From the link Floer complex of a link $K$, we extract a lower bound $t_q'(K)$ for the rational unknotting number of $K$ (i.e. the minimum number of rational replacements required to unknot $K$). Moreover, we show that the torsion obstruction $t_q(K)=\hat{t}(K)$ from an earlier paper of Alishahi and the author is a lower bound for the proper rational unknotting number. Moreover, $t_q(K\#K')=\max\{t_q(K),t_q(K')\}$ and $t'_q(K\#K')=\max\{t'_q(K),t'_q(K')\}$. For the torus knot $K=T_{p,pk+1}$ we compute $t'_q(K)=\lfloor p/2\rfloor$ and $t_q(K)=p-1$.

We describe a weighted $A_\infty$-algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fukaya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work.

A symplectic rational cuspidal curve with positive self-intersection number admits a concave neighborhood, and thus a corresponding contact manifold on the boundary. In this article, we study symplectic fillings of such contact manifolds, providing a complementary perspective to our earlier article on symplectic isotopy classes of rational cuspidal curves. We explore aspects of these symplectic fillings through Stein handlebodies and rational blow-downs. We give examples of such contact manifolds which are identifiable as links of normal surface singularities, other examples which admit no symplectic fillings, and further examples where the fillings can be fully classified.

In this paper we give an explicit description of the Floer cohomology of a composition of Dehn twists τ about disjoint Lagrangian spheres in a w + -monotone symplectic manifold whose symplectic class [ω] admits a rational representative. To do so, we generalize the approach developed in [Sei96] and [Gau03] and apply it to the modified Floer cohomology groups defined by K. Ono in [Ono95] which can be shown to be isomorphic to the standard ones. As a byproduct of this new framework, in a monotone manifold, we are able to prove that a certain class in the fixed point Floer cohomology group of a single Dehn twist τ V has to vanish. This class, which counts pseudo-holomorphic halfstrips bound to the Lagrangian sphere V , plays a role in a new geometric proof of the exactness of the triangle P. Seidel defined in [Sei03] which is the subject of subsequent ongoing work.

algebras of type A as Fukaya-Seidel categories

This paper is a sequel to [11] . We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of ∂-Laplacians, as well as the convergence result of quantum Hilbert spaces. We also consider the case of almost Kähler quantization for compatible almost complex structures, and show the analogous convergence results.

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