I have been looking around for an answer to this question, but I have not been able to find anything. My question is:


Is it known how to tell whether two elements $b\_1, b\_2 \in B\_n$ commute?


EDIT: Is there perhaps a nicer/cleaner rule for $B\_3$ and $B\_4$?



When do elements in the braid group $B_n$ commute?
Krammer ("[The braid group $B\_4$ is linear](https://link.springer.com/article/10.1007%2Fs002220000088)", *Invent. Math.* **142** (2000), 451–486 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=1804157))) constructed
a representation $\rho: B\_n \to {\rm GL}\_N({\bf Z}[q^{\pm 1}, t^{\pm 1}])$
with $N = {n \choose 2}$, and Bigelow ("[Braid groups are linear](https://www.ams.org/journals/jams/2001-14-02/S0894-0347-00-00361-1)", *J. AMS* **14** #2 (2000), 471–486 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=1815219)))
proved it is faithful for all $n$. Thus $b\_1,b\_2 \in B\_n$ commute
if and only if $\rho(b\_1)$ and $\rho(b\_2)$ commute in
${\rm GL}\_N({\bf Z}[q^{\pm 1}, t^{\pm 1}])$,
which is a finite computation.


