  We show that the 20 graph Heawood family, obtained by a combination of
triangle-Y and Y-triangle moves on $K_7$, is precisely the set of graphs of at
most 21 edges that are minor minimal for the property not $2$--apex. As a
corollary, this gives a new proof that the 14 graphs obtained by triangle-Y
moves on $K_7$ are the minor minimal intrinsically knotted graphs of 21 or
fewer edges. Similarly, we argue that the seven graph Petersen family, obtained
from $K_6$, is the set of graphs of at most 17 edges that are minor minimal for
the property not apex.
