One of the most interesting questions about a group is whether 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 geometric one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition in order to give a new approach for solving its word problem. Our algorithm, although not better in complexity, is faster in comparison with known algorithms for short braid words, 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.
Bibliographical noteFunding Information:
2Partially supported by the Emmy Noether Research Institute for Mathematics, Bar-Ilan University and the Minerva Foundation, Germany, and by the Excellency Center ‘‘Group theoretic methods in the study of algebraic varieties’’ of the National Science Foundation of Israel.
- Braid group
- Fundamental group
- Word problem