edges.