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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. 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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3. Exact Lagrangian tori in symplectic Milnor fibers constructed with fillings

   https://doi.org/10.2140/agt.2025.25.3225  

   Capovilla-Searle, Orsola (January 2025, Algebraic & Geometric Topology)

   We use exact Lagrangian fillings and Weinstein handlebody diagrams to construct infinitely many distinct exact Lagrangian tori in 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality. We also provide a generalization of a criterion for when the symplectic homology of a Weinstein 4-manifold is nonvanishing given an explicit Weinstein handlebody diagram. 

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4. A Kirby color for Khovanov homology

   https://doi.org/10.4171/JEMS/1589  

   Hogancamp, Matthew; Rose, David_E V; Wedrich, Paul (February 2025, Journal of the European Mathematical Society)

   We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we define a Kirby-colored Khovanov homology that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu–Neithalath 2-handle formula, Kirby-colored Khovanov homology agrees with the\mathfrak{gl}_{2}skein lasagna module, hence is an invariant of 4-dimensional 2-handlebodies. 

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5. 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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