An introduction to knot Floer homology and curved bordered algebras (Q2204125)

The paper under review is devoted to the study of an important topological invariant of knots called knot Floer homology. In fact, this notion was introduced in the paper \textit{P. Ozsváth} and \textit{Z. Szabó} [Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019)] (see also \textit{J. Rasmussen} [``Floer homology and knot complements'', Preprint, \url{arXiv:math/0306378}]). The authors proclaim that the main goal of the paper is to give an outline of Ozsváth-Szabó's approach discussed in their work [\textit{P. Ozsváth} and \textit{Z. Szabó}, Adv. Math. 328, 1088--1198 (2018; Zbl 1417.57015)]. This approach is based on the systematic use of various (more or less known) algebraic, analytic and topological concepts such as \(A_\infty\)-modules, flat \(D\)-structures, chain complexes, box tensor products, knot projections, Heegaard diagrams, Riemann surfaces, symplectic manifolds, Kähler forms, Kauffman states, boundary algebras, curvature, gluing, isotopy, etc. Probably, the key ingredient of the approach is an elegant construction of a chain complex associated with a flat \(D\)-structure over a certain curved associative algebra. Thus, in some cases, there is a possibility to compute the Floer homology explicitly. As a good example, for a knot in \(S^3\), the authors compute the differential operator of the corresponding complex, using its representation by a diagram obtained by concatenating some suitable elementary configurations. In the final remarks, they explain how such a chain complex can be constructed directly by taking the box tensor product of the \(DA\)-bimodules associated with a given diagram. They also emphasize that the corresponding combinatorial formulation of knot Floer homology can be viewed as an alternative to the grid homology from \textit{P. S. Ozsváth} et al. [Grid homology for knots and links. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1348.57002)] and that the obtained complexes are more ``economical''.\par Indeed, the paper is written in a very clear manner, the authors explain in detail necessary preliminaries, tools and techniques followed by nice pictures, exercises and interesting examples. Without a doubt, graduate students and beginners as well as advances researchers from other areas of mathematics will be able to understand the most part of basic ideas, relevant constructions, comments, remarks and useful applications discussed in this paper.