TITLE: When do elements in the braid group $B_n$ commute?
QUESTION [8 upvotes]: 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 $B_4$?

REPLY [4 votes]: You are asking: "Suppose that a and b are braids, with inverses A and B.  Is the element abAB trivial?"  
This is a special case of the word problem for the braid group, which has many solutions.  Perhaps Artin's 1947 paper (Theorems 17 and 18) is the first.  Many of the papers mentioned by other answers in the thread also "quickly" give solutions to the word problem (eg for linear groups over $\mathbb{Z}$, for groups acting on free groups, etc).