Involutive Yang–Baxter: cabling, decomposability, and Dehornoy class

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.