Observations on the Perturbed Wedge

Abstract

Santos' construction of the first known counterexample to the Hirsch conjecture, for bounded polytopes, follows the strategy of first finding a counterexample to the nonrevisiting conjecture. Santos constructs a $5$-dimensional all-but-simple spindle $(P,x,y)$ of length $6$, which is a counterexample to the nonrevisiting conjecture. For simple polytopes, if we had a counterexample to the nonrevisiting conjecture, we would produce the corresponding counterexample to the Hirsch conjecture through repeated wedging, over all the facets not incident to $x$ or $y$. However, Santos $5$-dimensional spindle is not simple. Every facet is incident to either $x$ or $y$, so we need an alternate method to produce the corresponding counterexample to the Hirsch conjecture. Santos has offered the perturbed wedge to accomplish this. In these working notes, we offer some technical details regarding the nonsimplicities under iterations of the perturbed wedge construction. NOTE: these are working notes about the construction.