Poset topology, moves, and Bruhat interval polytope lattices

Abstract We study the poset topology of lattices arising from orientations of 1‐skeleta of directionally simple polytopes, with Bruhat interval polytopes as our main example. We show that the order complex of an interval therein is homotopy equivalent to a sphere if is a face of and is otherwise contractible. This significantly generalizes the known case of the permutahedron. We also show that saturated chains from to in such lattices are connected, and in fact highly connected, under flipping across 2‐faces. When is a Grassmannian permutation, this implies a strengthening of the restriction of Postnikov's move‐equivalence theorem to the class of BCFW bridge decomposable plabic graphs.