About a question of Gateva-Ivanova and Cameron on square-free set-theoretic solutions of the Yang-Baxter equation

Abstract

In this article, we introduce a new sequence ${\overline{N}}_m$ to find a new estimation of the cardinality N_m of the minimal involutive square-free solution of level m. As an application, using the first values of ${\overline{N}}_m,$ we improve the estimations of N_m obtained by Gateva-Ivanova and Cameron and Lebed and Vendramin. Following the approach of the first part, in the last section we construct several new counterexamples to the Gateva-Ivanova’s Conjecture.

Keywords:

• Yang-Baxter equation
• set-theoretic solution
• cycle set

2010 MSC:

• 16T25

Acknowledgements

We thank the referee for careful reading of this article.