Rigidity and symmetry of cylindrical handlebody-knots

Abstract

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 conditions 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.