On Newton polytopes of augmentations of the Legendrian DGA

In joint work with Roger Casals we provide a new application of Newton polytopes to the classification of Lagrangian fillings of Legendrian submanifolds in the standard contact (2n + 1) sphere. In particular, we show that Newton polytopes can be used to distinguish infinitely many distinct Lagrangian fillings of Legendrian links in the standard contact 3-sphere and higher dimensional Legendrian spheres in the standard contact (2n + 1) sphere up to Hamiltonian isotopy. We provide the first examples of Legendrian links with infinitely many distinct non-orientable exact Lagrangian fillings