Recursion relations for open Gromov-Witten invariants

In this talk, I will explain how to use (relative) recursion relations in the HOMFLYPT-skein to study skein-valued open Gromov-Witten partition functions as defined by Ekholm and Shende. As a first application, I will prove a crossing formula for partition functions of basic holomorphic disks and all of their multiple covers which can be interpreted as a multi-cover wall-crossing identity at a hyperbolic singularity of an underlying basic holomorphic disk. Then I will discuss how to express the open Gromov-Witten partition function of different Lagrangian fillings of the unit conormal of the Hopf link in the cotangent bundle of a 3-sphere in terms of more basic partition functions. The obtained expressions can be viewed as an instance of a Gopakumar-Vafa type formula for skein-valued open Gromov-Witten invariants. The latter part of the talk is based on joint work in progress with T. Ekholm and P. Longhi.