Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology

Abstract

Given any polytope $P$ and any generic linear functional ${\bf c} $, one obtains a directed graph $G(P,{\bf c})$ from the 1-skeleton of $P$ by orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. For $P$ a simple polytope and $G(P,{\bf c})$ the Hasse diagram of a lattice $L$, the join of any collection $S$ of elements which all cover a common element $u$ in $L$ is proven to equal the sink of the smallest face of $P$ containing $u$ and all of the elements of $S$. The author conjectures for such $G(P,{\bf c})$ that no directed path in $G(P,{\bf c})$ ever revisits any facet of $P$. This would imply for such $P$ and ${\bf c}$ that the simplex method for linear programming is efficient under all possible pivot rules. This conjecture is proven for 3-polytopes and for spindles. For simple polytopes in which $G(P,{\bf c})$ is the Hasse diagram of a lattice $L$, the order complex of each open interval in $L$ is proven homotopy equivalent to a ball or a sphere. Applications are given to the weak Bruhat order, the Tamari lattice, and the Cambrian lattices. This paper concludes with an appendix by Dominik Preußproving the monotone Hirsch conjecture for $P$ a simple polytope and $G(P,{\bf c})$ the Hasse diagram of a lattice. This confirms one of the main consequences that the author's conjecture would have.