Non-ridge-chordal complexes whose clique complex has shellable Alexander dual

TL;DR

This paper disproves a conjecture linking shellable Alexander duals of clique complexes to ridge-chordality by providing an infinite family of counterexamples, thereby refining understanding of simplicial complex properties.

Contribution

It demonstrates that the conjecture connecting shellable Alexander duals to ridge-chordality is false through explicit counterexamples.

Findings

Counterexamples to the conjecture are constructed.

The conjecture that all complexes with shellable Alexander duals are ridge-chordal is false.

The result impacts the understanding of extendable shellability in simplicial complexes.

Abstract

A recent conjecture that appeared in three papers by Bigdeli--Faridi, Dochtermann, and Nikseresht, is that every simplicial complex whose clique complex has shellable Alexander dual, is ridge-chordal. This strengthens the long-standing Simon's conjecture that the $k$-skeleton of the simplex is extendably shellable, for any $k$. We show that the stronger conjecture has a negative answer, by exhibiting an infinite family of counterexamples.