Exotic elliptic surfaces without 1-handles

In this article, we consider a sufficient condition that a knot-surgery or log-transformation of $E(n)$ admits a handle decomposition without 1-handles. We show that if $K$ is a knot that the bridge number is $b(K)\le 9n$, then the knot-surgery $E(n{)}_K$ of the elliptic surface $E(n)$ admits a handle decomposition without 1-handles. This means that if $\gcd (p,q)=1$, and $\min \{p,q\}\le 9$, then $E(1{)}_{p,q}$ admits a handle decomposition without 1-handles. We also show that if $\gcd (p,q)=1$, $\min \{p,q\}\le 4$, then the double log-transformation $E(n{)}_{p,q}$ admits a handle decomposition without 1-handles for any positive integer $n$.