Danilov, Karzanov and Koshevoy introduced framings on a directed acyclic graph and used them to induce regular unimodular “framing triangulations” on the unit flow polytope. The dual graphs of these triangulations have been shown to have the structure of the Hasse diagram of a lattice, generalizing many classical and modern families of lattices in combinatorics. Through a recent/ongoing work of González D’León, Hanusa and Yip, this theory has led to a deeper understanding of h*-polynomials of certain flow polytopes.

 

Thus far, this study of framing triangulations and framing lattices has largely been limited to unit flow polytopes — i.e., from DAGs with one source, one sink and netflow vector (1,0,…,0,-1). I will talk about my recent efforts to generalize beyond the unit case. First, I will give a notion of framed DAGs inducing unimodular framing triangulations in the full generality of arbitrary integer flow polytopes. I will conclude by reducing to a special case of framed DAGs, containing all theories of framing triangulations existing in the literature, to which we can give a theory of framing posets generalizing framing lattices in the unit case.