  One of the most interesting questions about a group is if its word problem
can be solved and how. The word problem in the braid group is of particular
interest to topologists, algebraists and geometers, and is the target of
intensive current research. We look at the braid group from a topological point
of view (rather than a geometrical one). The braid group is defined by the
action of diffeomorphisms on the fundamental group of a punctured disk. We
exploit the topological definition of the braid group in order to give a new
approach for solving its word problem. Our algorithm is faster, in comparison
with known algorithms, for short braid words with respect to the number of
generators combining the braid, and it is almost independent of the number of
strings in the braids. Moreover, the algorithm is based on a new computer
presentation of the elements of the fundamental group of a punctured disk. This
presentation can be used also for other algorithms.
