Annulus configuration in handlebody-knot exteriors

TL;DR

This paper introduces an invariant called the annulus diagram, which helps distinguish handlebody-knots with homeomorphic exteriors, addressing a key challenge in knot theory.

Contribution

The paper demonstrates that the annulus diagram invariant can effectively differentiate handlebody-knots with identical exteriors, advancing knot classification methods.

Findings

Annulus diagram distinguishes handlebody-knots with homeomorphic exteriors.

The invariant differentiates members of specific handlebody-knot families.

The method leverages Johannson's theory and Koda-Ozawa classification.

Abstract

In contrast to classical knots, the knot type of a genus two handlebody-knot is not determined by its exterior, and it is often a challenging task to distinguish handlebody-knots with homeomorphic exteriors. The present paper considers an invariant (the annulus diagram), defined via Johannson's characteristic submanifold theory and the Koda-Ozawa classification for essential annuli, and demonstrates its capability to distinguish such handlebody-knots; particularly, the annulus diagram is able to differentiate members in the handlebody-knot families given by Motto and Lee-Lee.