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| Main Authors: | , |
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| Format: | Preprint |
| Udgivet: |
2026
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| Online adgang: | https://arxiv.org/abs/2602.17109 |
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| _version_ | 1866911456918765568 |
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| author | Ozawa, Makoto Wang, Yi-Sheng |
| author_facet | Ozawa, Makoto Wang, Yi-Sheng |
| contents | 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 $τ$- and $ρ$-tangles and good rectangles and annuli. By classifying $τ$- and $ρ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17109 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 3-decompositions of genus two handlebody-knots Ozawa, Makoto Wang, Yi-Sheng Geometric Topology Primary 57K12, 57K32, Secondary 57M15 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 $τ$- and $ρ$-tangles and good rectangles and annuli. By classifying $τ$- and $ρ$-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. |
| title | 3-decompositions of genus two handlebody-knots |
| topic | Geometric Topology Primary 57K12, 57K32, Secondary 57M15 |
| url | https://arxiv.org/abs/2602.17109 |