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

TL;DR

This paper proves that for any positive integer n, the knot surgery 4-manifold E(n)_K can be decomposed into handles without 1- and 3-handles, for specific classes of knots K.

Contribution

It establishes the existence of handle decompositions without 1- and 3-handles for E(n)_K with particular knots K, extending understanding of 4-manifold structures.

Findings

Handle decompositions without 1- and 3-handles are possible for E(n)_K.

Applicable to fibered two-bridge knots and Stallings knots.

Advances the understanding of 4-manifold handle structures.

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 $K_m$ ($m \in \mathbb{Z}$).