Let $Br_n$ be the braid group generated by $T_1, T_2, \ldots, T_{n-1}$ subject to the relations $ T_i T_j T_i = T_j T_i T_j $ if $|i-j|=1$ and $T_i T_j = T_j T_i$ if $|i-j|>1$.
Are $T_1 T_4 T_3 T_5 T_4 $ and $T_2 T_1 T_3 T_2 T_5 $ in $Br_6$ conjugate?
In the symmetric group $S_6$, we have $ (s_2 s_4) s_1 s_4 s_3 s_5 s_4 (s_2 s_4)^{-1} = s_2 s_1 s_3 s_2 s_5 $. But in $Br_6$, is there some $g \in Br_6$ such that $g T_1 T_4 T_3 T_5 T_4 g^{-1} = T_2 T_1 T_3 T_2 T_5$? Thank you very much.
I advise you to draw the two braids. You'll see that the first one is a braid on 2 strands on the left and a braid on 4 strands on the right, while the second one is the same 4-strand braid on the left and the same 2-strand braid on the right. They are thus conjugate by the braid where the two left-most strands go above the 4 remaining strands, i.e. by $T_2T_3T_4T_5T_1T_2T_3T_4$. They may well be conjugate by a simpler element, but this one comes from the picture.