I was looking for interesting questions pertaining to braid theory. I don't know if the following are considered important, but I'd like to ask:
(1) in relation with the following link, is it true that every subgroup of $B_n$ has a generating set of size at most $f(n)$, for some function $f$? This would imply that $B_n$ is a Noetherian group.
(2) are there positive algorithmic results for braid groups described by generators?
(3) it is known that every finite group of order $n$ can be embedded in $S_n$, so it is possible to define a notion of 'pseudo-order $n$' for an infinite group that allows embedding in $B_n$?
(4) are there interesting partial orders to study on the braids of fixed size? Dehornoy showed that $B_n$ admits a left-invariant total order $\leq$, but I'm looking for a partial order $\preceq$ that would have a more 'topological' flavor (possibly such that $\preceq \subseteq \leq$).
This list is certainly incomplete, so please feel free to point out other questions in the field that are deemed important.