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Given any polytope $P$ and any generic linear functional ${\bf c} $, oneobtains a directed graph $G(P,{\bf c})$ from the 1-skeleton of $P$ by orientingeach edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. For $P$ asimple polytope and $G(P,{\bf c})$ the Hasse diagram of a lattice $L$, the joinof 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 allof the elements of $S$. The author conjectures for such $G(P,{\bf c})$ that nodirected path in $G(P,{\bf c})$ ever revisits any facet of $P$. This wouldimply for such $P$ and ${\bf c}$ that the simplex method for linear programmingis efficient under all possible pivot rules. This conjecture is proven for3-polytopes and for spindles.  For simple polytopes in which $G(P,{\bf c})$ is the Hasse diagram of alattice $L$, the order complex of each open interval in $L$ is proven homotopyequivalent to a ball or a sphere. Applications are given to the weak Bruhatorder, the Tamari lattice, and the Cambrian lattices.  This paper concludes with an appendix by Dominik Preu\ss proving the monotoneHirsch conjecture for $P$ a simple polytope and $G(P,{\bf c})$ the Hassediagram of a lattice. This confirms one of the main consequences that theauthor's conjecture would have.

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