Unknotting annuli and handlebody-knot symmetry

Abstract

By Thurston's hyperbolization theorem, irreducible handlebody-knots are classified into three classes: hyperbolic, toroidal, and atoroidal cylindrical. It is known that a non-trivial handlebody-knot of genus two has a finite symmetry group if and only if it is atoroidal. The paper investigates the topology of cylindrical handlebody-knots of genus two that admit an unknotting annulus; we show that the symmetry group is trivial if the unknotting annulus is unique and of type $2$.