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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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.