Morse diagrams, Murasugi sums, and the mapping class group
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917376831782912 |
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| author | Brand, Jack Gay, David Licata, Joan |
| author_facet | Brand, Jack Gay, David Licata, Joan |
| contents | A combinatorial Morse structure encodes a mapping class for a surface with boundary, and the data may be efficiently represented via a Morse diagram. This diagram determines an open book decomposition of a 3-manifold, and hence, a contact structure on that 3-manifold. We examine how combinatorial Morse structures behave under the connect sum of open books, with particular attention paid to the case of negative stabilisation. This leads to a diagrammatic criterion for detecting overtwisted contact structures. Finally, in the case of open books with one-holed torus pages, we classify all the Morse diagrams associated to a fixed open book decomposition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_00448 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Morse diagrams, Murasugi sums, and the mapping class group Brand, Jack Gay, David Licata, Joan Geometric Topology 57K33(primary) 57M50 (secondary) A combinatorial Morse structure encodes a mapping class for a surface with boundary, and the data may be efficiently represented via a Morse diagram. This diagram determines an open book decomposition of a 3-manifold, and hence, a contact structure on that 3-manifold. We examine how combinatorial Morse structures behave under the connect sum of open books, with particular attention paid to the case of negative stabilisation. This leads to a diagrammatic criterion for detecting overtwisted contact structures. Finally, in the case of open books with one-holed torus pages, we classify all the Morse diagrams associated to a fixed open book decomposition. |
| title | Morse diagrams, Murasugi sums, and the mapping class group |
| topic | Geometric Topology 57K33(primary) 57M50 (secondary) |
| url | https://arxiv.org/abs/2604.00448 |