Chart descriptions of regular braided surfaces

Braided surfaces are surfaces embedded or immersed in the bidisk D2×D2 which are projected onto the second factor of the bidisk as branched covering maps. A simple and embedded braided surface is described by a graph in the 2-disk, called a chart. We generalize the chart description method so that one can consider braided surfaces which are not necessarily simple or embedded. We show that if a braided surface is regular, then it can be described by a chart called regular, and that such a chart is unique up to regular chart move equivalence.