Abstract
Using square bridge position, Akbulut-Ozbagci and later Arikan gave algorithms both of which construct an explicit compatible open book decomposition on a closed contact 3-manifold which results from a contact \((\pm 1)\)-surgery on a Legendrian link in the standard contact 3-sphere. In this article, we introduce the “generalized square bridge position” for a Legendrian link in the standard contact 5-sphere and partially generalize this result to the dimension five via an algorithm which constructs relative open book decompositions on relative contact pairs.
Article PDF
Similar content being viewed by others
References
S. Akbulut and B. Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol., 5 (2001), 319–334.
M. F. Arikan, On the support genus of a contact structure, J. Gökova Geom. Topol., 1 (2007), 92–115.
R. Avdek, Liouville hypersurfaces and connect sum cobordisms, J. Symplectic Geom., 19 (2021), 865–957.
R. Casals, E. Murphy and F. Presas, Geometric criteria for overtwistedness,J. Amer. Math. Soc., 32 (2015), 563–604.
J. Conway and J. Etnyre, Contact surgery and symplectic caps, Bull. London Math. Soc., 52 (2020), 379–394.
F. Ding and H. Geiges, A Legendrian surgery presentation of contact 3-manifolds, Math. Proc. Cambridge Philos. Soc., 136 (2004), 583–598.
T. Ekholm, J. Etnyre and M. Sullivan, Non-isotopic Legendrian submanifolds in \(\mathbb{R}^{2n+1}\), J. Differential Geom., 71 (2005), 85–128.
J. B. Etnyre, Lectures on open book decompositions and contact structures, in: Floer homology, gauge theory, and low-dimensional topology, Clay Math. Proc., vol. 5, Amer. Math. Soc. (Providence, RI, 2006), pp. 103–141.
J. B. Etnyre and B. Ozbagcı, Invariants of contact structures from open books, Trans. Amer. Math. Soc., 360 (2008), 3133–3151.
H. Geiges, An Introduction to Contact Topology, Cambridge University Press (New York, 2008).
E. Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, in: Proceedings of the International Congress of Mathematicians, vol. 2, Higher Education Press (Beijing, 2002), pp. 405-414.
R. E. Gompf, Handlebody construction of Stein Surfaces, Ann. of Math., 148 (1998), 619–693.
N.Goodman, Contact structures and open books, PhD thesis, University of Texas (Austin, 2003).
O. V. Koert, Open books on contact five manifolds, Ann. Inst. Fourier (Grenoble), 58 (2008), 139–157.
O. V. Koert, Lecture notes on stabilization of contact open books, Münster J. Math., 10 (2017), 425–455.
O. Lazarev, Maximal contact and symplectic structures, J. Topology, 13 (2020), 1058–1083.
H.Lyon, Torus knots in the complements of links and surfaces, Michigan Math. J., 27 (1980), 39–46.
D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press (1995).
B. Ozbagci and A. I. Stipsicz, Surgery on Contact 3-Manifolds and Stein Surfaces, Bolyai Soc. Math. Stud., vol. 13, Springer-Verlag (Budapest, 2004).
P. Seidel, Floer homology and the symplectic isotopy problem, PhD thesis, University of Oxford (1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Arıkan, M.F., Taşpınar, İ.Ö. Compatible relative open books on relative contact pairs via generalized square bridge diagrams. Acta Math. Hungar. 172, 80–118 (2024). https://doi.org/10.1007/s10474-024-01402-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-024-01402-5