Identifying Powers of Half-Twists and Computing its Root

In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin’s generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin’s generators in any power can be restored. The result is applicable to calculations of braid monodromy of branch curves and verification of Hurwitz equivalence of braid monodromy factorizations, which are essential in order to determine braid monodromy type of algebraic surfaces and symplectic 4-manifolds. Introduction During past decades braid groups have become important in many fields. Hence, a practical solution for its conjugacy problem has become extremely important. Although the groups conjugacy problem was first solved by Garside [4] (1969) and was addressed many times in the past (i.e., [5], [2]), still a practical polynomial algorithm for its solution is unknown. This has lead to the research for the solution of partial problems such as identification of special conjugacy classes. In [6], a random algorithm for the identifying half-twists in any power was given, and the aim of this paper is to give a deterministic algorithm for the problem, which although exponential is of interest because of the simplicity of the proofs involved, and the combination of techniques used in order to solve the problem. The algorithm presented here enables to test factors of braid monodromy type of surfaces, and is a partial solution for the conjugacy problem in the braid groups. Moreover, it enables to solve specific cases of quasi-positivity. Partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center ”Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and by EAGER (European Network in Algebraic Geometry).