  We are interested in the algebraic intersection of closed curves of a given
length on translation surfaces. Namely, we study the quantity KVol which
measures how many times can two closed curves of a given length intersect. In
this paper, we construct families of translation surfaces in each connected
component of the minimal stratum $\mathcal{H}(2g-2)$ of the moduli space of
translation surfaces of genus $g \geq 2$ such that KVol is arbitrarily close to
the genus of the surface, which is conjectured to be the infimum of KVol on
$\mathcal{H}(2g-2)$.
