# Knotted surfaces in 4-manifolds by Knot surgery and Stabilization

## TL;DR

This paper demonstrates that various knotted surfaces in 4-manifolds, constructed via knot surgery, become diffeomorphic after a single stabilization, revealing a unifying structure in their classification.

## Contribution

It proves that all known examples of surface knots from knot surgery become diffeomorphic after one stabilization, extending understanding of surface equivalence in 4-manifolds.

## Key findings

- Surfaces become diffeomorphic after stabilization with $S^2\tilde{\times}S^2$
- Results apply to surfaces with fundamental groups preserved by knot surgery
- Additional results for spin 4-manifolds with cyclic fundamental group

## Abstract

Given a simply-connected closed 4-manifold $X$ and a smoothly embedded oriented surface $\Sigma$, various constructions based on Fintushel-Stern knot surgery have produced new surfaces in $X$ that are pairwise homeomorphic to $\Sigma$, but not diffeomorphic. We prove that for all known examples of surface knots constructed from knot surgery operations that preserve the fundamental group of the complement of surface knots, they become pairwise diffeomorphic after stabilizing by connected summing with one $S^2\tilde{\times}S^2$. When $X$ is spin, we show in addition that any surfaces obtained by a knot surgery whose complements have cyclic fundamental group become pairwise diffeomorphic after one stabilization by $S^2\tilde{\times}S^2$.