Joint with Joe Breen, Luya Wang.
Abstract: We characterize regularity of Lagrangian submanifolds in Weinstein Lefschetz fibrations, establishing a conjecture of Giroux and Pardon. Our main result is the Weinstein analogue of a closed symplectic Lefschetz pencil result of Auroux, Muñoz, and Presas. As an application, given a Legendrian link in tight S^3 and an exact filling which is part of an arboreal skeleton for the
4-ball, we build a Lefschetz fibration such that the image of the filling and all of its mutations are arcs in the base.