Identifying Half-Twists Using Randomized Algorithm Methods

Since the braid group was discovered by E. Artin [1], the question of its conjugacy problem is open. Although different approaches and advances where made, still the solution is 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 primarity of a number had been 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 in any power. These elements of the braid group, the half-twists, are important for themselves, as they are the key players in some geometrical and algebraical methods, the building blocks of quasipositive braids and construct endless sets of generators for the group. Partially 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.