Monodromies of surfaces in 3-manifolds, right-veeringness, and primeness of links
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916946693324800 |
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| author | Feller, Peter Lewark, Lukas Rodriguez, Miguel Orbegozo |
| author_facet | Feller, Peter Lewark, Lukas Rodriguez, Miguel Orbegozo |
| contents | Extending the notion of monodromies associated with open books of $3$-manifolds, we consider monodromies for all incompressible surfaces in $3$-manifolds as partial self-maps of the arc set of the surfaces. We use them to develop a primeness criterion for incompressible surfaces constructed as iterative Murasugi sums in irreducible $3$-manifolds.
We also consider a suitable notion of right-veeringness for monodromies of incompressible surfaces. We show strongly quasipositive surfaces are right-veering, thereby generalizing the corresponding result for open books and providing a proof that does not draw on contact geometry. In fact, we characterize when all elements of a family of incompressible surfaces that is closed under positive stabilization are right-veering. The latter also offers a new perspective on the characterization of tight contact structures via right-veeringness as first established by Honda, Kazez, and Matić.
As an application to links in $S^3$, we prove visual primeness of a large class of links, the so-called alternative links. This subsumes all prior visual primeness results related to Cromwell's conjecture. The application is enabled by the fact that all links in $S^3$ arise as the boundary of incompressible surfaces, whereas classical open book theory is restricted to fibered links -- those links that arise as the boundary of the page of an open book. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monodromies of surfaces in 3-manifolds, right-veeringness, and primeness of links Feller, Peter Lewark, Lukas Rodriguez, Miguel Orbegozo Geometric Topology 57K10, 57K20, 57K30 Extending the notion of monodromies associated with open books of $3$-manifolds, we consider monodromies for all incompressible surfaces in $3$-manifolds as partial self-maps of the arc set of the surfaces. We use them to develop a primeness criterion for incompressible surfaces constructed as iterative Murasugi sums in irreducible $3$-manifolds. We also consider a suitable notion of right-veeringness for monodromies of incompressible surfaces. We show strongly quasipositive surfaces are right-veering, thereby generalizing the corresponding result for open books and providing a proof that does not draw on contact geometry. In fact, we characterize when all elements of a family of incompressible surfaces that is closed under positive stabilization are right-veering. The latter also offers a new perspective on the characterization of tight contact structures via right-veeringness as first established by Honda, Kazez, and Matić. As an application to links in $S^3$, we prove visual primeness of a large class of links, the so-called alternative links. This subsumes all prior visual primeness results related to Cromwell's conjecture. The application is enabled by the fact that all links in $S^3$ arise as the boundary of incompressible surfaces, whereas classical open book theory is restricted to fibered links -- those links that arise as the boundary of the page of an open book. |
| title | Monodromies of surfaces in 3-manifolds, right-veeringness, and primeness of links |
| topic | Geometric Topology 57K10, 57K20, 57K30 |
| url | https://arxiv.org/abs/2509.09615 |