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.
The Powell Conjecture for the genus-three Heegaard splitting of the $3$-sphere
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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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On symmetry and exterior problems of knotted handlebodies
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.