Elliptic surfaces without 1-handles

Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1){2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n){p,q} has a handle decomposition without 1-handles for $n\ge 1$ and (p,q)=(2,3),(2,5),(3,4),(4,5).