TITLE: Conjugacy problem for small braid groups
QUESTION [5 upvotes]: The conjugacy problem for braid groups $B_3$ and $B_4$ can be solved in polynomial time, it is noted in the paper by Birman, Ko and Lee(2001).
That was a result in 2001. Are there any new results on other small braid groups? Is the conjugacy problem in $B_5$ solvable in polynomial time?
Also, I'm still curious on exactly how fast the conjugacy problem in $B_3$ can be solved(polynomial time is too broad). The only paper that described such algorithm is behind a paywall.

REPLY [3 votes]: This 2008 paper (freely available) explains the fastest known solution of the conjugacy problem in $B_3$. I think the polynomial time complexity of that solution can be easily extracted from that paper.