# Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

## TL;DR

This paper establishes a criterion for identifying genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings, providing a new proof and detailed classification of such knots.

## Contribution

It introduces a necessary and sufficient condition for simple closed curves to decompose handlebodies, enabling classification of GOF-knots in specific 3-manifolds.

## Key findings

- Characterization of curves decomposing genus two handlebodies
- Determination of the number and positions of GOF-knots
- Alternative proof of Morimoto and Baker's results

## Abstract

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto and Baker.