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1. Some applications of Menke’s JSJ decomposition for symplectic fillings

   https://doi.org/10.1090/tran/8781  

   Christian, Austin; Li, Youlin (July 2023, Transactions of the American Mathematical Society)

   We apply Menke’s JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying exact symplectic fillings to the same problem for universally tight or canonical contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings. 

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2. The symplectic isotopy problem for rational cuspidal curves

   https://doi.org/10.1112/s0010437x2200762x  

   Golla, Marco; Starkston, Laura (July 2022, Compositio Mathematica)

   We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. 

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3. Bourgeois contact structures: Tightness, fillability and applications

   https://doi.org/10.1007/s00222-022-01131-y  

   Bowden, Jonathan; Gironella, Fabio; Moreno, Agustin (July 2022, Inventiones mathematicae)

   Abstract Given a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $$V \times {\mathbb {T}}^2$$ V × T 2 . We prove that all such structures are universally tight in dimension 5, independent of whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact 5-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $${\mathbb {S}}^1$$ S 1 -invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the n -torus has a unique symplectically aspherical strong filling up to diffeomorphism. 

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4. An elementary alternative to ECH capacities

   https://doi.org/10.1073/pnas.2203090119  

   Hutchings, Michael (August 2022, Proceedings of the National Academy of Sciences)

   The embedded contact homology (ECH) capacities are a sequence of numerical invariants of symplectic four-manifolds that give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg–Witten theory. In this paper we define a sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, namely convex and concave toric domains. The capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff and Siegel [D. McDuff, K. Siegel, arXiv [Preprint] (2021)], giving a similar elementary alternative to symplectic capacities from rational symplectic field theory (SFT). 

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5. Cubic Planar Graphs and Legendrian Surface Theory

   https://doi.org/http://dx.doi.org/10.4310/ATMP.2018.v22.n5.a5  

   Treumann, David; Zaslow, Eric (May 2018, Advances in theoretical and mathematical physics)

   We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same classical invariants. The Legendrians have no exact Lagrangian fillings, but have many interesting non-exact fillings. We obtain these results by studying sheaves on a three-ball with microsupport in the surface. The moduli of such sheaves has a concrete description in terms of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold of a symplectic period domain, a Lagrangian that has appeared in the work of Dimofte–Gabella–Goncharov. We exploit this structure to find conjectural open Gromov–Witten invariants for the non-exact filling, following Aganagic–Vafa. 

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