  In this paper, we prove that the trisection genus of the Akbulut cork is $3$
and construct infinitely many corks with trisection genus $3$. These results
give the first examples of contractible $4$-manifolds whose trisection genera
are determined except for the $4$-ball. We also give a lower bound for the
trisection genus of a $4$-manifold with boundary. In addition, we construct low
genus relative trisection diagrams of an exotic pair of simply-connected
$4$-manifolds with $b_2 = 1$.
