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 ⩾ 1 and (p, q) = (2, 3), (2, 5), (3, 4), (4, 5).