Topological Algorithm for Conjugation of Half-Twists in the Braid Group with Application to Hurwitz Equivalence

In this paper we present a topological algorithm for conjugating two powers of half-twists in the braid group. The algorithm allows us to compute the conjugation without knowing the algebraic structure of the half-twist's diffeomorphism. The motivation of using the graph structure is to ignore the longer algebraic structure, and more important, to allow us a trivial comparison of the results without using the solution of the word problem in the braid group. Using the topological properties of the half-twist, we present an efficient solution for the conjugation of high powers of half-twists. The algorithm is motivated by the Hurwitz equivalence problem which is based on conjugation and comparison of power of half-twists. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (First example by Manetti). We have constructed a new invariant (BMT invariant) based on Hurwitz equivalence class of factorizations. The invariant can distinguish among diffeomorphic surfaces which are not deformation of each other. The precise definition of the new invariant can be found at [4] or [5].