Obstructing Lagrangian concordance for closures of 3-braids

The study of knot concordance for smooth knots is a classical and essential problem in knot theory, an important field in topology since the mid 1800s. Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. This project studies the variant most relevant to symplectic geometry, called Lagrangian concordance, in which we ask for the knots to be Legendrian and for them to bound a Lagrangian surface. We ask which knots are Lagrangian concordant to and from the standard Legendrian unknot and find obstructions coming from the cyclic p-fold branched covers of these knots. We restrict large classes of closures of 3-braids from candidacy using a variety of techniques from smooth, symplectic, and contact topology. For the remaining family of braids, we draw Weinstein diagrams of symplectic fillings of their double covers. We use the Chekanov-Eliashberg differential graded algebra of the links in these diagrams to compute the symplectic homology of these fillings in order to obstruct the last of these 3-braids