Knotted handle decomposing spheres for handlebody-knots

TL;DR

This paper establishes the uniqueness of maximal knotted handle decomposing spheres for certain handlebody-knots and applies this to distinguish specific knots and analyze their exteriors.

Contribution

It introduces the concept of a unique maximal unnested set of knotted handle decomposing spheres for boundary-irreducible handlebody-knots and applies it to classify and distinguish particular knots.

Findings

Uniqueness of maximal unnested knotted handle decomposing spheres for boundary-irreducible handlebody-knots

Distinction of handlebody-knots 6_{14} and 6_{15} using their exteriors

Some genus two handlebody-knots can be uniquely determined by their exteriors

Abstract

We show that a handlebody-knot whose exterior is boundary-irreducible has a unique maximal unnested set of knotted handle decomposing spheres up to isotopies and annulus-moves. As an application, we show that the handlebody-knots $6_{14}$ and $6_{15}$ are not equivalent. We also show that some genus two handlebody-knots with a knotted handle decomposing sphere can be determined by their exteriors. As an application, we show that the exteriors of $6_{14}$ and $6_{15}$ are not homeomorphic.