Set theoretic solutions of the Yang-Baxter equation, graphs and computations

TL;DR

This paper advances the understanding of set-theoretic solutions to the Yang-Baxter equation by exploring automorphisms, unions, and graph-based computational methods, especially for solutions of multipermutation level 2.

Contribution

It introduces graph-theoretic methods and new results on automorphisms and unions of solutions, focusing on solutions of finite order and multipermutation level 2.

Findings

Automorphism groups of solutions are characterized.

Graph methods enable computation of solutions and automorphisms.

Detailed analysis of solutions with multipermutation level 2.

Abstract

We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2.