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 T. Our machinery yields an elementary proof of a recent decomposability theorem of Camp-Mora and Sastriques, as well as original decomposability results. It also provides a conceptual interpretation (using the language of braces) of the Dehornoy class, a combinatorial invariant naturally appearing in the Garsidetheoretic approach to involutive solutions.