On unique decomposition of knotted handlebodies

TL;DR

This paper investigates the uniqueness of decomposing knotted handlebodies in the 3-sphere, establishing new results on factorization, handlebody-knot symmetry, and constructing hyperbolic handlebody-knots with identical exteriors.

Contribution

It provides a new uniqueness theorem for handlebody decomposition along specific 2-spheres and applies this to determine the chirality of a particular handlebody-knot, also constructing an infinite family of hyperbolic knots with the same exterior.

Findings

Uniqueness of handlebody decomposition along certain 2-spheres

Determination of chirality of handlebody-knot 6_{10}

Construction of infinite hyperbolic handlebody-knots with homeomorphic exteriors

Abstract

The paper considers the uniqueness question of factorization of a knotted handlebody in the $3$-sphere along decomposing $2$-spheres. We obtain a uniqueness result for factorization along decomposing $2$-spheres meeting the handlebody at three parallel disks. The result is used to examine handlebody-knot symmetry; particularly, the chirality of $6_{10}$ in the handlebody-knot table, previously unknown, is determined. In addition, an infinite family of hyperbolic handlebody-knots with homeomorphic exteriors is constructed.