A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the F -polynomials of the corresponding cluster algebras.

For A a gentle algebra, and X and Y string modules, we construct a combinatorial basis for Hom(X, τ Y ). We use this to describe support τ -tilting modules for A. We give a combinatorial realization of maps in both directions realizing the bijection between support τ -tilting modules and functorially finite torsion classes. We give an explicit basis of Ext 1 (Y, X) as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville [McC], showing that many but not all of them can be extended to general gentle algebras.

We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant associated to an $\textrm{SL}_3$ web diagram, with respect to a particular term order.

We use Khovanov and Kuperberg's web growth rules to identify the minimal term in the invariant associated to an SL 3 web diagram, with respect to a particular term order.