# On dissolving knot surgery $4$-manifolds under a $\mathbb{CP}^2$-connected sum

## TL;DR

This paper proves that knot surgery 4-manifolds become diffeomorphic after a connected sum with ^2, and demonstrates that certain elliptic surfaces are almost completely decomposable after knot surgery.

## Contribution

It establishes a diffeomorphism classification of knot surgery 4-manifolds under ^2-sum and shows their almost complete decomposability for elliptic surfaces.

## Key findings

- All knot surgery 4-manifolds are mutually diffeomorphic after a ^2-sum.
- Knot surgery elliptic surfaces are almost completely decomposable.
- The results apply to simply connected elliptic surfaces E(n).

## Abstract

In this article we prove that, if $X$ is a smooth $4$-manifold containing an embedded double node neighborhood, all knot surgery $4$-manifolds $X_K$ are mutually diffeomorphic to each other after a connected sum with $\mathbb{CP}^2$. Hence, by applying to the simply connected elliptic surface $E(n)$, we also show that every knot surgery $4$-manifold $E(n)_K$ is almost completely decomposable.