Figure 1:
Different aspects of the topology, or “braiding,” of virtual worldlines in two spatial dimensions. The horizontal direction represents time, the vertical and out of page directions represent the two spatial dimensions available to the particles. Thus our three spatial dimensions allow a convenient representation of the worldlines for particles moving in two spatial dimensions. (a) Two indistinguishable particles can change places with either one passing over the other. These two kinds of histories are topologically distinct; neither can be continuously deformed into the other. (b) There is a natural multiplication operation on braids, which turns the topologically distinct classes of braids into a group (the braid group). The multiplication operation illustrated here is simply to glue the separate braids together.