Two Observations on the Perturbed Wedge

TL;DR

This paper discusses the perturbed wedge construction by Santos, showing it can produce counterexamples to the nonrevisiting conjecture and identifying conditions where it does not increase diameter.

Contribution

It provides two new insights into Santos's perturbed wedge construction, including its implications for the nonrevisiting conjecture and diameter bounds.

Findings

An all-but-simple spindle of dimension d and length d+1 is a counterexample to the nonrevisiting conjecture.

Conditions are identified under which the perturbed wedge does not increase the diameter.

The paper offers theoretical observations on the effects of the perturbed wedge construction.

Abstract

Francisco Santos has described a new construction, per- turbing apart a non-simple face, to offer a counterexample to the Hirsch Conjecture. We offer two observations about this perturbed wedge con- struction, regarding its effect on edge-paths. First, that an all-but- simple spindle of dimension d and length d + 1 is a counterexample to the nonrevisiting conjecture. Second, that there are conditions under which the perturbed wedge construction does not increase the diameter. NOTE: These are simply working notes, offering two observations on the construction identified by Santos.