📝 SummarXiv

Abstract

Consider an $m$-strand braid $x$ which is rigid in the sense of Garside-theory. Let $SC(x)$ be the set of rigid conjugates of $x$ -- this is a well-known characteristic subset of the conjugacy class of $x$. We present computational evidence that the sequence $(|SC(x^n)|)\_{n\in\mathbb N}$ is not only bounded, but in fact periodic, and that the length of the period can be bounded in terms of the number of strands $m$. We prove this result in the special case of the 3-strand braid group (where we prove that the sequence is always constant) and of the dual $4$-strand braid group.