Counterexample to a conjecture about dihedral quandle

TL;DR

This paper presents a counterexample to a conjecture about the structure of the augmentation ideal in dihedral quandles of even order, challenging previous assumptions in algebraic topology.

Contribution

It provides the first known counterexample disproving the conjecture about the augmentation ideal in dihedral quandles of even order.

Findings

Counterexample disproves the conjecture for even order dihedral quandles

The structure of the augmentation ideal differs from the conjectured pattern

Implications for algebraic topology and quandle theory

Abstract

It was conjectured that the augmentation ideal of a dihedral quandle of even order $n>2$ satisfies $|\Delta^k(\text{R}_n)/\Delta^{k+1}(\text{R}_{n})|=n$ for all $k\geq 2$. In this article we provide a counterexample against this conjecture.