Analysis reveals that non-separating weak reducing pairs in Heegaard surfaces admit reducing spheres, suggesting current understanding of genus limits.
Given a Heegaard surface in the $3$-sphere, we show that any non-separating weak reducing pair for the surface admits a reducing sphere that separates the two disks of the pair if and only if the genus of the surface is at most $3$.
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Cho et al. (2025) studied this question.
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