So, I have been reading Chapter 6 of "A Study of Braids" by Murasugi and Kurpita. The main goal of the chapter is to prove that many notions of equivalence of braids actually coincide. I have had issues with various arguments made by the authors; in particular, I can't see how one of the notions of equivalence is actually an equivalence relation on the set of $n$-braids (the proof of this fact is given as an exercise).
To be more specific, the authors define a braid (ambient) isotopy as follows:
Moreover, $H$ is said to be a $s$-isotopy if the following property is also satisfied:
and in this case $\beta$ and $\beta'$ are said $s$-isotopic.
First of all, I assume there is a typo in this sentence, once "$h_t$ is always an $n$-braid" makes no sense. Instead, I interpreted the property as "$h_t(\beta)$ is always an $n$-braid", as it seems to be the most appropriate alternative.
However, I can't see how one proves that the relation of $s$-isotopy of braids is symmetric and transitive. For example, if $\beta$ and $\beta'$ are $s$-isotopic via $H$, the natural candidate for an $s$-isotopy between $\beta'$ and $\beta$ would be $H^{-1}$ (which works just fine for the case of a simple isotopy). But $(3)$ only guarantees that $h_t(\beta)$ is a braid, which, at an arbitrary instant $t$, need not have anything to do with $\beta'$, making it hard to prove $(3)$ for $H^{-1}$.
To be honest, I am not even 100% sure if this is actually true (the only other place I found this concept of $s$-isotopy was Birman's book, but she doesn't give any proofs), but I would appreciate any help understanding how one could construct the $s$-isotopies we need, or how to see that it can't be done.