3-decompositions of genus two handlebody-knots

Makoto Ozawa; Yi-Sheng Wang

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arXiv · 2602.17109

3-decompositions of genus two handlebody-knots

Makoto Ozawa·Yi-Sheng Wang

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Abstract

We investigate the class of $3$-decomposable genus two handlebody-knots and provide a complete classification of essential annuli in their exteriors. We introduce the notion of $\tau$- and $\rho$-tangles and good rectangles and annuli. By classifying $\tau$- and $\rho$-tangles whose exteriors admit a good rectangle or annulus, we categorize atoroidal $3$-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 $3$-decomposable are constructed with explicit examples provided.

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Makoto Ozawa, Yi-Sheng Wang. 2026-02-19. 3-decompositions of genus two handlebody-knots. https://arxiv.org/abs/2602.17109

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