  Harer, Kas and Kirby have conjectured that every handle decomposition of the
elliptic surface $E(1)_{2,3}$ requires both 1- and 3-handles. In this article,
we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as
$E(1)_{2,3}$ and admits neither 1- nor 3-handles, by using rational blow-downs
and Kirby calculus. Our manifold gives the first example of either a
counterexample to the Harer-Kas-Kirby conjecture or a homeomorphic but
non-diffeomorphic pair of simply connected closed smooth 4-manifolds with the
same non-vanishing Seiberg-Witten invariants.
