# On 2-knots and connected sums with projective planes

## TL;DR

This paper extends Satoh's result on 2-knots, showing that certain connected sums with unknotted projective planes in 4-sphere simplify to the unknotted projective plane, and corrects an error in Satoh's proof.

## Contribution

It generalizes Satoh's theorem to all odd natural numbers and provides a correction to a previous proof regarding 2-bridge knots.

## Key findings

- Connected sum of n-twist spun sphere and unknotted projective plane simplifies to the unknotted projective plane for odd n.
- Provides a corrected proof for the case when the knot is a 2-bridge knot.
- Extends understanding of 2-knot connected sums in 4-sphere.

## Abstract

In this paper, we generalize a result of Satoh to show that for any odd natural $n$, the connected sum of the $n$-twist spun sphere of a knot $K$ and an unknotted projective plane in the 4-sphere is equivalent to the same unknotted projective plane. We additionally provide a fix to a small error in Satoh's proof of the case that $K$ is a 2-bridge knot.