Braids and the Configuration Space of Points
One can view a n string braid as a loop in the configuration space of n points in the plane. I made a quickie screencast to illustrate this.
Actually I was trying to see if I could get SketchUp to animate the moving points seen in the first half of the screencast. In the end I did it manually with the assistance of section planes. The second half however is done automatically. Get the model and click on the beginsweep/endsweep buttons after opening it in SketchUp.