Lagrangian cobordisms and Legendrian invariants in knot Floer homology
A fundamental question in contact and symplectic geometry is whether given two Legendrian links in \((\mathbb{R}^3,\xi_{std})\) there is an exact Lagrangian cobordism between them. There are many obstructions known towards this end, both in terms of classical invariants [\textit{B. Chantraine}, Algebr. Geom. Topol. 10, No. 1, 63--85 (2010; Zbl 1203.57010)] and more subtle invariants such as the Chekanov-Eliashberg DGA [\textit{T. Ekholm} et al., J. Eur. Math. Soc. (JEMS) 18, No. 11, 2627--2689 (2016; Zbl 1357.57044)] and knot Floer homology. In this paper, the authors show that the GRID invariants \(\widehat{\lambda}^{\pm}\) of \textit{P. Ozsváth} et al. [Geom. Topol. 12, No. 2, 941--980 (2008; Zbl 1144.57012)] can be used to obstruct the existence of \textit{decomposable} cobordisms, i.e. a special class of exact Lagrangian cobordisms that corresponds to combinatorial moves on front projections. This is very interesting in light of the following open question: is every exact Lagrangian cobordism between non-empty links decomposable? The authors exhibit concrete examples to show that \(\widehat{\lambda}^{\pm}\) are effective (i.e. they give obstructions in examples for which classical invariants do not). They also provide a very useful discussion on the strengths and weaknesses of their invariant in comparison to other ones in the literature. In particular, their invariant is the first one in knot Floer homology that allows to obstruct positive genus cobordisms (with the assumption that they are decomposable).
scientific article