English

3-decompositions of genus two handlebody-knots

Geometric Topology 2026-02-20 v1

Abstract

We investigate the class of 33-decomposable genus two handlebody-knots and provide a complete classification of essential annuli in their exteriors. We introduce the notion of τ\tau- and ρ\rho-tangles and good rectangles and annuli. By classifying τ\tau- and ρ\rho-tangles whose exteriors admit a good rectangle or annulus, we categorize atoroidal 33-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 33-decomposable are constructed with explicit examples provided.

Keywords

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