A note on reducing spheres for the genus-4 Heegaard surface in the 3-sphere

TL;DR

This paper provides a sufficient condition for separating weak reducing pairs in genus-4 Heegaard surfaces in the 3-sphere, simplifying the connectivity analysis of the reducing sphere complex.

Contribution

It introduces a criterion that reduces the connectivity problem in the reducing sphere complex to a more manageable disjointness-based problem.

Findings

Established a sufficient condition for separating weak reducing pairs.

Reduced the connectivity problem to a disjointness condition.

Simplified the analysis of the reducing sphere complex for genus-4 surfaces.

Abstract

For the genus-$4$ Heegaard surface in the $3$-sphere, we present a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere for the surface. As a consequence, we reduce the connectivity problem in the reducing sphere complex for the surface to the problem of showing that any two vertices, whose representative reducing spheres are disjoint from a fixed non-separating compressing disk for the surface, are connected in the complex.