JSJ decomposition for handlebody-knots

TL;DR

This paper introduces a novel method for analyzing handlebody-knots using JSJ decomposition and annulus diagrams, providing a new classification framework and insights into their symmetries.

Contribution

It develops the concept of annulus diagrams and applies JSJ decomposition to classify genus two handlebody-knots and study their symmetries.

Findings

Classified genus two handlebody-knots via annulus diagrams.

Provided methods to construct handlebody-knots with specific annulus configurations.

Discussed applications to handlebody-knot symmetries.

Abstract

The paper applies the JSJ decomposition and Koda-Ozawa's annulus classification to analyze the annulus configuration in a handlebody-knot exterior. We introduce the notion of the annulus diagram, to pack the configuration into a labeled graph, and classify genus two handlebody-knots in terms of their annulus diagrams. Applications to handlebody-knot symmetries are discussed; methods to produce handlebody-knots with various types of annulus diagrams are also presented.