I have been thinking about braids as permutations. A braid can be thought of as performing a permutation on a number of elements repeatedly. Note that this permutation is just how the elements have been rearranged after each complete repeat of moves to make the braid. It doesn't describe which direction the elements took to get there or which crossed over another.
Together, all of the permutations that the elements go through, in the process of making the braid, form what is called in mathematics a group. The permutation is said to be the generator of this group. And this group is a sub-group of the group consisting of all possible permutations of the elements.
I think is would be interesting to count the size of the subgroup generated by the permutation corresponding to some braids that I'm familiar with.