How essential is a spanning surface?
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912008544190464 |
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| author | Kindred, Thomas |
| author_facet | Kindred, Thomas |
| contents | Gabai proved that any plumbing, or Murasugi sum, of $π_1$-essential Seifert surfaces is also $π_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show that the analogous statement about geometrically essential surfaces is untrue. We then introduce new numerical invariants, the algebraic and geometric essence of a spanning surface $F\subset S^3$, which measure how far $F$ is from being compressible, and we extend Ozawa's theorem by showing that plumbing respects the algebraic version of this new invariant. We also introduce a ``twisted'' generalization of plumbing and use it to compute essence for many examples, including checkerboard surfaces from reduced alternating diagrams. Finally, we extend all of these results to plumbings and twisted plumbings of spanning surfaces in arbitrary 3-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16948 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | How essential is a spanning surface? Kindred, Thomas Geometric Topology 57K10, 57K12 Gabai proved that any plumbing, or Murasugi sum, of $π_1$-essential Seifert surfaces is also $π_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show that the analogous statement about geometrically essential surfaces is untrue. We then introduce new numerical invariants, the algebraic and geometric essence of a spanning surface $F\subset S^3$, which measure how far $F$ is from being compressible, and we extend Ozawa's theorem by showing that plumbing respects the algebraic version of this new invariant. We also introduce a ``twisted'' generalization of plumbing and use it to compute essence for many examples, including checkerboard surfaces from reduced alternating diagrams. Finally, we extend all of these results to plumbings and twisted plumbings of spanning surfaces in arbitrary 3-manifolds. |
| title | How essential is a spanning surface? |
| topic | Geometric Topology 57K10, 57K12 |
| url | https://arxiv.org/abs/2408.16948 |