Abstract
In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.
Similar content being viewed by others
References
Akbulut, S.: A solution to a conjecture of Zeeman. Topology 30(3), 513–515 (1991)
Akbulut, S.: 4-manifolds, Oxford Graduate Texts in Mathematics, vol. 25. Oxford University Press, Oxford (2016)
Akbulut, S., Yildiz, E.Z.: Knot concordances in \(S^1\times S^2\) and exotic smooth 4-manifolds. J. Gökova Geom. Topol. GGT 13, 41–52 (2019)
Auroux, D.: Factorizations in \(SL(2, {\mathbb{Z}})\) and simple examples of inequivalent Stein fillings. J. Symplectic Geom. 13(2), 261–277 (2015)
Baumslag, G., Solitar, D.: Some two-generator one-relator non-Hopfian groups. Bull. Am. Math. Soc. 68, 199–201 (1962)
Cao, C., Gallup, N., Hayden, K., Sabloff, J.M.: Topologically distinct Lagrangian and symplectic fillings. Math. Res. Lett. 21(1), 85–99 (2014)
Chantraine, B.: Lagrangian concordance of Legendrian knots. Algebr. Geom. Topol. 10(1), 63–85 (2010)
Chantraine, B.: Some non-collarable slices of Lagrangian surfaces. Bull. Lond. Math. Soc. 44(5), 981–987 (2012)
Chantraine, B.: Lagrangian concordance is not a symmetric relation. Quantum Topol. 6(3), 451–474 (2015)
Conway, J., Etnyre, J. B., Tosun, B.: Symplectic fillings, contact surgeries, and Lagrangian disks.arXiv:1712.07287v2
Cornwell, C., Ng, L., Sivek, S.: Obstructions to Lagrangian concordance. Algebr. Geom. Topol. 16(2), 797–824 (2016)
Dimitroglou Rizell, G.: Lifting pseudo-holomorphic polygons to the symplectisation of \(P\times {\mathbb{R}}\) and applications. Quantum Topol. 7(1), 29–105 (2016)
Ding, F., Geiges, H.: A Legendrian surgery presentation of contact 3-manifolds. Math. Proc. Camb. Philos. Soc. 136, 583–598 (2004)
Ekholm, T.: Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology. Perspectives in Analysis, Geometry, and Topology, pp. 109–145. Birkhäuser, New York (2012)
Ekholm, T.: Non-loose Legendrian spheres with trivial contact homology DGA. J. Topol. 9(3), 826–848 (2016)
Ekholm, T.: Corrigendum: Non-loose Legendrian spheres with trivial contact homology DGA. J. Topol. 11(4), 1133–1135 (2018)
Ekholm, T., Honda, K., Kálmán, T.: Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS) 18(11), 2627–2689 (2016)
Gompf, R.: Handlebody construction of Stein surfaces. Ann. Math. 148(2), 619–693 (1998)
Gompf, R.E., Stipsicz, A.I.: \(4\)-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, vol. 20. American Mathematical Society, Providence (1999)
Hayden, K., Sabloff, J.M.: Positive knots and Lagrangian fillability. Proc. Am. Math. Soc. 143(4), 1813–1821 (2015)
Lisca, P., Matić, G.: Tight contact structures and Seiberg-Witten invariants. Invent. Math. 129(3), 509–525 (1997)
Rudolph, L.: A congreunce between link polynomials. Math. Proc. Camb. Philos. Soc. 107, 319–327 (1990)
Shende, V., Treumann, D., Williams, H., Zaslow, E.: Cluster varieties from Legendrian knots. Duke Math. J. 168(15), 2801–2871 (2019)
Acknowledgements
The authors would like to thank John Etnyre and Honghao Gao for useful conversations. We are also grateful to the referee(s) for valuable suggestions. Part of this work was carried out while the first author was visiting University of Tsukuba and he would like to thank for their hospitality. The first author was partially supported by Grant No. 11871332 of the National Natural Science Foundation of China. The second author was partially supported by JSPS KAKENHI Grant Number 17K14180.
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
About this article
Cite this article
Li, Y., Tange, M. Smoothly non-isotopic Lagrangian disk fillings of Legendrian knots. Geom Dedicata 213, 211–225 (2021). https://doi.org/10.1007/s10711-020-00575-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-020-00575-x