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1. Naturality and functoriality in involutive Heegaard Floer homology

   https://doi.org/10.4171/qt/210  

   Hendricks, Kristen; Hom, Jennifer; Stoffregen, Matthew; Zemke, Ian (April 2024, Quantum Topology)

   We prove first-order naturality of involutive Heegaard Floer homology, and furthermore, construct well-defined maps on involutive Heegaard Floer homology associated to cobordisms between three-manifolds. We also prove analogous naturality and functoriality results for involutive Floer theory for knots and links. The proof relies on the doubling model for the involution, as well as several variations. 

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2. Decomposing sutured monopole and Instanton Floer homologies

   Ghosh, Sudipta; Li, Zhenkun (November 2019, ArXivorg)

   In this paper generalizes the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bodies. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for monopole and instanton knot Floer homologies, which were introduced by Kronheimer and Mrowka. The same result was originally proved by Ozsváth and Szabó for link Floer homology. Last, we generalize the construction of minus versions of monopole and instanton knot Floer homology, which was initially done for knots by the second author, to the case of links. Along with the construction of polytopes, we also proved that, for a balanced sutured manifold with vanishing second homology, the rank of the sutured monopole or instanton Floer homology bounds the depth of the balanced sutured manifold. As a corollary, we obtain an independent proof that monopole and instanton knot Floer homologies, as mentioned above, both detect fibred knots in S3. This result was originally achieved by Kronheimer and Mrowka. 

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3. A remark on the geography problem in Heegaard Floer homology

   https://doi.org/10.1090/pspum/102/08  

   Hanselman, Jonathan; Kutluhan, Çagatay; Lidman, Tye (July 2019, Proceedings of symposia in pure mathematics)

   Gay, David; Wu, Weiwei (Ed.)

   We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this corollary to show that the chain complex depicted by Ozsváth, Stipsicz, and Szabó to argue that there is no algebraic obstruction to the existence of knots with trivial epsilon invariant and non-trivial upsilon invariant cannot be realized as the knot Floer complex of a knot. 

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4. Floer homology and right-veering monodromy

   https://doi.org/10.1515/crelle-2024-0079  

   Baldwin, John A; Ni, Yi; Sivek, Steven (October 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove that the knot Floer complex of a fibered knot detects whether the monodromy of its fibration is right-veering.In particular, this leads to a purely knot Floer-theoretic characterization of tight contact structures, by the work of Honda–Kazez–Matić.Our proof makes use of the relationship between the Heegaard Floer homology of mapping tori and the symplectic Floer homology of area-preserving surface diffeomorphisms.We describe applications of this work to Dehn surgeries and taut foliations. 

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5. Lattice homology, formality, and plumbed L-space links

   https://doi.org/10.4171/JEMS/1562  

   Borodzik, Maciej; Liu, Beibei; Zemke, Ian (December 2024, Journal of the European Mathematical Society)

   We define a link lattice complex for plumbed links, generalizing constructions of Ozsváth, Stipsicz and Szabó, and of Gorsky and Némethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is homotopy equivalent to the link Floer complex as anA_{\infty}-module. Additionally, we prove that the link Floer complex of a plumbed L-space link is a free resolution of its homology. As a consequence, we give an algorithm to compute the link Floer complexes of plumbed L-space links, in particular of algebraic links, from their multivariable Alexander polynomial. 

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