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.
Essential annuli in genus two handlebody exteriors
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abstract
We classify all potential configurations of essential annuli in a genus two atoroidal handlebody exterior in the $3$-sphere, building on two recent classifications: the classification of the JSJ-graph of the exterior and the classification of essential annuli in the exterior. In contrast to knots, genus two handlebody exteriors may contain infinitely many non-isotopic essential annuli, due to the JSJ-graph classification. Our main result characterizes the numbers of different types of essential annuli in such an infinite family.
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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.