Involutive Yang-Baxter: cabling, decomposability, Dehornoy class

TL;DR

This paper introduces new methods for testing decomposability of involutive solutions to the Yang-Baxter equation, utilizing cabling operations and the braces framework, leading to new theoretical insights and proofs.

Contribution

It develops a novel machinery based on cabling and the decomposability theorem, providing elementary proofs and a conceptual interpretation of the Dehornoy class.

Findings

Elementary proof of a recent decomposability theorem.

New decomposability results for involutive solutions.

A conceptual interpretation of the Dehornoy class.

Abstract

We develop new machinery for producing decomposability tests for involutive solutions to the Yang-Baxter equation. It is based on the seminal decomposability theorem of Rump, and on "cabling" operations on solutions and their effect on the diagonal map. Our machinery yields an elementary proof of a recent decomposability theorem of Camp-More and Sastriques, as well as original decomposability results. It also provides a conceptual interpretation (using the braces language) of the Dehornoy class, a combinatorial invariant naturally appearing in the Garside-theoretic approach to involutive solutions.