Unknotting annuli and handlebody-knot symmetry

TL;DR

This paper studies the symmetry properties of genus two cylindrical handlebody-knots with unknotting annuli, showing that uniqueness and specific annulus types imply trivial symmetry groups.

Contribution

It provides new insights into the relationship between unknotting annuli and symmetry groups in genus two cylindrical handlebody-knots.

Findings

Symmetry group is trivial if the unknotting annulus is unique and of type 2.

Classifies cylindrical handlebody-knots based on their unknotting annuli.

Connects topological properties with symmetry group triviality.

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 $2$.