Identifying Half-Twists Using Randomized Algorithm Methods

Since the braid group was discovered by Artin (1947), the question of its conjugacy problem has been solved by Garside (1969) and Birmanet al.(1998). However, the solutions given thus far are difficult to compute with a computer, since the number of operations needed is extremely large. Meanwhile, random algorithms used to solve difficult problems such as the primality of a number were developed, and the random practical methods have become an important tool. We give a random algorithm, along with a conjecture of how to improve its convergence speed, in order to identify elements in the braid group, which are conjugated to its generators (sayσ1k ) for a given power k. These elements of the braid group, the half-twists, are important in themselves, as they are the key players in some geometrical and algebraical methods, the building blocks of quasipositive braids and they construct endless sets of generators for the group.