Endocabling of involutive solutions to the Yang-Baxter equation, with an application to solutions whose diagonal is a cyclic permutation

In this article, we introduce endocabling as a technique to deform involutive, non-degenerate set-theoretic solutions to the Yang-Baxter equation (``solutions'', for short) by means of $\lambda$-endomorphisms of their associated permutation brace, thus generalizing the cabling method by Lebed, Vendramin and Ram\'{i}rez. In the first part of the article, we define endocabling and investigate the behaviour of solutions and their invariants under endocabling. In the second part, we apply our findings to solutions of size $n$ whose diagonal map is an $n$-cycle: we will prove that solutions with this property whose size is an odd prime power, are of finite multipermutation level. Furthermore, solutions with this property whose size is a power of $2$, will be proven either to be of finite multipermutation level or to admit an iterated retraction onto a unique solution of size $4$. We formulate our results in the language of cycle sets.