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1. Weinstein Handlebodies for Complements of Smoothed Toric Divisors

   https://doi.org/10.1090/memo/1561  

   Acu, Bahar; Capovilla-Searle, Orsola; Gadbled, Agnès; Marinković, Aleksandra; Murphy, Emmy; Starkston, Laura; Wu, Angela (May 2025, Memoirs of the American Mathematical Society)

   We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centeredness condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the total toric divisor supports an explicit Weinstein structure. Many examples which fail this condition also fail to have Weinstein (or even exact) complement to the partially smoothed divisor. We investigate the combinatorial possibilities of Delzant polytopes that realize such Weinstein domain complements. We also develop an algorithm to construct a Weinstein handlebody diagram in Gompf standard form for the complement of such a partially smoothed total toric divisor. The algorithm we develop more generally outputs a Weinstein handlebody diagram for any Weinstein 4-manifold constructed by attaching 2-handles to the disk cotangent bundle of any surface  $F$, where the 2-handles are attached along the co-oriented conormal lifts of curves on  $F$. We discuss how to use these diagrams to calculate invariants and provide numerous examples applying this procedure. For example, we provide Weinstein handlebody diagrams for the complements of the smooth and nodal cubics in $\mathbb{C}{\mathbb{P}}^2$. 

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2. An introduction to Weinstein handlebodies for complements of smoothed toric divisors.

   https://doi.org/10.1007/978-3-030-80979-9_4  

   Acu, Bahar; Capovilla-Searle, Orsola; Gadbled, Agnes; Marinkovic, Aleksandra; Murphy, Emmy; Starkston, Laura (January 2021, Association for Women in Mathematics series)

   Acu, Bahar; Cannizzo, Catherine; McDuff, Dusa; Myer, Ziva; Traynor, Lisa; Pan, Yu (Ed.)

   In this article, we provide an introduction to an algorithm for constructing Weinstein handlebodies for complements of certain smoothed toric divisors using explicit coordinates and a simple example. This article also serves to welcome newcomers to Weinstein handlebody diagrams and Weinstein Kirby calculus. Finally, we include several complicated examples at the end of the article to showcase the algorithm and the types of Weinstein Kirby diagrams it produces. 

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3. Lagrangian skeleta and plane curve singularities

   https://doi.org/10.1007/s11784-022-00939-8  

   Casals, Roger (June 2022, Journal of Fixed Point Theory and Applications)

   Abstract We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs $$(\mathbb {C}^2,\Lambda )$$ ( C 2 , Λ ) and Weinstein 4-manifolds $$W(\Lambda )$$ W ( Λ ) associated to max-tb Legendrian representatives of algebraic links $$\Lambda \subseteq (\mathbb {S}^3,\xi _\text {st})$$ Λ ⊆ ( S 3 , ξ st ) . We provide computations of Legendrian and Weinstein invariants, and discuss the contact topological nature of the Fomin–Pylyavskyy–Shustin–Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems. 

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4. Sectorial descent for wrapped Fukaya categories

   https://doi.org/10.1090/jams/1035  

   Ganatra, Sheel; Pardon, John; Shende, Vivek (October 2023, Journal of the American Mathematical Society)

   We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful. 

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5. Multisections with divides and Weinstein 4-manifolds

   https://doi.org/10.4310/jsg.240921005038  

   Islambouli, Gabriel; Starkston, Laura (January 2024, Journal of Symplectic Geometry)

   We introduce a new decomposition of Weinstein 4-manifolds called multisections with divides and show these can be encoded diagrammatically by a sequence of cut systems on a surface, together with a separating collection of curves. We give two algorithms to construct a multisection with divides for a Weinstein 4-manifold, one starting with a Kirby-Weinstein handle decomposition and the other starting with a positive, allowable Lefschetz fibration (PALF). Through the connections with PALFs, we define a monodromy of a multisection and show how to symplectically carry out monodromy substitution on multisections with divides. 

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