Some questions (still) unresolved about braids?

Question

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.

Answer 1

The following notion of 'tameness' seems interesting. Say that a braid $\beta \in B_n$ is tamed if there is a set of $n$ rectilinear curves $\cal{C}$ over the 3-dimensional cylinder $S_1 \times [0,n]$ such that when projecting $\cal{C}$ on the $x$ plane, we obtain the graphic representation of $\beta$. Clearly, this notion is closed by inversion and composition, and we can therefore define the notion of tamed braid group (TBG). This leads to the following definition: a pair $(G,H)$ of braid groups is a tamed pair if the shuffle product $G \| H$ is a TBG.

Let us precise this notation. Given a braid $\beta \in B_n$, and given a $k$-subset $I \subseteq B_n$, we denote by $\beta | I$ the braid in $B_k$ obtained by keeping the strands of $\beta$ starting at an index of $I$. Now, if $G \subseteq B_m$ and $H \subseteq B_n$, we define $G \| H$ is the subset of $B_{m+n}$ containing the braids $\beta$ such that $\beta | I \in G, \beta | J \in H$ for some partition $(I,J)$ of $[n+m]$.

This definition seems to have some interesting properties, in particular $G$ and $H$ play symmetric roles. There might be an alternate way to define 'tameness' though, but this is currently the only one I can think of. Note that there is a different notion of tame/wild knot in knot theory, but it doesn't seem very relevant to us as it only makes sense for infinite knots.