  We show that the 14 graphs obtained by $\nabla\mathrm{Y}$ moves on K_7
constitute a complete list of the minor minimal intrinsically knotted graphs on
21 edges. We also present evidence in support of a conjecture that the 20 graph
Heawood family, obtained by a combination of $\nabla\mathrm{Y}$ and
$\mathrm{Y}\nabla$ moves on K_7, is the list of graphs of size 21 that are
minor minimal with respect to the property not 2--apex.
