H0kay, try this little exercise. Draw a diagram of a crossing from two points to two points, that is, draw a planar representation of a braid group generator. Now if you draw the same diagram rotated by 90°, it's equivalent to the inverse of the original generator. Another way to get the inverse is a reflection through a horizontal or a vertical line. The original is invariant under rotations of 180°, but the points (say you label them with a, b, c, d) are permuted. Suppose you want to use these switching elements to make a set of nonintersecting paths, so the only important function is the permutation of paths, not points. That's a way to quotient a braid group--make the two orientations of each crossing generator equivalent; in effect you choose two inputs and two outputs, and put the crossing in a black box. The only other kind of operations you allow are merging of two tracks (paths), and branching one track into two.