The braid group $B_n$ on $n$ strands fits into a short exact sequence of groups:

$$ 1 \longrightarrow P_n \longrightarrow B_n \longrightarrow S_n \longrightarrow 1,$$

where $S_n$ is the symmetric group on the strands, and $P_n$ is the normal subgroup of braids that do not permute the strands.

Since symmetric groups are, in some sense, ``general linear groups over the field with one element,'' perhaps there is some corresponding short exact sequence ending with $GL_n(F_q)$ that specializes to the exact sequence above as $q \rightarrow 1$.

In the spirit of the exact sequence above, is there a $q$-analog to the braid group?