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The Powell Conjecture for the genus-three Heegaard splitting of the $3$-sphere

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arxiv 2402.07438 v2 pith:IYLL3YZO submitted 2024-02-12 math.GT

Sangbum Cho , Yuya Koda , Jung Hoon Lee This is my paper · ORCID
classification math.GT
keywords conjecturesphereheegaardpowellgroupreducingsplittingadditionally
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abstract

The Powell Conjecture states that the Goeritz group of the Heegaard splitting of the $3$-sphere is finitely generated; furthermore, four specific elements suffice to generate the group. Zupan demonstrated that the conjecture holds if and only if the reducing sphere complexes are all connected. In this work, we establish the connectivity of the reducing sphere complex for the genus-$3$ case, thereby confirming the Powell Conjecture in genus $3$. Additionally, we propose a potential framework for extending this approach to Heegaard splittings of higher genera.

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  1. On symmetry and exterior problems of knotted handlebodies

    math.GT 2025-06 conditional novelty 7.0 of 10

    For handlebody-knots with type 4-1 annuli, the symmetry group is Z2 or Z2 x Z2, and the exterior determines the handlebody-knot up to isotopy via annulus slopes.

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