Knot surgery $4$-manifolds $E(n)_K$ without $1$- and $3$-handles

Abstract

In this article, we demonstrate that for any positive integer $n$, the knot surgery $4$-manifold $E(n)_K$ has a handle decomposition without $1$- and $3$-handles. Here, $K$ represents either a fibered two-bridge knot $C(2\epsilon_1, 2\epsilon_2,\cdots, 2\epsilon_{2g})$ ($\epsilon_i \in \{ 1, -1\}$) in Conway's notation or a Stallings knot $K_m$ ($m \in \mathbb{Z}$).