  A recent result of Funayoshi-Koda shows that a handlebody-knot of genus two
has a finite symmetry group if and only if it is hyperbolic -- the exterior
admits a hyperbolic structure with totally geodesic boundary -- or irreducible,
atoroidal, cylindrical -- the exterior contains no essential disks or tori but
contains an essential annulus. Based on the Koda-Ozawa classification theorem,
essential annuli in an irreducible, atoroidal handlebody-knots of genus two are
classified into four classes: type $2$, type $3$-$2$, type $3$-$3$ and type
$4$-$1$. We show that under mild condition most genus two cylindrical
handlebody-knot exteriors contain no essential disks or tori, and when a type
$3$-$3$ annulus exists, it is often unique up to isotopy; a classification
result for symmetry groups of such cylindrical handlebody-knots is also
obtained.

Geometric Topology