3-decompositions of genus two handlebody-knots
Abstract
We investigate the class of -decomposable genus two handlebody-knots and provide a complete classification of essential annuli in their exteriors. We introduce the notion of - and -tangles and good rectangles and annuli. By classifying - and -tangles whose exteriors admit a good rectangle or annulus, we categorize atoroidal -decomposable genus two handlebody-knots into distinct classes, based on the number of essential annuli. As an application, the hyperbolicity of all genus two handlebody-knots with up to six crossings are determined, and numerous hyperbolic handlehody-knots with seven crossings identified. Furthermore, obstructions for a handlebody-knot to be -decomposable are constructed with explicit examples provided.
Cite
@article{arxiv.2602.17109,
title = {3-decompositions of genus two handlebody-knots},
author = {Makoto Ozawa and Yi-Sheng Wang},
journal= {arXiv preprint arXiv:2602.17109},
year = {2026}
}
Comments
26 pages, 16 figures