Lagrangian zigzag cobordisms

One key difference between Lagrangian concordances of Legendrian links and smooth concordances of smooth links is that Lagrangian concordances are not invertible in general. In other words, smooth concordance is an equivalence relation while Lagrangian concordance is not; this means that interesting concepts such as the smooth concordance group and the smooth four-genus (as a metric on the smooth concordance group) do not have a natural analogue in contact geometry. In this talk, we explore a new notion that bridges smooth and Lagrangian concordance, known as Lagrangian zigzag concordance, which is an equivalence relation. We will compare it with smooth and Lagrangian concordance, and discuss some structural properties, as well as an application to Poincaré–Chekanov polynomials. This is joint work with Josh Sabloff, Shea Vela-Vick, and Angela Wu.