Rigidity and symmetry of cylindrical handlebody-knots

TL;DR

This paper classifies the symmetry groups of genus two cylindrical handlebody-knots, showing most lack essential disks or tori and that certain annuli are often unique, advancing understanding of their geometric structures.

Contribution

It provides a classification of symmetry groups for genus two cylindrical handlebody-knots and analyzes the uniqueness of certain essential annuli.

Findings

Most genus two cylindrical handlebody-knot exteriors contain no essential disks or tori.

When a type 3-3 annulus exists, it is often unique up to isotopy.

A classification of symmetry groups for these handlebody-knots is achieved.

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 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.